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<div class="stackedit__html"><h1 id="window-clipping-algorithm">Window Clipping Algorithm</h1> |
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<p><img src="https://i.imgur.com/5stAw9T.png" alt="https://i.imgur.com/5stAw9T.png"></p> |
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<p>어떤 하나의 점 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">(x, y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="mclose">)</span></span></span></span></span>가 window 내에 속하는지를 알아내기는 간단하다. 그저 다음을 만족하면 된다.</p> |
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<p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>X</mi><msub><mi>W</mi><mtext>min</mtext></msub><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>X</mi><msub><mi>W</mi><mtext>max</mtext></msub><mo>)</mo><mi mathvariant="normal">&amp;</mi><mi mathvariant="normal">&amp;</mi><mo>(</mo><mi>Y</mi><msub><mi>W</mi><mtext>min</mtext></msub><mo>≤</mo><mi>y</mi><mo>≤</mo><mi>X</mi><msub><mi>W</mi><mtext>max</mtext></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">(XW_{\text{min}} \le x \le XW_{\text{max}}) \&amp; \&amp; (YW_{\text{min}} \le y \le XW_{\text{max}})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span class="mord mathit" style="margin-right: 0.07847em;">X</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">min</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.77194em; vertical-align: -0.13597em;"></span><span class="mord mathit">x</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathit" style="margin-right: 0.07847em;">X</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">max</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord">&</span><span class="mord">&</span><span class="mopen">(</span><span class="mord mathit" style="margin-right: 0.22222em;">Y</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">min</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.83041em; vertical-align: -0.19444em;"></span><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathit" style="margin-right: 0.07847em;">X</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">max</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span></p> |
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<p>그럼 선은 어떻게 진행할 수 있을까?</p> |
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<h2 id="line-clipping">Line Clipping</h2> |
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<p>선이 window 내에 속하는지를 알아보기에 앞서, 먼저 경우의 수를 생각해보도록 하자. 다음과 같다.</p> |
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<ul> |
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<li>양 끝 점 모두 window 밖에 있음</li> |
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<li>양 끝 점 모두 window 안에 있음</li> |
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<li>한 쪽의 점만 window 안에 있음</li> |
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</ul> |
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<p>이러한 것들에 대해 Cohen-Sutherland Algorithm과 Liang-Barsky Algorithm을 통해 line clipping을 진행할 수 있다.</p> |
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<h3 id="pre-defined">Pre-defined</h3> |
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<p>그 전에, 먼저 어떤 것이 경계 밖에 있는지를 정확히 정의하고 가자.</p> |
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<p>window의 무한한 경계선 하나는 평면을 둘로 나눈다. 이 때, window가 있는 쪽을 <strong>경계선 안 쪽</strong>, 그 반대쪽을 <strong>경계선 바깥 쪽</strong> 이라고 정의한다. 즉, 다음과 같다.</p> |
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<ul> |
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<li> |
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<p>왼쪽 경계선(<span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>X</mi><msub><mi>W</mi><mtext>min</mtext></msub></mrow><annotation encoding="application/x-tex">XW_{\text{min}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.07847em;">X</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">min</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>)에 대해</p> |
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<ul> |
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<li>왼쪽(<span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>X</mi><msub><mi>W</mi><mtext>min</mtext></msub><mo>&gt;</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">XW_{\text{min}} \gt x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.07847em;">X</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">min</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">></span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathit">x</span></span></span></span></span>)은 경계의 밖</li> |
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<li>오른쪽(<span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>X</mi><msub><mi>W</mi><mtext>min</mtext></msub><mo>≤</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">XW_{\text{min}} \le x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.07847em;">X</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">min</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathit">x</span></span></span></span></span>)은 경계의 안</li> |
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</ul> |
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</li> |
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<li> |
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<p>오른쪽 경계선(<span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>X</mi><msub><mi>W</mi><mtext>max</mtext></msub></mrow><annotation encoding="application/x-tex">XW_{\text{max}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.07847em;">X</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">max</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>)에 대해</p> |
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<ul> |
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<li>왼쪽(<span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>X</mi><msub><mi>W</mi><mtext>max</mtext></msub><mo>≥</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">XW_{\text{max}} \ge x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.07847em;">X</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">max</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathit">x</span></span></span></span></span>)은 경계의 안</li> |
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<li>오른쪽(<span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>X</mi><msub><mi>W</mi><mtext>max</mtext></msub><mo>&lt;</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">XW_{\text{max}} \lt x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.07847em;">X</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">max</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathit">x</span></span></span></span></span>)은 경계의 밖</li> |
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</ul> |
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</li> |
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<li> |
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<p>아래쪽 경계선(<span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>Y</mi><msub><mi>W</mi><mtext>min</mtext></msub></mrow><annotation encoding="application/x-tex">YW_{\text{min}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.22222em;">Y</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">min</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>)에 대해</p> |
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<ul> |
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<li>아래쪽(<span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>Y</mi><msub><mi>W</mi><mtext>min</mtext></msub><mo>&gt;</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">YW_{\text{min}} \gt y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.22222em;">Y</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">min</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">></span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord mathit" style="margin-right: 0.03588em;">y</span></span></span></span></span>)은 경계의 밖</li> |
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<li>위쪽(<span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>Y</mi><msub><mi>W</mi><mtext>min</mtext></msub><mo>≤</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">YW_{\text{min}} \le y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.22222em;">Y</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">min</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord mathit" style="margin-right: 0.03588em;">y</span></span></span></span></span>)은 경계의 안</li> |
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</ul> |
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</li> |
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<li> |
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<p>위쪽 경계선(<span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>Y</mi><msub><mi>W</mi><mtext>max</mtext></msub></mrow><annotation encoding="application/x-tex">YW_{\text{max}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.22222em;">Y</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">max</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>)에 대해</p> |
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<ul> |
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<li>아래쪽(<span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>Y</mi><msub><mi>W</mi><mtext>max</mtext></msub><mo>≥</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">YW_{\text{max}} \ge y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.22222em;">Y</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">max</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord mathit" style="margin-right: 0.03588em;">y</span></span></span></span></span>)은 경계의 안</li> |
|
<li>위쪽(<span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>Y</mi><msub><mi>W</mi><mtext>max</mtext></msub><mo>&lt;</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">YW_{\text{max}} \lt y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.22222em;">Y</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">max</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord mathit" style="margin-right: 0.03588em;">y</span></span></span></span></span>)은 경계의 밖</li> |
|
</ul> |
|
</li> |
|
</ul> |
|
<p>원점 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(0, 0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span></span>이 <em>좌측 하단</em> 에 위치한다는것을 주의하자. 예시를 하나 보자면 다음과 같다.</p> |
|
<p><img src="https://i.imgur.com/7ZxlZCj.png" alt="https://i.imgur.com/7ZxlZCj.png"></p> |
|
<p>여기서 각각 다음과 같다.</p> |
|
<ul> |
|
<li> |
|
<p><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathit">a</span></span></span></span></span></p> |
|
<ul> |
|
<li>왼쪽 경계 안</li> |
|
<li>오른쪽 경계 안</li> |
|
<li>아래쪽 경계 안</li> |
|
<li>위쪽 경계 밖</li> |
|
</ul> |
|
</li> |
|
<li> |
|
<p><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.69444em; vertical-align: 0em;"></span><span class="mord mathit">b</span></span></span></span></span></p> |
|
<ul> |
|
<li>왼쪽 경계 안</li> |
|
<li>오른쪽 경계 밖</li> |
|
<li>아래쪽 경계 밖</li> |
|
<li>위쪽 경계 안</li> |
|
</ul> |
|
</li> |
|
</ul> |
|
<p>이를 line으로 확장시키면 다음과 같다.</p> |
|
<p><img src="https://i.imgur.com/qLwxNew.png" alt="https://i.imgur.com/qLwxNew.png"></p> |
|
<ul> |
|
<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathit">a</span></span></span></span></span>: 두 끝점이 모두 (왼쪽)경계 밖에 있기에 <em>window의 밖에 있다.</em></li> |
|
<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.69444em; vertical-align: 0em;"></span><span class="mord mathit">b</span></span></span></span></span>: 두 끝점이 모두 경계 안에 있기에 <em>window의 안에 있다.</em></li> |
|
<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathit">c</span></span></span></span></span>: 시작점은 (왼쪽) 경계 밖에, 끝점은 경계 안에 있기에 <em>window 밖에서 안으로 들어온다</em> 고 할 수 있다.</li> |
|
<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.69444em; vertical-align: 0em;"></span><span class="mord mathit">d</span></span></span></span></span>: 양 끝점 모두 경계 밖이지만, <em>선분 중 일부분이 window에 걸친다.</em></li> |
|
<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathit">e</span></span></span></span></span>: 두 점이 모두 경계 밖이지만, <em>선분 중 일부분이 window에 걸친다.</em></li> |
|
</ul> |
|
<p>자. 그럼 진행해보자.</p> |
|
<h3 id="cohen-sutherland-line-clipping">Cohen-Sutherland Line Clipping</h3> |
|
<p>이 알고리즘은 먼저 다음과 같이 경계로 나뉘는 구역들에 대해 <em>4-bit로 나타낸다.</em></p> |
|
<p><img src="https://i.imgur.com/ki13ijr.png" alt="https://i.imgur.com/ki13ijr.png"></p> |
|
<ul> |
|
<li>bit-1: 왼쪽 경계 벗어났는지 여부 <em>(1: true)</em></li> |
|
<li>bit-2: 오른쪽 경계 벗어났는지 여부</li> |
|
<li>bit-3: 아래쪽 경계 벗어났는지 여부</li> |
|
<li>bit-4: 위쪽 경계 벗어났는지 여부</li> |
|
</ul> |
|
<p>이 bit들은 다음과 같이 계산이 가능하다. (<em><code>1</code>: positive</em>)</p> |
|
<ul> |
|
<li><code>bit_1 = sign_bit(x - XW_min)</code></li> |
|
<li><code>bit_2 = sign_bit(XW_max - x)</code></li> |
|
<li><code>bit_3 = sign_bit(y - YW_min)</code></li> |
|
<li><code>bit_4 = sign_bit(YW_max - y)</code></li> |
|
</ul> |
|
<p>이들을 통해 선이 <strong>완전히</strong> window 내부에 있는지, 또는 <strong>완전히</strong> window 외부에 있는지 여부를 알아낼 수 있다.</p> |
|
<ul> |
|
<li>완전히 window 외부에 위치: 양 끝점의 bit에 대해 같은 위치의 bit 값이 모두 1인 경우 |
|
<ul> |
|
<li>이는 & 연산으로 어렵지 않게 구할 수 있을 것이다.</li> |
|
</ul> |
|
</li> |
|
<li>완전히 window 내부에 위치: 두 bit 모두 <code>0000</code>일 경우</li> |
|
</ul> |
|
<p>이렇게 진행하면 된다. 그럼, 만약 다음과 같이 window에 일부분이 걸치는 경우는 어떻게 진행해야 할까?</p> |
|
<p><img src="https://i.imgur.com/EzPIXKf.png" alt="https://i.imgur.com/EzPIXKf.png"></p> |
|
<p>위 그림과 같은 선에 대한 이야기인데, 이는 line과 경계가 만나는 점을 구해 경계 밖의 부분을 <em>잘라버리는</em> 방식으로 진행할 수 있다. 참고로 보통 경계 테스트는 <strong>left, right, bottom, top</strong> 순으로 진행한다는 것을 알아두자.</p> |
|
<p>위 그림의 선 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mover accent="true"><mrow><mi>P</mi><mn>1</mn><mi>P</mi><mn>2</mn></mrow><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline{P1P2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.88333em; vertical-align: 0em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.88333em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">P</span><span class="mord">1</span><span class="mord mathit" style="margin-right: 0.13889em;">P</span><span class="mord">2</span></span></span><span class="" style="top: -3.80333em;"><span class="pstrut" style="height: 3em;"></span><span class="overline-line" style="border-bottom-width: 0.04em;"></span></span></span></span></span></span></span></span></span></span>를 대상으로 진행해보면 다음과 같다.</p> |
|
<ul> |
|
<li> |
|
<p><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>P</mi><mn>1</mn></mrow><annotation encoding="application/x-tex">P1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.68333em; vertical-align: 0em;"></span><span class="mord mathit" style="margin-right: 0.13889em;">P</span><span class="mord">1</span></span></span></span></span>: <code>(0100)</code> 이기에 검사를 진행해야 함</p> |
|
<ol> |
|
<li>left boundary 안에 위치함: <em>PASS</em></li> |
|
<li>right boundary 안에 위치함: <em>PASS</em></li> |
|
<li>bottom boundary <strong>외부에 위치함</strong>: <em>CLIP</em><br> |
|
a. 아래쪽 경계 안에 위치하도록 잘라 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>P</mi><msup><mn>1</mn><mo mathvariant="normal">′</mo></msup></mrow><annotation encoding="application/x-tex">P1&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.751892em; vertical-align: 0em;"></span><span class="mord mathit" style="margin-right: 0.13889em;">P</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.751892em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span>을 구한다.<br> |
|
b. <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>P</mi><msup><mn>1</mn><mo mathvariant="normal">′</mo></msup></mrow><annotation encoding="application/x-tex">P1&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.751892em; vertical-align: 0em;"></span><span class="mord mathit" style="margin-right: 0.13889em;">P</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.751892em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span>의 bit가 <code>(0000)</code> 이기 때문에, <em>검사 종료. 더 할 필요가 없다.</em></li> |
|
</ol> |
|
</li> |
|
<li> |
|
<p><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>P</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">P2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.68333em; vertical-align: 0em;"></span><span class="mord mathit" style="margin-right: 0.13889em;">P</span><span class="mord">2</span></span></span></span></span>: <code>(1001)</code> 이기 때문에 검사를 진행해야 함</p> |
|
<ol> |
|
<li>left boundary <strong>외부에 위치함</strong>: <em>CLIP</em><br> |
|
a. 왼쪽 경계 안에 위치하도록 잘라 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>P</mi><msup><mn>2</mn><mo mathvariant="normal">′</mo></msup></mrow><annotation encoding="application/x-tex">P2&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.751892em; vertical-align: 0em;"></span><span class="mord mathit" style="margin-right: 0.13889em;">P</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.751892em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span>를 구한다.<br> |
|
b. <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>P</mi><msup><mn>2</mn><mo mathvariant="normal">′</mo></msup></mrow><annotation encoding="application/x-tex">P2&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.751892em; vertical-align: 0em;"></span><span class="mord mathit" style="margin-right: 0.13889em;">P</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.751892em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span>의 bit가 <code>(1000)</code> 이기 때문에, <em>계속 검사를 진행한다.</em><br> |
|
c. 이 때, <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>P</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">P2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.68333em; vertical-align: 0em;"></span><span class="mord mathit" style="margin-right: 0.13889em;">P</span><span class="mord">2</span></span></span></span></span>가 아닌, <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>P</mi><msup><mn>2</mn><mo mathvariant="normal">′</mo></msup></mrow><annotation encoding="application/x-tex">P2&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.751892em; vertical-align: 0em;"></span><span class="mord mathit" style="margin-right: 0.13889em;">P</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.751892em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span>에 대해 검사를 진행해야 함을 주의</li> |
|
<li>right boundary 안에 위치함: <em>PASS</em></li> |
|
<li>bottom boundary 안에 위치함: <em>PASS</em></li> |
|
<li>top boundary <strong>외부에 위치함</strong>: <em>CLIP</em><br> |
|
a. 위쪽 경계 안에 위치하도록 잘라 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>P</mi><msup><mn>2</mn><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup></mrow><annotation encoding="application/x-tex">P2&#x27;&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.751892em; vertical-align: 0em;"></span><span class="mord mathit" style="margin-right: 0.13889em;">P</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.751892em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span>를 구한다.<br> |
|
b. <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>P</mi><msup><mn>2</mn><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup></mrow><annotation encoding="application/x-tex">P2&#x27;&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.751892em; vertical-align: 0em;"></span><span class="mord mathit" style="margin-right: 0.13889em;">P</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.751892em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span>의 bit가 <code>(0000)</code>이니, 검사 종료. 사실 더 할 수도 없다.</li> |
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</ol> |
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</li> |
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</ul> |
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<p>이렇게 line에 대한 clipping을 진행할 수 있다. 다른 예제 하나를 더 볼까.</p> |
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<p><img src="https://i.imgur.com/QNfVbej.png" alt="https://i.imgur.com/QNfVbej.png"></p> |
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<ul> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>P</mi><mn>3</mn></mrow><annotation encoding="application/x-tex">P3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.68333em; vertical-align: 0em;"></span><span class="mord mathit" style="margin-right: 0.13889em;">P</span><span class="mord">3</span></span></span></span></span>: <code>(0001)</code> 이기 때문에 검사 진행 |
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<ol> |
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<li>left boundary <strong>외부에 위치</strong>: <em>CLIP</em><br> |
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a. 왼쪽 경계선 안으로 들어오도록 잘라 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>P</mi><msup><mn>3</mn><mo mathvariant="normal">′</mo></msup></mrow><annotation encoding="application/x-tex">P3&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.751892em; vertical-align: 0em;"></span><span class="mord mathit" style="margin-right: 0.13889em;">P</span><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.751892em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span>를 구한다.<br> |
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b. 여기서 구해지는 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>P</mi><msup><mn>3</mn><mo mathvariant="normal">′</mo></msup></mrow><annotation encoding="application/x-tex">P3&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.751892em; vertical-align: 0em;"></span><span class="mord mathit" style="margin-right: 0.13889em;">P</span><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.751892em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span>의 bit는 <code>(0100)</code>이 되는데, 이 때 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>P</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">P4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.68333em; vertical-align: 0em;"></span><span class="mord mathit" style="margin-right: 0.13889em;">P</span><span class="mord">4</span></span></span></span></span>의 bit도 <code>(0100)</code> 이기 때문에, <em>선이 완전히 window 외부에 위치한다고 할 수 있다.</em></li> |
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</ol> |
|
</li> |
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</ul> |
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<p><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>P</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">P4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.68333em; vertical-align: 0em;"></span><span class="mord mathit" style="margin-right: 0.13889em;">P</span><span class="mord">4</span></span></span></span></span>는 진행할 것도 없다. 어차피 line이 window 외부에 위치한다는 것을 알아냈으니 말이다.</p> |
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<p>자 그럼 이 intersection point는 어떻게 구할 수 있을까. 어렵지 않게 다음으로 구할 수 있다.</p> |
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<ul> |
|
<li> |
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<p>left boundary IP</p> |
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<ul> |
|
<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>=</mo><mi>X</mi><msub><mi>W</mi><mtext>min</mtext></msub></mrow><annotation encoding="application/x-tex">x = XW_{\text{min}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathit">x</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.07847em;">X</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">min</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></li> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>y</mi><mo>=</mo><msub><mi>y</mi><mn>1</mn></msub><mo>+</mo><mfrac><mrow><mi>X</mi><msub><mi>W</mi><mtext>min</mtext></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow></mfrac><mi mathvariant="normal">Δ</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">y = y_1 + \frac{XW_{\text{min}} - x_1}{\Delta x} \Delta y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.77777em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1.23343em; vertical-align: -0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.888431em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Δ</span><span class="mord mathit mtight">x</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.4101em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight" style="margin-right: 0.07847em;">X</span><span class="mord mtight"><span class="mord mathit mtight" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.334043em;"><span class="" style="top: -2.357em; margin-left: -0.13889em; margin-right: 0.0714286em;"><span class="pstrut" style="height: 2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">min</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.143em;"><span class=""></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight"><span class="mord mathit mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317314em;"><span class="" style="top: -2.357em; margin-left: 0em; margin-right: 0.0714286em;"><span class="pstrut" style="height: 2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.143em;"><span class=""></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord">Δ</span><span class="mord mathit" style="margin-right: 0.03588em;">y</span></span></span></span></span></li> |
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</ul> |
|
</li> |
|
<li> |
|
<p>right boundary IP</p> |
|
<ul> |
|
<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>=</mo><mi>X</mi><msub><mi>W</mi><mtext>max</mtext></msub></mrow><annotation encoding="application/x-tex">x = XW_{\text{max}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathit">x</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.07847em;">X</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">max</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></li> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>y</mi><mo>=</mo><msub><mi>y</mi><mn>1</mn></msub><mo>+</mo><mfrac><mrow><mi>X</mi><msub><mi>W</mi><mtext>max</mtext></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow></mfrac><mi mathvariant="normal">Δ</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">y = y_1 + \frac{XW_{\text{max}} - x_1}{\Delta x} \Delta y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.77777em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1.23343em; vertical-align: -0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.888431em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Δ</span><span class="mord mathit mtight">x</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.4101em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight" style="margin-right: 0.07847em;">X</span><span class="mord mtight"><span class="mord mathit mtight" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.164543em;"><span class="" style="top: -2.357em; margin-left: -0.13889em; margin-right: 0.0714286em;"><span class="pstrut" style="height: 2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">max</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.143em;"><span class=""></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight"><span class="mord mathit mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317314em;"><span class="" style="top: -2.357em; margin-left: 0em; margin-right: 0.0714286em;"><span class="pstrut" style="height: 2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.143em;"><span class=""></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord">Δ</span><span class="mord mathit" style="margin-right: 0.03588em;">y</span></span></span></span></span></li> |
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</ul> |
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</li> |
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<li> |
|
<p>bottom boundary IP</p> |
|
<ul> |
|
<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>=</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mfrac><mrow><mi>Y</mi><msub><mi>W</mi><mtext>min</mtext></msub><mo>−</mo><msub><mi>y</mi><mn>1</mn></msub></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>y</mi></mrow></mfrac><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">x = x_1 + \frac{YW_{\text{min}} - y_1}{\Delta y} \Delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathit">x</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.73333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1.40555em; vertical-align: -0.481108em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.924439em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Δ</span><span class="mord mathit mtight" style="margin-right: 0.03588em;">y</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.44611em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight" style="margin-right: 0.22222em;">Y</span><span class="mord mtight"><span class="mord mathit mtight" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.334043em;"><span class="" style="top: -2.357em; margin-left: -0.13889em; margin-right: 0.0714286em;"><span class="pstrut" style="height: 2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">min</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.143em;"><span class=""></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight"><span class="mord mathit mtight" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317314em;"><span class="" style="top: -2.357em; margin-left: -0.03588em; margin-right: 0.0714286em;"><span class="pstrut" style="height: 2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.143em;"><span class=""></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.481108em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord">Δ</span><span class="mord mathit">x</span></span></span></span></span></li> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>y</mi><mo>=</mo><mi>Y</mi><msub><mi>W</mi><mtext>min</mtext></msub></mrow><annotation encoding="application/x-tex">y = YW_{\text{min}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.22222em;">Y</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">min</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></li> |
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</ul> |
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</li> |
|
<li> |
|
<p>top boundary IP</p> |
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<ul> |
|
<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>=</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mfrac><mrow><mi>Y</mi><msub><mi>W</mi><mtext>max</mtext></msub><mo>−</mo><msub><mi>y</mi><mn>1</mn></msub></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>y</mi></mrow></mfrac><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">x = x_1 + \frac{YW_{\text{max}} - y_1}{\Delta y} \Delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathit">x</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.73333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1.40555em; vertical-align: -0.481108em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.924439em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Δ</span><span class="mord mathit mtight" style="margin-right: 0.03588em;">y</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.44611em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight" style="margin-right: 0.22222em;">Y</span><span class="mord mtight"><span class="mord mathit mtight" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.164543em;"><span class="" style="top: -2.357em; margin-left: -0.13889em; margin-right: 0.0714286em;"><span class="pstrut" style="height: 2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">max</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.143em;"><span class=""></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight"><span class="mord mathit mtight" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317314em;"><span class="" style="top: -2.357em; margin-left: -0.03588em; margin-right: 0.0714286em;"><span class="pstrut" style="height: 2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.143em;"><span class=""></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.481108em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord">Δ</span><span class="mord mathit">x</span></span></span></span></span></li> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>y</mi><mo>=</mo><mi>Y</mi><msub><mi>W</mi><mtext>max</mtext></msub></mrow><annotation encoding="application/x-tex">y = YW_{\text{max}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.22222em;">Y</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">max</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></li> |
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</ul> |
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</li> |
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</ul> |
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<p>이렇게 진행하면 된다.</p> |
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<h3 id="liang-barsky-line-clipping">Liang-Barsky Line Clipping</h3> |
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<p>솔직히 말해서 개인적으로는 이 방법이 좀 더 편하다.</p> |
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<p>line의 두 끝점 사이의 점 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">(x, y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="mclose">)</span></span></span></span></span>를 구하는 다음의 식</p> |
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<ul> |
|
<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>=</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mi>u</mi><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">x = x_1 + u \Delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathit">x</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.73333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.68333em; vertical-align: 0em;"></span><span class="mord mathit">u</span><span class="mord">Δ</span><span class="mord mathit">x</span></span></span></span></span></li> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>y</mi><mo>=</mo><msub><mi>y</mi><mn>1</mn></msub><mo>+</mo><mi>u</mi><mi mathvariant="normal">Δ</mi><mi>y</mi><mspace width="1em"></mspace><mo>(</mo><mn>0</mn><mo>≤</mo><mi>u</mi><mo>≤</mo><mn>1</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">y = y_1 + u \Delta y \quad (0 \le u \le 1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.77777em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathit">u</span><span class="mord">Δ</span><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="mspace" style="margin-right: 1em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.77194em; vertical-align: -0.13597em;"></span><span class="mord mathit">u</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span></li> |
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</ul> |
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<p>을 이용해 진행하는 알고리즘이다. 참고로 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>u</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">u = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathit">u</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.64444em; vertical-align: 0em;"></span><span class="mord">0</span></span></span></span></span>인 경우는 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>=</mo><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x = x_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathit">x</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>, <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>y</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">y = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.64444em; vertical-align: 0em;"></span><span class="mord">1</span></span></span></span></span>인 경우는 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>=</mo><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x = x_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathit">x</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span> 이렇게 될 것이다. <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord mathit" style="margin-right: 0.03588em;">y</span></span></span></span></span>도 마찬가지고. <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>[</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo>]</mo></mrow><annotation encoding="application/x-tex">[0, 1]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord">1</span><span class="mclose">]</span></span></span></span></span> 범위를 벗어난 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathit">u</span></span></span></span></span>는 line의 범위를 벗어났다고도 할 수 있겠다.</p> |
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<p><img src="https://i.imgur.com/MJD3rk9.png" alt="https://i.imgur.com/MJD3rk9.png"></p> |
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<p>이 식과 앞서 보았던 하나의 점이 window 내부에 위치하고 있을 조건을 이용하면 다음과 같은 식을 얻을 수 있다.</p> |
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<p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>X</mi><msub><mi>W</mi><mtext>min</mtext></msub><mo>≤</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mi>u</mi><mi mathvariant="normal">Δ</mi><mi>x</mi><mo>≤</mo><mi>X</mi><msub><mi>W</mi><mtext>max</mtext></msub><mo>)</mo><mi mathvariant="normal">&amp;</mi><mi mathvariant="normal">&amp;</mi><mo>(</mo><mi>Y</mi><msub><mi>W</mi><mtext>min</mtext></msub><mo>≤</mo><msub><mi>y</mi><mn>1</mn></msub><mo>+</mo><mi>u</mi><mi mathvariant="normal">Δ</mi><mi>y</mi><mo>≤</mo><mi>Y</mi><msub><mi>W</mi><mtext>max</mtext></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">(XW_{\text{min}} \le x_1 + u \Delta x \le XW_{\text{max}}) \&amp;\&amp; (YW_{\text{min}} \le y_1 + u \Delta y \le YW_{\text{max}})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span class="mord mathit" style="margin-right: 0.07847em;">X</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">min</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.73333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.8193em; vertical-align: -0.13597em;"></span><span class="mord mathit">u</span><span class="mord">Δ</span><span class="mord mathit">x</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathit" style="margin-right: 0.07847em;">X</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">max</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord">&</span><span class="mord">&</span><span class="mopen">(</span><span class="mord mathit" style="margin-right: 0.22222em;">Y</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">min</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.77777em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.87777em; vertical-align: -0.19444em;"></span><span class="mord mathit">u</span><span class="mord">Δ</span><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathit" style="margin-right: 0.22222em;">Y</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">max</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span></p> |
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<p>이 식은 다음과 같이 네 개의 관계식으로 분리가 가능할 것이다.</p> |
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<ul> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>X</mi><msub><mi>W</mi><mtext>min</mtext></msub><mo>≤</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mi>u</mi><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">XW_{\text{min}} \le x_1 + u \Delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.07847em;">X</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">min</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.73333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.68333em; vertical-align: 0em;"></span><span class="mord mathit">u</span><span class="mord">Δ</span><span class="mord mathit">x</span></span></span></span></span>: left boundary 안에 위치하기 위한 조건</li> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mi>u</mi><mi mathvariant="normal">Δ</mi><mi>x</mi><mo>≤</mo><mi>X</mi><msub><mi>W</mi><mtext>max</mtext></msub></mrow><annotation encoding="application/x-tex">x_1 + u \Delta x \le XW_{\text{max}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.73333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.8193em; vertical-align: -0.13597em;"></span><span class="mord mathit">u</span><span class="mord">Δ</span><span class="mord mathit">x</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.07847em;">X</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">max</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>: right boundary 안에 위치하기 위한 조건</li> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>Y</mi><msub><mi>W</mi><mtext>min</mtext></msub><mo>≤</mo><msub><mi>y</mi><mn>1</mn></msub><mo>+</mo><mi>u</mi><mi mathvariant="normal">Δ</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">YW_{\text{min}} \le y_1 + u \Delta y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.22222em;">Y</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">min</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.77777em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.87777em; vertical-align: -0.19444em;"></span><span class="mord mathit">u</span><span class="mord">Δ</span><span class="mord mathit" style="margin-right: 0.03588em;">y</span></span></span></span></span>: bottom boundary 안에 위치하기 위한 조건</li> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>y</mi><mn>1</mn></msub><mo>+</mo><mi>u</mi><mi mathvariant="normal">Δ</mi><mi>y</mi><mo>≤</mo><mi>Y</mi><msub><mi>W</mi><mtext>max</mtext></msub></mrow><annotation encoding="application/x-tex">y_1 + u \Delta y \le YW_{\text{max}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.77777em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.87777em; vertical-align: -0.19444em;"></span><span class="mord mathit">u</span><span class="mord">Δ</span><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.22222em;">Y</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">max</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>: top boundary 안에 위치하기 위한 조건</li> |
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</ul> |
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<p>이 네 조건이 모두 성립해야 점 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">(x, y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="mclose">)</span></span></span></span></span>가 window 내에 위치하게 되는 것이다.</p> |
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<p>얘를 좀 더 써먹기 쉽도록 다음과 같이 나타낼 수가 있다.</p> |
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<ul> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>u</mi><mo>(</mo><mo>−</mo><mi mathvariant="normal">Δ</mi><mi>x</mi><mo>)</mo><mo>≤</mo><msub><mi>x</mi><mn>1</mn></msub><mo>−</mo><mi>X</mi><msub><mi>W</mi><mtext>min</mtext></msub></mrow><annotation encoding="application/x-tex">u (-\Delta x) \le x_1 - XW_{\text{min}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathit">u</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">Δ</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.73333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.07847em;">X</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">min</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>: left</li> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>u</mi><mo>(</mo><mi mathvariant="normal">Δ</mi><mi>x</mi><mo>)</mo><mo>≤</mo><mi>X</mi><msub><mi>W</mi><mtext>max</mtext></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">u (\Delta x) \le XW_{\text{max}} - x_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathit">u</span><span class="mopen">(</span><span class="mord">Δ</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.07847em;">X</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">max</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>: right</li> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>u</mi><mo>(</mo><mo>−</mo><mi mathvariant="normal">Δ</mi><mi>y</mi><mo>)</mo><mo>≤</mo><msub><mi>y</mi><mn>1</mn></msub><mo>−</mo><mi>Y</mi><msub><mi>W</mi><mtext>min</mtext></msub></mrow><annotation encoding="application/x-tex">u (-\Delta y) \le y_1 - YW_{\text{min}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathit">u</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">Δ</span><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.77777em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.22222em;">Y</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">min</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>: bottom</li> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>u</mi><mo>(</mo><mi mathvariant="normal">Δ</mi><mi>y</mi><mo>)</mo><mo>≤</mo><mi>Y</mi><msub><mi>W</mi><mtext>max</mtext></msub><mo>−</mo><msub><mi>y</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">u (\Delta y) \le YW_{\text{max}} - y_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathit">u</span><span class="mopen">(</span><span class="mord">Δ</span><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.22222em;">Y</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">max</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>: top</li> |
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</ul> |
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<p>여기서 다음과 같이 치환해주면…</p> |
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<p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>u</mi><msub><mi>p</mi><mi>k</mi></msub><mo>≤</mo><msub><mi>q</mi><mi>k</mi></msub><mspace width="1em"></mspace><mo>(</mo><mi>k</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo separator="true">,</mo><mn>4</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">u p_k \le q_k \quad (k = 1, 2, 3, 4)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83041em; vertical-align: -0.19444em;"></span><span class="mord mathit">u</span><span class="mord"><span class="mord mathit">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right: 0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right: 0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 1em;"></span><span class="mopen">(</span><span class="mord mathit" style="margin-right: 0.03148em;">k</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord">4</span><span class="mclose">)</span></span></span></span></span></span></p> |
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<ul> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>p</mi><mn>1</mn></msub><mo>=</mo><mo>−</mo><mi mathvariant="normal">Δ</mi><mi>x</mi><mo separator="true">,</mo><mtext> </mtext><msub><mi>q</mi><mn>1</mn></msub><mo>=</mo><msub><mi>x</mi><mn>1</mn></msub><mo>−</mo><mi>X</mi><msub><mi>W</mi><mtext>min</mtext></msub></mrow><annotation encoding="application/x-tex">p_1 = -\Delta x, \space q_1 = x_1 - XW_{\text{min}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.87777em; vertical-align: -0.19444em;"></span><span class="mord">−</span><span class="mord">Δ</span><span class="mord mathit">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mspace"> </span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.73333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.07847em;">X</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">min</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>: left</li> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>p</mi><mn>2</mn></msub><mo>=</mo><mi mathvariant="normal">Δ</mi><mi>x</mi><mo separator="true">,</mo><mtext> </mtext><msub><mi>q</mi><mn>2</mn></msub><mo>=</mo><mi>X</mi><msub><mi>W</mi><mtext>max</mtext></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p_2 = \Delta x, \space q_2 = XW_{\text{max}} - x_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.87777em; vertical-align: -0.19444em;"></span><span class="mord">Δ</span><span class="mord mathit">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mspace"> </span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.07847em;">X</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">max</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>: right</li> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>p</mi><mn>3</mn></msub><mo>=</mo><mo>−</mo><mi mathvariant="normal">Δ</mi><mi>y</mi><mo separator="true">,</mo><mtext> </mtext><msub><mi>q</mi><mn>3</mn></msub><mo>=</mo><msub><mi>y</mi><mn>1</mn></msub><mo>−</mo><mi>Y</mi><msub><mi>W</mi><mtext>min</mtext></msub></mrow><annotation encoding="application/x-tex">p_3 = -\Delta y, \space q_3 = y_1 - YW_{\text{min}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.87777em; vertical-align: -0.19444em;"></span><span class="mord">−</span><span class="mord">Δ</span><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mspace"> </span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.77777em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.22222em;">Y</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">min</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>: bottom</li> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>p</mi><mn>4</mn></msub><mo>=</mo><mi mathvariant="normal">Δ</mi><mi>y</mi><mo separator="true">,</mo><mtext> </mtext><msub><mi>q</mi><mn>4</mn></msub><mo>=</mo><mi>Y</mi><msub><mi>W</mi><mtext>max</mtext></msub><mo>−</mo><msub><mi>y</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p_4 = \Delta y, \space q_4 = YW_{\text{max}} - y_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.87777em; vertical-align: -0.19444em;"></span><span class="mord">Δ</span><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mspace"> </span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.22222em;">Y</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">max</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>: top</li> |
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</ul> |
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<p>다음의 조건에 따라 line이 어디에 위치하는지 판별이 가능해진다.</p> |
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<h4 id="i-p_k--0">i) <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>p</mi><mi>k</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">p_k = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right: 0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.64444em; vertical-align: 0em;"></span><span class="mord">0</span></span></span></span></span></h4> |
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<p><em>해당 경계선에 대해 평행한 경우</em> 를 의미한다. 여기서 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>q</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">q_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right: 0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>의 값에 따라 해당 line이 어디에 위치할지를 알아낼 수 있다.</p> |
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<ul> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>q</mi><mi>k</mi></msub><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">q_k \lt 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.73354em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right: 0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.64444em; vertical-align: 0em;"></span><span class="mord">0</span></span></span></span></span>: 완전히 해당 경계선 외부에 위치</li> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>q</mi><mi>k</mi></msub><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">q_k \ge 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83041em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right: 0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.64444em; vertical-align: 0em;"></span><span class="mord">0</span></span></span></span></span>: 완전히 해당 경계선 안에 위치</li> |
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</ul> |
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<h4 id="ii-p_k-mathllap-0">ii) <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>p</mi><mi>k</mi></msub><mo>=</mo><mpadded lspace="-1width" width="0px"><mrow><mi mathvariant="normal">/</mi>&ThinSpace;</mrow></mpadded><mn>0</mn></mrow><annotation encoding="application/x-tex">p_k {=}\mathllap{/\,} 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord"><span class="mord mathit">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right: 0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mrel">=</span></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.75em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="llap"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="inner"><span class="mord"><span class="mord">/</span><span class="mspace" style="margin-right: 0.166667em;"></span></span></span><span class="fix"></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.25em;"><span class=""></span></span></span></span></span><span class="mord">0</span></span></span></span></span></h4> |
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<p>이 말은 곧 <em>평행하지 않음</em> 을 의미한다. 다시말해 window 네 개의 경계선과 어디선가 교차한다는 말.</p> |
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<ul> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>p</mi><mi>k</mi></msub><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">p_k \lt 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.73354em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right: 0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.64444em; vertical-align: 0em;"></span><span class="mord">0</span></span></span></span></span>: 해당 경계선을 <strong>밖에서 안으로</strong> 뚫고 들어옴</li> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>p</mi><mi>k</mi></msub><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">p_k \gt 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.73354em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right: 0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">></span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.64444em; vertical-align: 0em;"></span><span class="mord">0</span></span></span></span></span>: 해당 경계선을 <strong>안에서 밖으로</strong> 뚫고 나감</li> |
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</ul> |
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<p>따라서, <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>p</mi><mi>k</mi></msub><mo>=</mo><mpadded lspace="-1width" width="0px"><mrow><mi mathvariant="normal">/</mi>&ThinSpace;</mrow></mpadded><mn>0</mn></mrow><annotation encoding="application/x-tex">p_k {=}\mathllap{/\,} 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord"><span class="mord mathit">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right: 0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mrel">=</span></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.75em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="llap"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="inner"><span class="mord"><span class="mord">/</span><span class="mspace" style="margin-right: 0.166667em;"></span></span></span><span class="fix"></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.25em;"><span class=""></span></span></span></span></span><span class="mord">0</span></span></span></span></span>인 경우, 경계선들 중 <em>두 개는 뚫고 들어오고, 두 개는 둟고 나간다</em> 는 것을 의미한다. 여기서 두개두개는 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>−</mo><mi mathvariant="normal">Δ</mi></mrow><annotation encoding="application/x-tex">-\Delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.76666em; vertical-align: -0.08333em;"></span><span class="mord">−</span><span class="mord">Δ</span></span></span></span></span>와 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>+</mo><mi mathvariant="normal">Δ</mi></mrow><annotation encoding="application/x-tex">+\Delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.76666em; vertical-align: -0.08333em;"></span><span class="mord">+</span><span class="mord">Δ</span></span></span></span></span> 두개를 의미.</p> |
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<p>참고로, 여기서 계속 언급되는 '들어옴’과 '나감’은 다음과 같다.</p> |
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<p><img src="https://i.imgur.com/04zDZ4V.png" alt="https://i.imgur.com/04zDZ4V.png"></p> |
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<p>시작점 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>1</mn></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">(x_1, y_1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>에서 끝 점 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>2</mn></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">(x_2, y_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>로 진행할 때 말이다.</p> |
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<h4 id="parameter-u">parameter ‘u’</h4> |
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<p>아무튼 이 경우의 IP는 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>u</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">u_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right: 0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span> 값으로 구할 수 있다. 이를 위해 위에서 봤던 식을 이용하면 다음과 같은 식을 얻을 수 있다.</p> |
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<p><span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>u</mi><mi>k</mi></msub><mo>=</mo><mfrac><msub><mi>q</mi><mi>k</mi></msub><msub><mi>p</mi><mi>k</mi></msub></mfrac></mrow><annotation encoding="application/x-tex">u_k = \frac{q_k}{p_k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right: 0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1.988em; vertical-align: -0.88044em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.10756em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathit">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right: 0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right: 0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.88044em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span></p> |
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<p>이렇게 총 네 개의 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathit">u</span></span></span></span></span>가 구해질 것이고, 여기서…</p> |
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<ul> |
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<li>밖에서 안으로 뚫고 들어오는(<span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>p</mi><mi>k</mi></msub><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">p_k \lt 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.73354em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right: 0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.64444em; vertical-align: 0em;"></span><span class="mord">0</span></span></span></span></span>) <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathit">u</span></span></span></span></span> 값: <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>u</mi><mtext>in</mtext></msub></mrow><annotation encoding="application/x-tex">u_{\text{in}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">in</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></li> |
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<li>안에서 밖으로 뚫고 나가는(<span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>p</mi><mi>k</mi></msub><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">p_k \ge 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83041em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right: 0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.64444em; vertical-align: 0em;"></span><span class="mord">0</span></span></span></span></span>) <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathit">u</span></span></span></span></span> 값: <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>u</mi><mtext>out</mtext></msub></mrow><annotation encoding="application/x-tex">u_{\text{out}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.280556em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">out</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></li> |
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</ul> |
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<p>이라고 하면, line의 시작 위치 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>u</mi><mtext>start</mtext></msub></mrow><annotation encoding="application/x-tex">u_{\text{start}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.280556em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">start</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>와 끝 위치 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>u</mi><mtext>end</mtext></msub></mrow><annotation encoding="application/x-tex">u_{\text{end}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">end</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>는 다음과 같이 구할 수 있게 된다.</p> |
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<ul> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>u</mi><mtext>start</mtext></msub><mo>=</mo><mtext>Max</mtext><mo>(</mo><msub><mi>u</mi><msub><mtext>in</mtext><mn>1</mn></msub></msub><mo separator="true">,</mo><msub><mi>u</mi><msub><mtext>in</mtext><mn>2</mn></msub></msub><mo separator="true">,</mo><mn>0</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">u_{\text{start}} = \text{Max}(u_{\text{in}_1}, u_{\text{in}_2}, 0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.280556em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">start</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1.0001em; vertical-align: -0.2501em;"></span><span class="mord text"><span class="mord">Max</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">in</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317314em;"><span class="" style="top: -2.357em; margin-right: 0.0714286em;"><span class="pstrut" style="height: 2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.143em;"><span class=""></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2501em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">in</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317314em;"><span class="" style="top: -2.357em; margin-right: 0.0714286em;"><span class="pstrut" style="height: 2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.143em;"><span class=""></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2501em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span></span></li> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>u</mi><mtext>end</mtext></msub><mo>=</mo><mtext>Min</mtext><mo>(</mo><msub><mi>u</mi><msub><mtext>out</mtext><mn>1</mn></msub></msub><mo separator="true">,</mo><msub><mi>u</mi><msub><mtext>out</mtext><mn>2</mn></msub></msub><mo separator="true">,</mo><mn>1</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">u_{\text{end}} = \text{Min}(u_{\text{out}_1}, u_{\text{out}_2}, 1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">end</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1.0001em; vertical-align: -0.2501em;"></span><span class="mord text"><span class="mord">Min</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.280556em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">out</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317314em;"><span class="" style="top: -2.357em; margin-right: 0.0714286em;"><span class="pstrut" style="height: 2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.143em;"><span class=""></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2501em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.280556em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">out</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317314em;"><span class="" style="top: -2.357em; margin-right: 0.0714286em;"><span class="pstrut" style="height: 2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.143em;"><span class=""></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2501em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span></li> |
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</ul> |
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<p><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>u</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">u = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathit">u</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.64444em; vertical-align: 0em;"></span><span class="mord">0</span></span></span></span></span>은 line의 시작 지점, <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>u</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">u = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathit">u</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.64444em; vertical-align: 0em;"></span><span class="mord">1</span></span></span></span></span>은 line의 끝 지점이기에 위와 같이 들어가는 것임을 놓치지 말자.</p> |
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<p>왜 이렇게 계산되는지 이유는… 그냥 최소한의 범위로 잡아야 하기 때문이라 생각하자.</p> |
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<p>마지막으로 이렇게 구한 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>u</mi><mtext>start</mtext></msub></mrow><annotation encoding="application/x-tex">u_{\text{start}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.280556em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">start</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>와 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>u</mi><mtext>end</mtext></msub></mrow><annotation encoding="application/x-tex">u_{\text{end}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">end</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>에 대해, <strong><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>u</mi><mtext>start</mtext></msub></mrow><annotation encoding="application/x-tex">u_{\text{start}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.280556em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">start</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>가 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>u</mi><mtext>end</mtext></msub></mrow><annotation encoding="application/x-tex">u_{\text{end}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">end</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>보다 큰 값은 가질 수 없다</strong> 는 것을 주의하자. 당연한 말이기도 하고, 예제에서도 보겠지만, 이 경우는 window에서 그릴 부분이 없음을 의미한다.</p> |
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<p>아래의 그림을 보면 지금까지 진행했던 것들에 대해 좀 이해가 될 것이다.</p> |
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<p><img src="https://i.imgur.com/1LxToIP.png" alt="https://i.imgur.com/1LxToIP.png"></p> |
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<p><img src="https://i.imgur.com/YQZfVpa.png" alt="https://i.imgur.com/YQZfVpa.png"></p> |
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<p>이를 알고리즘으로 정리해보면 다음과 같다.</p> |
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<p><strong>Liang-Barsky Line Clipping Algorithm</strong></p> |
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<ol> |
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<li> |
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<p>주어진 line의 두 끝점 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>1</mn></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">(x_1, y_1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>, <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>2</mn></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">(x_2, y_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>로부터 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>p</mi><mo separator="true">,</mo><mtext> </mtext><mi>q</mi></mrow><annotation encoding="application/x-tex">p, \space q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord mathit">p</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mspace"> </span><span class="mord mathit" style="margin-right: 0.03588em;">q</span></span></span></span></span>를 구한다.</p> |
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<ul> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>p</mi><mn>1</mn></msub><mo>=</mo><mo>−</mo><mo>(</mo><msub><mi>x</mi><mn>2</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub><mo>)</mo><mo separator="true">,</mo><mtext> </mtext><msub><mi>q</mi><mn>1</mn></msub><mo>=</mo><msub><mi>x</mi><mn>1</mn></msub><mo>−</mo><mi>X</mi><msub><mi>W</mi><mtext>min</mtext></msub></mrow><annotation encoding="application/x-tex">p_1 = -(x_2 - x_1), \space q_1 = x_1 - XW_{\text{min}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord">−</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mspace"> </span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.73333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.07847em;">X</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">min</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></li> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>p</mi><mn>2</mn></msub><mo>=</mo><mo>(</mo><msub><mi>x</mi><mn>2</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub><mo>)</mo><mo separator="true">,</mo><mtext> </mtext><msub><mi>q</mi><mn>2</mn></msub><mo>=</mo><mi>X</mi><msub><mi>W</mi><mtext>max</mtext></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p_2 = (x_2 - x_1), \space q_2 = XW_{\text{max}} - x_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mspace"> </span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.07847em;">X</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">max</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></li> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>p</mi><mn>3</mn></msub><mo>=</mo><mo>−</mo><mo>(</mo><msub><mi>y</mi><mn>2</mn></msub><mo>−</mo><msub><mi>y</mi><mn>1</mn></msub><mo>)</mo><mo separator="true">,</mo><mtext> </mtext><msub><mi>q</mi><mn>3</mn></msub><mo>=</mo><msub><mi>y</mi><mn>1</mn></msub><mo>−</mo><mi>Y</mi><msub><mi>W</mi><mtext>min</mtext></msub></mrow><annotation encoding="application/x-tex">p_3 = -(y_2 - y_1), \space q_3 = y_1 - YW_{\text{min}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord">−</span><span class="mopen">(</span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mspace"> </span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.77777em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.22222em;">Y</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">min</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></li> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>p</mi><mn>4</mn></msub><mo>=</mo><mo>(</mo><msub><mi>y</mi><mn>2</mn></msub><mo>−</mo><msub><mi>y</mi><mn>1</mn></msub><mo>)</mo><mo separator="true">,</mo><mtext> </mtext><msub><mi>q</mi><mn>4</mn></msub><mo>=</mo><mi>Y</mi><msub><mi>W</mi><mtext>max</mtext></msub><mo>−</mo><msub><mi>y</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p_4 = (y_2 - y_1), \space q_4 = YW_{\text{max}} - y_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mspace"> </span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.22222em;">Y</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">max</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></li> |
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<li></li> |
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</ul> |
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</li> |
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<li> |
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<p>다음 조건에 부합하는지 판별</p> |
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<ol> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">dx</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.69444em; vertical-align: 0em;"></span><span class="mord mathit">d</span><span class="mord mathit">x</span></span></span></span></span>가 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.64444em; vertical-align: 0em;"></span><span class="mord">0</span></span></span></span></span>일 경우 (<span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><msub><mi>p</mi><mn>1</mn></msub><mo>=</mo><mo>=</mo><mn>0</mn><mo>)</mo><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mo>(</mo><msub><mi>p</mi><mn>2</mn></msub><mo>=</mo><mo>=</mo><mn>0</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(p_1 == 0) || (p_2 == 0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathit">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span></span><span class="base"><span class="strut" style="height: 0.36687em; vertical-align: 0em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mord">∣</span><span class="mord">∣</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span></span><span class="base"><span class="strut" style="height: 0.36687em; vertical-align: 0em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span></span>) |
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<ul> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><msub><mi>q</mi><mn>1</mn></msub><mo>&lt;</mo><mn>0</mn><mo>)</mo><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mo>(</mo><msub><mi>q</mi><mn>2</mn></msub><mo>&lt;</mo><mn>0</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(q_1 \lt 0) || (q_2 \lt 0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mord">∣</span><span class="mord">∣</span><span class="mopen">(</span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span></span>인 경우, <strong>Exit</strong></li> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>max</mtext><mo>(</mo><msub><mi>y</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>2</mn></msub><mo>)</mo><mo>&lt;</mo><mi>Y</mi><msub><mi>W</mi><mtext>min</mtext></msub></mrow><annotation encoding="application/x-tex">\text{max} (y_1, y_2) \lt YW_{\text{min}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord text"><span class="mord">max</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.22222em;">Y</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">min</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span> 인 경우, <strong>Exit</strong></li> |
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<li>그 외의 경우, line과 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>Y</mi><mi>W</mi></mrow><annotation encoding="application/x-tex">YW</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.68333em; vertical-align: 0em;"></span><span class="mord mathit" style="margin-right: 0.22222em;">Y</span><span class="mord mathit" style="margin-right: 0.13889em;">W</span></span></span></span></span> 사이의 교차를 구해, 교차하는 경우 해당 지점을 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord mathit" style="margin-right: 0.03588em;">y</span></span></span></span></span> 값으로 정의</li> |
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</ul> |
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</li> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>d</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">dy</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.88888em; vertical-align: -0.19444em;"></span><span class="mord mathit">d</span><span class="mord mathit" style="margin-right: 0.03588em;">y</span></span></span></span></span>가 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.64444em; vertical-align: 0em;"></span><span class="mord">0</span></span></span></span></span>일 경우 (<span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><msub><mi>p</mi><mn>3</mn></msub><mo>=</mo><mo>=</mo><mn>0</mn><mo>)</mo><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mo>(</mo><msub><mi>p</mi><mn>4</mn></msub><mo>=</mo><mo>=</mo><mn>0</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(p_3 == 0) || (p_4 == 0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathit">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span></span><span class="base"><span class="strut" style="height: 0.36687em; vertical-align: 0em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mord">∣</span><span class="mord">∣</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span></span><span class="base"><span class="strut" style="height: 0.36687em; vertical-align: 0em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span></span>) |
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<ul> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><msub><mi>q</mi><mn>3</mn></msub><mo>&lt;</mo><mn>0</mn><mo>)</mo><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mo>(</mo><msub><mi>q</mi><mn>4</mn></msub><mo>&lt;</mo><mn>0</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(q_3 \lt 0) || (q_4 \lt 0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mord">∣</span><span class="mord">∣</span><span class="mopen">(</span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span></span>인 경우, <strong>Exit</strong></li> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>max</mtext><mo>(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo>)</mo><mo>&lt;</mo><mi>X</mi><msub><mi>W</mi><mtext>min</mtext></msub></mrow><annotation encoding="application/x-tex">\text{max} (x_1, x_2) \lt XW_{\text{min}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord text"><span class="mord">max</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord mathit" style="margin-right: 0.07847em;">X</span><span class="mord"><span class="mord mathit" style="margin-right: 0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: -0.13889em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">min</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span> 인 경우, <strong>Exit</strong></li> |
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<li>그 외의 경우, line <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>X</mi><mi>W</mi></mrow><annotation encoding="application/x-tex">XW</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.68333em; vertical-align: 0em;"></span><span class="mord mathit" style="margin-right: 0.07847em;">X</span><span class="mord mathit" style="margin-right: 0.13889em;">W</span></span></span></span></span> 사이의 교차를 구해, 교차하는 경우 해당 지점을 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathit">x</span></span></span></span></span> 값으로 정의</li> |
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</ul> |
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</li> |
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</ol> |
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</li> |
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<li> |
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<p>이 외의 경우 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>u</mi><mtext>in</mtext></msub><mo separator="true">,</mo><mtext> </mtext><msub><mi>u</mi><mtext>out</mtext></msub></mrow><annotation encoding="application/x-tex">u_{\text{in}}, \space u_{\text{out}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">in</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mspace"> </span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.280556em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">out</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>이 존재하는 line임</p> |
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<ol> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>u</mi><mi>k</mi></msub><mo>=</mo><mfrac><msub><mi>q</mi><mi>k</mi></msub><msub><mi>p</mi><mi>k</mi></msub></mfrac></mrow><annotation encoding="application/x-tex">u_k = \frac{q_k}{p_k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right: 0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1.22861em; vertical-align: -0.481108em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.7475em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3448em;"><span class="" style="top: -2.34877em; margin-left: 0em; margin-right: 0.0714286em;"><span class="pstrut" style="height: 2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathit mtight" style="margin-right: 0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.151229em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.44611em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight" style="margin-right: 0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3448em;"><span class="" style="top: -2.34877em; margin-left: -0.03588em; margin-right: 0.0714286em;"><span class="pstrut" style="height: 2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathit mtight" style="margin-right: 0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.151229em;"><span class=""></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.481108em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>에 의해 구해지는 네 개의 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathit">u</span></span></span></span></span> 중 |
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<ul> |
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<li>경계를 뚫고 들어오는(<span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>p</mi><mi>k</mi></msub><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">p_k \lt 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.73354em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right: 0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.64444em; vertical-align: 0em;"></span><span class="mord">0</span></span></span></span></span>) <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>u</mi><mtext>in</mtext></msub></mrow><annotation encoding="application/x-tex">u_{\text{in}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">in</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></li> |
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<li>경계를 뚫고 나가는 (<span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>p</mi><mi>k</mi></msub><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">p_k \ge 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.83041em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right: 0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.64444em; vertical-align: 0em;"></span><span class="mord">0</span></span></span></span></span>) <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>u</mi><mtext>out</mtext></msub></mrow><annotation encoding="application/x-tex">u_{\text{out}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.280556em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">out</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span> 을 구한다.</li> |
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</ul> |
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</li> |
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<li>다음으로 <span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>u</mi><mtext>start</mtext></msub><mo separator="true">,</mo><mtext> </mtext><msub><mi>u</mi><mtext>end</mtext></msub></mrow><annotation encoding="application/x-tex">u_{\text{start}}, \space u_{\text{end}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.280556em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">start</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mspace"> </span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">end</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>를 구한다. |
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<ul> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>u</mi><mtext>start</mtext></msub><mo>=</mo><mtext>max</mtext><mo>(</mo><msub><mi>u</mi><mtext>in</mtext></msub><mo separator="true">,</mo><mn>0</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">u_{\text{start}} = \text{max} (u_{\text{in}}, 0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.280556em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">start</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord text"><span class="mord">max</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.317502em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">in</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span></span></li> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>u</mi><mtext>end</mtext></msub><mo>=</mo><mtext>min</mtext><mo>(</mo><msub><mi>u</mi><mtext>out</mtext></msub><mo separator="true">,</mo><mn>0</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">u_{\text{end}} = \text{min} (u_{\text{out}}, 0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">end</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord text"><span class="mord">min</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.280556em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">out</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span></span></li> |
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</ul> |
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</li> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>u</mi><mtext>start</mtext></msub><mo>&gt;</mo><msub><mi>u</mi><mtext>end</mtext></msub></mrow><annotation encoding="application/x-tex">u_{\text{start}} \gt u_{\text{end}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.6891em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.280556em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">start</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">></span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.58056em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">end</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>인 경우 <strong>Exit</strong></li> |
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<li>그 외의 경우, 다음의 식으로 새로운 좌표를 구한다. |
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<ul> |
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<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mi>x</mi><mn>1</mn><mo mathvariant="normal">′</mo></msubsup><mo>=</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>u</mi><mtext>start</mtext></msub><mi mathvariant="normal">Δ</mi><mi>x</mi><mo separator="true">,</mo><mtext> </mtext><msubsup><mi>y</mi><mn>1</mn><mo mathvariant="normal">′</mo></msubsup><mo>=</mo><msub><mi>y</mi><mn>1</mn></msub><mo>+</mo><msub><mi>u</mi><mtext>start</mtext></msub><mi mathvariant="normal">Δ</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">x_1&#x27; = x_1 + u_{\text{start}} \Delta x, \space y_1&#x27; = y_1 + u_{\text{start}} \Delta y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.248108em;"></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.751892em;"><span class="" style="top: -2.45189em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.248108em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.73333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.248108em;"></span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.280556em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">start</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord">Δ</span><span class="mord mathit">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mspace"> </span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.751892em;"><span class="" style="top: -2.45189em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.248108em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.77777em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.87777em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.280556em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">start</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord">Δ</span><span class="mord mathit" style="margin-right: 0.03588em;">y</span></span></span></span></span></li> |
|
<li><span class="katex--inline"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mi>x</mi><mn>2</mn><mo mathvariant="normal">′</mo></msubsup><mo>=</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>u</mi><mtext>end</mtext></msub><mi mathvariant="normal">Δ</mi><mi>x</mi><mo separator="true">,</mo><mtext> </mtext><msubsup><mi>y</mi><mn>2</mn><mo mathvariant="normal">′</mo></msubsup><mo>=</mo><msub><mi>y</mi><mn>1</mn></msub><mo>+</mo><msub><mi>u</mi><mtext>end</mtext></msub><mi mathvariant="normal">Δ</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">x_2&#x27; = x_1 + u_{\text{end}} \Delta x, \space y_2&#x27; = y_1 + u_{\text{end}} \Delta y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.248108em;"></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.751892em;"><span class="" style="top: -2.45189em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.248108em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.73333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.248108em;"></span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">end</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord">Δ</span><span class="mord mathit">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mspace"> </span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.751892em;"><span class="" style="top: -2.45189em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.248108em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.77777em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit" style="margin-right: 0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.301108em;"><span class="" style="top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 0.87777em; vertical-align: -0.19444em;"></span><span class="mord"><span class="mord mathit">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.336108em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">end</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord">Δ</span><span class="mord mathit" style="margin-right: 0.03588em;">y</span></span></span></span></span></li> |
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</ul> |
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</li> |
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</ol> |
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</li> |
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</ol> |
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<p>이렇게 Liang-Barsky Line Clipping을 진행할 수 있다.</p> |
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<h1 id="abbr.">Abbr.</h1> |
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<ul> |
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<li>CAD: Computer Aided Design |
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<ul> |
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<li>컴퓨터를 이용해 전문적인 설계(design)를 하는 것을 의미</li> |
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</ul> |
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</li> |
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<li>CAM: Computer Aided Manufacturing |
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<ul> |
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<li>컴퓨터를 이용해 제품 생산 준비 전반을 하는 것을 의미</li> |
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</ul> |
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</li> |
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<li>WCS: World Coordinate System |
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<ul> |
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<li>실제 세계에서의 좌표 시스펨</li> |
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</ul> |
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</li> |
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<li>NCS: Normalized Coordinate System |
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<ul> |
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<li>단위를 규격화한 좌표 시스템</li> |
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</ul> |
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</li> |
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<li>DCS: Device Coordinate System |
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<ul> |
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<li>출력 디바이스의 좌표 시스템</li> |
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</ul> |
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</li> |
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<li>CRT: Cathode Ray Tube |
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<ul> |
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<li>전자를 인광물질이 코팅된 화면에 충돌시켜 디스플레이하는 장치</li> |
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</ul> |
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</li> |
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<li>TFED: Thin Film Electroluminescent Display |
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<ul> |
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<li>발광 물질을 얇게 발라 발광시키는 디스플레이 장치</li> |
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</ul> |
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</li> |
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<li>PDP: Plasma Display Panel |
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<ul> |
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<li>가스를 방전시켜 발광시키는 디스플레이 정치</li> |
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</ul> |
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</li> |
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<li>DVST: Direct Video Storage Tube |
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<ul> |
|
<li>한 번 그리면 계속 남아있으며, 방전시켜야 refresh되는 디스플레이 장치</li> |
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</ul> |
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</li> |
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<li>LCD: Liquid Crystal Display |
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<ul> |
|
<li>액정이 전기신호에 따라 빛을 차단하거나 통과시키는 방식으로 디스플레이 하는 장치</li> |
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</ul> |
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</li> |
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<li>LED: Light Emitting Diode |
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<ul> |
|
<li>전압을 가했을 때 발광하는 반도체 소자</li> |
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</ul> |
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</li> |
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<li>OLED: Organic Light Emitting Diode |
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<ul> |
|
<li>전류에 반응해 빛을 내는 유기 화합물의 필름으로 이루어진 발광 다이오드</li> |
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</ul> |
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</li> |
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<li>DDA: Digital Differential Analyzer |
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<ul> |
|
<li>기울기와 반올림, 그리고 덧셈만을 이용해 선을 그리는 알고리즘</li> |
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</ul> |
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</li> |
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<li>CG: Computer Graphics |
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<ul> |
|
<li>컴퓨터를 이용해 실제 세계의 영상을 조작하거나 만들어내는 기술</li> |
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</ul> |
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</li> |
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<li>CA: Computer Animation |
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<ul> |
|
<li>화면 상에서 시각적 변화에 대한 연속적인 배열</li> |
|
</ul> |
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</li> |
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<li>GT: Graphical Transformation |
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<ul> |
|
<li>시각적인 그래픽 변화</li> |
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</ul> |
|
</li> |
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<li>RC: Refreshing Cycle |
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<ul> |
|
<li>CA에서 각 장면 사이의 텀</li> |
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</ul> |
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</li> |
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<li>SB: StoryBoard |
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<ul> |
|
<li>영상에 대한 흐름을 설명하기 위한 문서</li> |
|
</ul> |
|
</li> |
|
<li>KF: Key Frame |
|
<ul> |
|
<li>변화(transition)의 시작점 또는 끝점을 나타낸(mark) 것</li> |
|
</ul> |
|
</li> |
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<li>SA: Simulating Acceration |
|
<ul> |
|
<li>in-between 갯수를 통해 애니메이션의 속도를 지정하는 것</li> |
|
<li>여기서 in-between은 KF 사이의 장면들을 의미함</li> |
|
</ul> |
|
</li> |
|
<li>DMS: Direct Motion Specification |
|
<ul> |
|
<li>실제 물리법칙을 적용하는 것</li> |
|
</ul> |
|
</li> |
|
<li>GDS: Goal Directed System |
|
<ul> |
|
<li>가상의 물리법칙을 적용하는 것</li> |
|
</ul> |
|
</li> |
|
</ul> |
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<h3 id="안나올-것-같은-것들">안나올 것 같은 것들</h3> |
|
<ul> |
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<li>TC: Training Center |
|
<ul> |
|
<li>훈련소</li> |
|
</ul> |
|
</li> |
|
<li>CVP: Camera ViewPoint |
|
<ul> |
|
<li>카메라의 시점</li> |
|
</ul> |
|
</li> |
|
<li>LE: Lighting Effect |
|
<ul> |
|
<li>조명 효과</li> |
|
</ul> |
|
</li> |
|
<li>ETS, EITS: Equal Interval Time Sequence |
|
<ul> |
|
<li>등속도</li> |
|
</ul> |
|
</li> |
|
<li>SG: Silicon Graphics |
|
<ul> |
|
<li>회사이름임</li> |
|
</ul> |
|
</li> |
|
<li>FR: Fully Rendered |
|
<ul> |
|
<li>렌더링이 모두 완료된 것을 의미함</li> |
|
</ul> |
|
</li> |
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</ul> |
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</div> |
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</body> |
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</html> |
