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quadratic bezier curve length in javascript
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 /* * http://en.wikipedia.org/wiki/B%C3%A9zier_curve * http://www.malczak.linuxpl.com/blog/quadratic-bezier-curve-length/g */ function Point(x, y) { this.x = x; this.y = y; } function quadraticBezierLength(p0, p1, p2) { var a = new Point( p0.x - 2 * p1.x + p2.x, p0.y - 2 * p1.y + p2.y ); var b = new Point( 2 * p1.x - 2 * p0.x, 2 * p1.y - 2 * p0.y ); var A = 4 * (a.x * a.x + a.y * a.y); var B = 4 * (a.x * b.x + a.y * b.y); var C = b.x * b.x + b.y * b.y; var Sabc = 2 * sqrt(A+B+C); var A_2 = sqrt(A); var A_32 = 2 * A * A_2; var C_2 = 2 * sqrt(C); var BA = B / A_2; return (A_32 * Sabc + A_2 * B * (Sabc - C_2) + (4 * C * A - B * B) * log((2 * A_2 + BA + Sabc) / (BA + C_2))) / (4 * A_32); }

### ayadirh commented Apr 6, 2021 • edited

how do you calculate the length for Bezier Curve in three dimensions? Is this the right approach? @steveruizok @chknoflach

```    function quadraticBezierLength(x1, y1, z1, x2, y2, z2, x3, y3, z3) {
let a, b, c, u
let v1x = x2 * 2
let v1y = y2 * 2
let v1z = z2 * 2
let d = x1 - v1x + x3
let d1 = y1 - v1y + y3
let d2 = z1 - v1z + z3
let e = v1x - 2 * x1
let e1 = v1y - 2 * y1
let e2 = v1z - 2 * z1
let c1 = (a = 4 * (d * d + d1 * d1 + d2 * d2))
c1 += b = 4 * (d * e + d1 * e1 + d2 * e2)
c1 += c = e * e + e1 * e1 + e2 * e2
c1 = 2 * Math.sqrt(c1)
let a1 = 2 * a * (u = Math.sqrt(a))
let u1 = b / u
a = 4 * c * a - b * b
c = 2 * Math.sqrt(c)
return (
(a1 * c1 + u * b * (c1 - c) + a * Math.log((2 * u + u1 + c1) / (u1 + c))) /
(4 * a1)
)
}```

### bczhc commented May 17, 2021

@chknoflach Yes but in the first case you mentioned, if P0 is (10, 10), P1 is (10.00001, 10.00012), P2 is (20, 20), the divisor will be so small and the division result will be so big, makes a poor precision as a `double` number. It's annoying for me to process these points data from users drawing on a GUI.

### herrstrietzel commented Sep 22, 2023 • edited

As commented before, we should also check for actually linear commands:

```/**
* p0: previous command's final point
* cp1: quadratic bézier control point
* p: final point
*/

function quadraticBezierLength(p0, cp1, p, t = 1) {
if (t === 0) {
return 0;
}

const interpolate = (p1, p2, t) => {
let pt = { x: (p2.x - p1.x) * t + p1.x, y: (p2.y - p1.y) * t + p1.y };
return pt;
}
const getLineLength = (p1, p2) => {
return Math.sqrt(
(p2.x - p1.x) * (p2.x - p1.x) + (p2.y - p1.y) * (p2.y - p1.y)
);
}

// is flat/linear
let l1 = getLineLength(p0, cp1) + getLineLength(cp1, p);
let l2 = getLineLength(p0, p);
if (l1 === l2) {
let m1 = interpolate(p0, cp1, t);
let m2 = interpolate(cp1, p, t);
p = interpolate(m1, m2, t);
let lengthL;
lengthL = Math.sqrt((p.x - p0.x) * (p.x - p0.x) + (p.y - p0.y) * (p.y - p0.y));
return lengthL;
}

let a, b, c, d, e, e1, d1, v1x, v1y;

v1x = cp1.x * 2;
v1y = cp1.y * 2;
d = p0.x - v1x + p.x;
d1 = p0.y - v1y + p.y;
e = v1x - 2 * p0.x;
e1 = v1y - 2 * p0.y;
a = 4 * (d * d + d1 * d1);
b = 4 * (d * e + d1 * e1);
c = e * e + e1 * e1;

const bt = b / (2 * a),
ct = c / a,
ut = t + bt,
k = ct - bt ** 2;

return (
(Math.sqrt(a) / 2) *
(ut * Math.sqrt(ut ** 2 + k) -
bt * Math.sqrt(bt ** 2 + k) +
k *
Math.log((ut + Math.sqrt(ut ** 2 + k)) / (bt + Math.sqrt(bt ** 2 + k))))
);
}```