Sampling a point on a unit sphere

sample-from-unit-sphere.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Three ways of sampling a random point on a unit sphere's surface"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.141592653589793"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import random\n",
    "import math\n",
    "π = math.pi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The incorrect (biased one) one;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "823 ns ± 16.9 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "θ = 2 * π * random.random()\n",
    "φ = π * random.random()\n",
    "\n",
    "x = math.sin(θ) * math.cos(φ)\n",
    "y = math.sin(θ) * math.sin(φ)\n",
    "z = math.cos(θ)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The Fermat Library one;\n",
    "[Source](https://twitter.com/fermatslibrary/status/1339922144080367619)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "862 ns ± 13.1 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "u = random.random()\n",
    "v = random.random()\n",
    "θ = 2 * π * u\n",
    "φ = math.acos(2*v-1)\n",
    "\n",
    "sinθ = math.sin(θ)\n",
    "x = sinθ * math.cos(φ)\n",
    "y = sinθ * math.sin(φ)\n",
    "z = math.cos(θ)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The one by @norpadon\n",
    "[Source](https://twitter.com/norpadon/status/1339923332796432388)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.72 µs ± 158 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "x = random.normalvariate(0, 1)\n",
    "y = random.normalvariate(0, 1)\n",
    "z = random.normalvariate(0, 1)\n",
    "\n",
    "r = math.sqrt(x**2 + y**2 + z **2)\n",
    "x /= r\n",
    "y /= r\n",
    "z /= r"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# And with NumPy:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10000"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "N = 10000"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Fermat Library implementation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.32 ms ± 14 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "u = np.random.random(N)\n",
    "v = np.random.random(N)\n",
    "θ = 2 * π * u\n",
    "φ = np.arccos(2*v-1)\n",
    "\n",
    "sinθ = np.sin(θ)\n",
    "x = sinθ * np.cos(φ)\n",
    "y = sinθ * np.sin(φ)\n",
    "z = np.cos(θ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.06 ms ± 13 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "x = np.random.normal(size=N)\n",
    "y = np.random.normal(size=N)\n",
    "z = np.random.normal(size=N)\n",
    "\n",
    "r = np.sqrt(x**2 + y**2 + z **2)\n",
    "x /= r\n",
    "y /= r\n",
    "z /= r"
   ]
  }
 ],
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