gauravsdeshmukh/MonteCarloSimulation_PutnamProblem.ipynb

## MonteCarloSimulation_PutnamProblem.ipynb
{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Monte-Carlo Simulation to solve a Putnam Competition problem\n",
    "\n",
    "### Created by: Gaurav S. Deshmukh\n",
    "\n",
    "This notebook details a Monte-Carlo simulation used to solve the following problem:\n",
    ">If you choose 4 random points on a sphere and join them, what is the probability that the resulting tetrahedron contains the centre of the sphere?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Import Required Modules"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import scipy as sci\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy.linalg\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Define Sample Space"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Define number of points and create empty arrays to store coordinates"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "points=10000\n",
    "x=sci.zeros((points,3))\n",
    "y=sci.zeros((points,3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Store all 3D vectors on the surface of a sphere of radius 1 in 'y'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "for i in range(points):\n",
    "    vector=sci.random.rand(1,3)-0.5\n",
    "    if(sci.linalg.norm(vector)!=0 and sci.linalg.norm(vector)<=1.0):\n",
    "        x[i,:]=vector\n",
    "        y[i,:]=x[i,:]/sci.linalg.norm(x[i,:])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Check if all the norms of vectors in 'y' are equal to 1; if \"All Clear\" then proceed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "All Clear\n"
     ]
    }
   ],
   "source": [
    "y_norms=sci.zeros(points)\n",
    "for i in range(points):\n",
    "    y_norms[i]=sci.linalg.norm(y[i,:])\n",
    "    tol=1e-10\n",
    "    norm_diff=abs(y_norms-1)\n",
    "    danger_array=y_norms[norm_diff>tol]\n",
    "    \n",
    "if(len(danger_array)==0):\n",
    "    print(\"All Clear\")\n",
    "else:\n",
    "    print(\"Danger\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. Create functions to determine whether the centre lies inside a tetrahedron"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The CheckSide function checks if the fourth vertex and a stray point lie on the same side of the plane formed by the remaining three vertices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def CheckSide(vertices,point):\n",
    "    t1,t2,t3,t4=vertices\n",
    "    p=point\n",
    "    side_1=t2-t1\n",
    "    side_2=t3-t1\n",
    "    normal=sci.cross(side_1,side_2)\n",
    "    ref_vector=t4-t1\n",
    "    ref_sign=sci.dot(normal,ref_vector)\n",
    "    point_vector=p-t1\n",
    "    point_sign=sci.dot(normal,point_vector)\n",
    "    if(sci.sign(ref_sign)==sci.sign(point_sign)):\n",
    "        return 1\n",
    "    else:\n",
    "        return 0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The CheckTetrahedron function calls the CheckSide function 4 times and concludes whether the stray point lies inside or outside the tetrahedron"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def CheckTetrahedron(vertices,point):\n",
    "    vert=sci.copy(vertices)\n",
    "    check_1=CheckSide(vert,point)\n",
    "    vert=sci.roll(vert,1,axis=0)\n",
    "    check_2=CheckSide(vert,point)\n",
    "    vert=sci.roll(vert,1,axis=0)\n",
    "    check_3=CheckSide(vert,point)\n",
    "    vert=sci.roll(vert,1,axis=0)\n",
    "    check_4=CheckSide(vert,point)\n",
    "    sum_check=check_1+check_2+check_3+check_4\n",
    "    if(sum_check==4.):\n",
    "        return 1\n",
    "    else:\n",
    "        return 0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4. Initialization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "centre=[0,0,0] #Centroid of the sphere\n",
    "number_of_samples=50000 #Number of samples of tetrahedra to be taken\n",
    "sample_span=sci.arange(0,number_of_samples,1) #Array for plotting\n",
    "check_point=sci.zeros(number_of_samples) #Array to store 0s and 1s based on success or failure\n",
    "prob=sci.zeros(number_of_samples) #Array to store resultant probabilities "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5. The Run"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "for i in range(number_of_samples):\n",
    "    indices=sci.random.randint(0,points,4) #Get any 4 random numbers as indices for the 'y' array\n",
    "    vertex_list=y[indices] #Get 4 random points from 'y' based on indices obtained\n",
    "    check_point[i]=CheckTetrahedron(vertex_list,centre) #Return 1 for success and 0 for failure\n",
    "    prob[i]=len(check_point[check_point==1.])/(i+1) #Store resultant probability of run"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 6. Plotting"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'Variation of resultant probability with increase in the number of iterations')"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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YQZJ+kvODq42kBdbaH4wxI+UMOrJRzt33WyRt8QjGv5b0gDHmv3Lu4I6S8+M/0D5ZY8wO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\ntGjhPgcbNmyo3r17691339U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FAQAAAADSq7Xh7zFJhyXv3yJpppktl3SPuMA7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CcGku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\nzMzMOsB/AV/aS5opdK6KAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1c9e8710eb8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#Plot blank figure\n",
    "plt.figure(figsize=(15,10))\n",
    "#Plot resultant probability from simulation\n",
    "plt.plot(sample_span,prob,color=\"navy\",linestyle=\"-\",label=\"Resultant probability from simulation\")\n",
    "#Plot resultant probability from analytical solution\n",
    "plt.plot(sample_span,[0.125]*len(sample_span),color=\"red\",linestyle=\"-\",label=\"Actual resultant probability from analytical solution (0.125)\")\n",
    "#Plot value of final resultant probability in text\n",
    "plt.text(sample_span[int(number_of_samples/2)],0.05,f\"The final probability is {prob[-1]:.4f}\",fontsize=16)\n",
    "#Display axis labels\n",
    "plt.xlabel(\"Number of iterations\",fontsize=14)\n",
    "plt.ylabel(\"Probability\",fontsize=14)\n",
    "#Display legend\n",
    "plt.legend(loc=\"upper right\",fontsize=14)\n",
    "#Display title of the plot\n",
    "plt.title(\"Variation of resultant probability with increase in the number of iterations\",fontsize=14)"
   ]
  }
 ],
 "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": "3.6.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}