thouis/Sphere sampling.ipynb

## Sphere sampling.ipynb
{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    }
   ],
   "source": [
    "% pylab inline\n",
    "import numpy as np\n",
    "\n",
    "U = np.random.uniform(-1, 1, 100000)\n",
    "theta = np.random.uniform(0, 2 * np.pi, 100000)\n",
    "X = np.sqrt(1 - U**2) * np.cos(theta)\n",
    "Y = np.sqrt(1 - U**2) * np.sin(theta)\n",
    "Z = U"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1076e18d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "midX = X / 2\n",
    "midY = Y / 2\n",
    "midZ = (Z + 1) / 2\n",
    "dist = np.sqrt(midX ** 2 + midY ** 2 + midZ ** 2)\n",
    "hist(dist, 'auto');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x108ffa6d8>]"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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2txHakfA04CQqWJcMp0MfwBhzIfAGZbtvyjMWuNFaW8lax7ExxrQgdDe/xzRr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\nEYW+iIiLKPRFRFxEoS8i4iL/DzekOZBPRyBnAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109002860>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# random points in sphere\n",
    "A = np.random.uniform(-1, 1, 100000)\n",
    "B = np.random.uniform(-1, 1, 100000)\n",
    "C = np.random.uniform(-1, 1, 100000)\n",
    "dist = np.sqrt(A**2 + B**2 + C**2)\n",
    "dist = dist[dist < 1]\n",
    "a, b, c = hist(dist, 'auto', normed=True)\n",
    "plot(np.linspace(0, 1, 10),  3 * np.linspace(0, 1, 10)**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "229.28148638736621"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dist.shape[0]**.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.0434106486638322"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.max()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2341.0055151165357"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(1/(b[1] - b[0]))**2/2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2269.0"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.max()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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2NMpd/s4HNcr3DDgWSdIAxjWo7m6Uu7wL1bzV1+xLkjSHxjWomgsilnVou1ejfOuAY5nK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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1080a3710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "U = np.random.uniform(-1, 1, 10000)\n",
    "R = np.random.uniform(0, 1, 10000)**(1/3)\n",
    "hist(np.sqrt(R**2 * (1 - U**2)), 'auto');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1094d4550>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "hist(R, 'auto', normed=True);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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O0/ql/DXhAABDNGv4RMQ9kh6u7Xp7zoVtL9b0Ndlubta0ab5R236VqvHnOY6rbT8h6fsD\ntKFVtv/EdnQ+E6NuD8af7UW297e9r+1dR90e7Pxsb6h9Ty1v89q5w5brryY41PbRGee8S1L9H8iV\n2a3a0XVJ+U9nO8H2PpJOqO36l5zVuAs6YfZDMN91gmbC9l2qup8fVbVs1DO2b7W91vYrRttK7Ixs\n7yvp14d1/dzw+XRSXjdT15ftV0v6aG3X3RGxsWnjpkTEBk1fofqPbB/U6/iOT0h6ea18eb/1t832\naySdOOp2YLzZPkXVc8qPSVqh6ZOkd5O0UtIlku6zfdyOVwAGco6m30C0Kit8IuJOSV+r7Vot6fxu\nx3buOK6WVH9t9kSPY9fUbunC9k0zNOOTte2Fkr5ie+8e1/2wti+TI0lfjYixWI6k013yGQ3xfyrm\nPtunS1qvvKkFSyXdYPv44bYK84XtgyWdN8w6mqwWsFbTJ2l+xPZNtk+1vcL2kbY/pGqAQr1b7tqI\nWD9oQyPiKklfr+16o6R7bX/Q9krbh9r+A9vfkHRh7binJX1g0PoHZXuh7d+VdJPocsMMOv/wL9f0\nf59flnSMqjDaW9JRkr5Q+/lCSV+0nbWuFtCN7V+xvVbSdzV9jmTrctZ2kyRFxP22z5T0JW1fsuaY\nzqeXTZLe03/zdnCGqld4T71jaImkv53h+BcknRYRP2qxDdlsnyPpLaqC8s1iVW3kuVjTV+ZYFxEf\nT47ZIGmD7U3a/sfWXp3tM4bfROwMOs91PqjqbdWHqXrhZj9rcDbWaJ20iLhG0vHa8T093VwlaVVE\nPNVPw3rUv0XSKklfzTj8QUnHRcTXZz1yeN6vqvvvaBE8yGB7haSTaru+2SV4XhIRF2n624VPs31A\nr+OBxAGS/lLSH6r6o75I8EjNF+lURNwo6XWS3qeqG+xxVRNIn1P1cPQySUdFxDsjIl0aJ73WFRHh\n2ufYjPqfiIh3qBpGfYWk+1WtdL1N1fI510s6S9IhEcG8Hsw1ZyXlyYxzLq5tL5C0prXWAEPi8Rp9\nvHOxfbK6PzB+g6S/qJUnWSIfkmT7AVVdIFK1HNRBs00RsL2nqnlsU68fuTnnDzmgM0Ds5B4/PlvS\nW2vlAyNic1t1Zz/zQXMRcW23/baP1fTwAdSZxFefQvCtnLlpEfGzzrOfqTkZK20viojn228ldiYR\n8aSqHqQddL6n3trtZ21o3O0GYGjSpatu63pUd/V5cAslHTh4c4DhIXyA8XFIUv5hg3MfTMrporrA\nWCF8gPGRztF5qMG56QjUpQO2BRgqwgcYH0uS8pYG56YjSxnaj7FG+ADjI33t/NauR3X381muBYwV\nwgcYHwuTchooM9mWlItNFgT6QfgA4+OFpNzk3+fLknLjFzcCJc3b8LF9bLKidhuf5aP+78Kc9mxS\nXtTg3LSbLb0WMFbmbfgAYygdNJDzOoUpi5PyYwO2BRgqwgcYH5uT8n4Nzk2PTa8FjJX5vLzOnapW\n6G7TT1q+HuaXHyTl10v6z8xz65NKX5T0QCstAoZk3oZPZ02jG0bdDqBmU1I+QtLnM889vLZ9R0Qw\n4ABjjW43YExExD2SHq7tenvOebYXq3pZ4ZSb22wXMAyEDzBerq9tH2r76J5HbvcuSbvWyle22ySg\nfYQPMF4+nZTX2e45YdT2qyV9tLbr7ojYOJSWAS0ifIAxEhF3SvpabddqSed3O7bzIrCrVb0KecrE\n0BoHtGjeDjgAxthaSau0fXHQj9g+StKnJN2nakLpqs5x9ZWwr42I9SUbCvSL8AHGTETcb/tMSV/S\n9mVzjul8etkk6T3DbhvQFrrdgDEUEdeomoeWvqenm6skrYqIp4bbKqA9znhFPIARsf1ySWsknaJq\nLs8+qhYg/bGkWyR9NiI2jKyBQJ8IHwBAcXS7AQCKI3wAAMURPgCA4ggfAEBxhA8AoDjCBwBQHOED\nACiO8AEAFEf4AACKI3wAAMURPgCA4ggfAEBxhA8AoDjCBwBQ3P8D3CYFVQJBLSIAAAAASUVORK5C\nYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1090f5240>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "theta = np.random.uniform(0, 2 * np.pi, 10000)\n",
    "X = R * np.sqrt((1 - U**2)) * cos(theta)\n",
    "Y = R * np.sqrt((1 - U**2)) * sin(theta)\n",
    "hist(X, 'auto', normed=True);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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HTIOfLwzb0H/G6oTPpqS8d4P7p+em96pjkPoXaep/Yz/1N1Y+OwKGgp8vDFuOn7E64ZNu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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10911e9e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "hist(A[np.sqrt(A**2 + B**2 + C**2) < 1], 'auto', normed=True);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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oM08FTh2lUdIkRcRlmXnS5m6HVreV8ns4qwNAHirKo/w7n1CU7x+zLZKkKZvVMLuvKK8d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\nklS9/w86mxv+xNaliAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109d44898>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "hist(np.sqrt(X**2 + Y**2 + (U*R)**2), 'auto');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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rusx8EFhTnDqiu2hekjQhBmA7z2y0r+/Zq7ebiuOlwP7DD0eSNF/uBNPOIY32HS2uvavR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\n20IDph8wkgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a6a2a20>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Random point in sphere\n",
    "A = np.random.uniform(-1, 1, 100000)\n",
    "B = np.random.uniform(-1, 1, 100000)\n",
    "C = np.random.uniform(-1, 1, 100000)\n",
    "dist = np.sqrt(A**2 + B**2 + C**2)\n",
    "X = A[dist < 1]\n",
    "Y = B[dist < 1]\n",
    "Z = C[dist < 1]\n",
    "\n",
    "# Random unit normal\n",
    "N = X.shape[0]\n",
    "dZ = np.random.uniform(-1, 1, N)\n",
    "dT = np.random.uniform(0, 2 * np.pi, N)\n",
    "dX = np.sqrt(1 - dZ**2) * cos(dT)\n",
    "dY = np.sqrt(1 - dZ**2) * sin(dT)\n",
    "\n",
    "# Find closest point on line through P in direction D to origin\n",
    "P = np.stack((X, Y, Z))\n",
    "D = np.stack((dX, dY, dZ))\n",
    "proj = (P * D).sum(axis=0, keepdims=True)\n",
    "closest = P - proj * D\n",
    "len_closest = np.sqrt((closest ** 2).sum(axis=0))\n",
    "hist(len_closest, 'auto');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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mFR9Tdi7KN8yzL7OZ19ShJFVg5H/HJjWoNhXlBw9wbXlu2dbQdL/nkqRFayH+jk1qUP2w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\nAxBvuxYAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a775be0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "N = 10000\n",
    "\n",
    "# Random point on sphere\n",
    "dZ = np.random.uniform(-1, 1, N)\n",
    "dT = np.random.uniform(0, 2 * np.pi, N)\n",
    "dX = np.sqrt(1 - dZ**2) * cos(dT)\n",
    "dY = np.sqrt(1 - dZ**2) * sin(dT)\n",
    "\n",
    "# 2nd Random point on sphere\n",
    "dZ2 = np.random.uniform(-1, 1, N)\n",
    "dT2 = np.random.uniform(0, 2 * np.pi, N)\n",
    "dX2 = np.sqrt(1 - dZ2**2) * cos(dT2)\n",
    "dY2 = np.sqrt(1 - dZ2**2) * sin(dT2)\n",
    "\n",
    "# closest approach to origin is midpoint\n",
    "midX = (dX + dX2) / 2\n",
    "midY = (dY + dY2) / 2\n",
    "midZ = (dZ + dZ2) / 2\n",
    "\n",
    "len_closest = np.sqrt((midX ** 2 + midY ** 2 + midZ **2))\n",
    "hist(len_closest, 'auto');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "https://gist.github.com/c9db8fd723fb8a5ea26289ca0d6b1cbc\r\n"
     ]
    }
   ],
   "source": [
    "!gist -u https://gist.github.com/c9db8fd723fb8a5ea26289ca0d6b1cbc ./Sphere\\ sampling.ipynb"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}