Intersection of Ellipse and Circle.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Intersection points of an ellipse centered at origin and a circle centered at (X,0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()\n",
    "x,y,H,W,X,R = symbols('x y H W X R')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Equation of circle of radius R centered at (X,0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAL0AAAAaCAYAAADizl1mAAAABHNCSVQICAgIfAhkiAAABYdJREFU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\nTkSuQmCC\n",
      "text/latex": [
       "$$y^{2} + \\left(- X + x\\right)^{2} = R^{2}$$"
      ],
      "text/plain": [
       " 2           2    2\n",
       "y  + (-X + x)  = R "
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e1 = Eq((x-X)**2 + y**2, R**2)\n",
    "e1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Equation of ellipse with radius W along x-axis and radius H along y-axis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGsAAAAoCAYAAAD5cwTOAAAABHNCSVQICAgIfAhkiAAAA+hJREFU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\n",
      "text/latex": [
       "$$\\frac{x^{2}}{W^{2}} + \\frac{y^{2}}{H^{2}} = 1$$"
      ],
      "text/plain": [
       " 2    2    \n",
       "x    y     \n",
       "── + ── = 1\n",
       " 2    2    \n",
       "W    H     "
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e2 = Eq((x/W)**2 + (y/H)**2, 1)\n",
    "e2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Solve for y."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAARsAAAAmCAYAAAD0vj/mAAAABHNCSVQICAgIfAhkiAAACIdJREFU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\nFEVRFEVRFEVRlMbxPwh5E5MKYKsBAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{H^{2}}{W^{2}} \\left(W^{2} - x^{2}\\right) + \\left(- X + x\\right)^{2} = R^{2}$$"
      ],
      "text/plain": [
       " 2 ⎛ 2    2⎞                 \n",
       "H ⋅⎝W  - x ⎠           2    2\n",
       "──────────── + (-X + x)  = R \n",
       "      2                      \n",
       "     W                       "
      ]
     },
     "execution_count": 118,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ysq = solve(e2, y**2)[0]\n",
    "e4 = e1.subs(y**2, ysq)\n",
    "e4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Expand out into quadratic equation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAlkAAAAWCAYAAADzaYnXAAAABHNCSVQICAgIfAhkiAAACdRJREFU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\niZTj/2e04cnGdySTAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$H^{4} W^{2} - H^{4} x^{2} - H^{2} R^{2} W^{2} + H^{2} W^{2} X^{2} - 2 H^{2} W^{2} X x + H^{2} W^{2} x^{2} = 0$$"
      ],
      "text/plain": [
       " 4  2    4  2    2  2  2    2  2  2      2  2        2  2  2    \n",
       "H ⋅W  - H ⋅x  - H ⋅R ⋅W  + H ⋅W ⋅X  - 2⋅H ⋅W ⋅X⋅x + H ⋅W ⋅x  = 0"
      ]
     },
     "execution_count": 119,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e5 = Eq(expand(Add(e4.lhs,-e4.rhs)*H**2*W**2),0)\n",
    "e5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Has solutions only if..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAcwAAAAYCAYAAAB3A80LAAAABHNCSVQICAgIfAhkiAAACZhJREFU\neJztnX2MXUUVwH+UYruW2qU1BKsEpEKRoLb1A/GjrqhAjV8RFEn8eCpUFNAoYEWDLB/GpjTYIgI2\nqBVBhNTQpAJaqF2VBKF82Wgo1Y2rhlaLq20pXesurH+cM77bu3Pvnffum7l32fklL2/fnLlz5pw3\n8+bOmZm7EIlEIpFIZFxxMTAKXBtI1yZgN/AUsA44PoDePELa32lC+LOO31k7VGlH9GG9dPimKhue\nD76zMqnqCihvBBYDmwPp6wGuA94EnASMAPcCMwPpTxPa/k7Tg39/htARgh6qs6NK3Z2kh9jeXOih\nGhuq0jshmAH0A28H+qhmhnUw8Czw3pw8PwR2ANM6rLsO9ncaF3+OBx0hqNKO6MN66fBNVTZk6X0t\nElU7K3B92qZohvlRxKA8o9arvCennBs0z2KLbBWwBtiYce3Beu2fM+TzEnV8R0aezSp/VYZ8OuKL\nf2fIXw98DFgKPJOS+bY/JGVtMRT50+hJvnYA9wEfcqxrkQ7f+PZVHdq9b0K0t25gJzCo+dJMQvrf\nKHBjmzpC4KvPVNnOHgbWAldoPariZcD3gW3APmAAWAEc0kohhyMN7WnyB8xBZMqdN/N6VMtYkEo/\nG3HaQfq5D/sMaxj4V0bZP6b5hX7QIj9VZXfm1O92reOBGfL1iC+6LLIQ9oeijC1Jivw5CDwHXAb0\nAlciP1ojWvYXHepapMM3IXxVdbv3Taj2drmWcbFFdq3K1uVc76LDNz77TJXt7A16/VdzrvfJHOAf\nWoe1yKTol/p5CzDLpZADkJhzP3AV2QPmUSr7XU5ZL0S+kP/QHBgA5iILwnMTaX3YB4ynkIaR5ggt\n+y9aj09Z8mxQ2dsy6nc1cmdxVIb8GKShrrLIQtkfgjK2JCnyp9GzxSI7R2UDBXUt0uGbUL6qst0X\n0aB4dphHKB+CzBR2Af9k/5nM17QO96uuLOriK199pup29rjqKIp4fg/4PHB0Qb5W+AVS//NT6Vdr\n+g0uhXwBGSQWInczWQPmh1VmG0wMb9U8D6bSG5o+kniNqt4RYEoi71aVpaft19D8Im13WQs0/bcZ\ndfsWsB04Nqf+S8kOR4SyPwRlbDG4+NPoucUiO1Jle3Oud9Hhm1C+qrLdF9Gg3CAQyoeGK7WsJfr5\nk7jNIurkK199pup2dqmWc0pBPlPPUeBPwLeBRdgjfy7MoRmOTg/W04E9yBJc7r6VVwJDiLGQP2Au\nU9mnc8q7SPNcl0rvRrYaJ1+bkBDA8cgs1/CglvHSRNosNWYjEoMeRcIVSX5CdihhJfB3xN48HiI7\nbBTK/hCUsQXc/Wn0fMkie6fKHs641lWHb0L5qsp2X0SDcoNAKB8aZiHLSzuQAWgYmfkckXNN3Xzl\nq89U3c6MDcsL8nUB70aicP00B88h4G5an32epdd/N0NuZp//nyxNTmWYDPwI+CtuMeXX6fsZyC5P\nGyfo+0Op9J36SvIMEkv/fSrdLBbPAJ7Uv89DwihLkXCLkRteDpyO3JWsTZX3HWQTzwe07MM0fY++\nDNOQRe/HGbvZB8LZH4Iytrj6M6knXcaLaXaYpRbdrejwTShfVdXuQxDKh4ZB5If2K8BtyBnBRUgo\n0EYdfeWrz1Tdzjbp+0KLLMkQcJe+QGaui5BB9CRkPXUlMpjeDdwMPJBTnlkO25oh/yNwMrIst8GW\n4XJk+++JibRe7DPMA5Af/FHH16tzKm7ow76GZ+5kTL26kLj7I4m6PAusTlxjwglnW8rLqmNvKt8x\nmr7eUkZI+31T1hZXfyb1LFf5FcBNmr4b+GxGHV11+CaUr6C6du9Cg/ZnTSF9mOTNibzvKchbR1/5\n6jN1aGdDyIy0XaYhM8zBhL6bC65ZhX1sM3xD5bbNYpyAhB6XpdJ7Mwo1g8n9ORU6RPPspdzuMrO1\n/FT9fK5+PiORZydwh/49E7mT2U65tcATVc9tFllI+9MM4P5j49JwQtli9Nhee2h+v51kIEdnnX0F\n1bX7NAO05sPVBeVV0Xdms78dX+5AmTaSOjrpK599pg7t7EnsG4/yeAESLl0O/IGmX/6LTD5OL7i+\n5QFzcuL9JmRqeoljZU2Y4JGcPGYr+GPIHUq7mJBBN9J5LkCm3WsSeXapHOQLn4bcie0roXdI36da\nZCHtT9OP7Bx0ZVuBPJQtRs8PaO64mwl8AtmVdisS6kmHqsswXn0F1bX7NCsSOgzzgPcjD/QYSMke\nKygvdN/pBn6OrFdegvwAXohEc/I2y7SDL1/57DN1aGddNH9v8zgSGdgXIYOl2VuyDTlLeRdwDzLz\nLsIWak5i0sf4thv3O6IVeo3ZduuyaH+NQ+XzWKLlfAb4SOLvJJuRDjgVOVezi2xHuDJbdd1nkYW0\n3zehbDF6PmeR3a6y9PbuujER2r0LDdoPM4b04VTgV1qW2bRijspdWLJsVxqU95XPPlN1O5uEnA7o\nL8jXR3McGgZ+g9z8zGtTb9ubfvYhZ1xsLADmI4PGEzTDKC53ifP1Pb1Y3SrJRenFSKx7dSrPbuAl\nyJ3XoUin2EU5tjP2rKQhpP2+CWWL0WPb0Xcj8sSSM5Ht4nVlIrR734Ty4YHIrvOFSPjtUk1fhgxA\nFyE7cDs9y+wkIfpM1e1sLrJOWjTbnqL1MrPIspGojfp+Ms1B2zAdWfPeS/axGSu9jI3zTkK2aO9D\n4shZbNFrj2tFoQVzDukBfV9iyXMnssN0q9ZrdkmdBvPYrFck0kLb75NQthg9w9hD3AchHfc5Ovfd\ndZqJ1O6LaNDerCmkD83a3B2MXQc1s0zbUY1O06Ccr3z3marbmTkTe14Hy3SlIw8uSNLL2AHzOE3L\nu0M0D9x9mvL/FeVdNKfiO4EXWfIkH+GUNVtuhzO1zHMTaaHt90koW4yevLvIWxjr6zoxkdp9EQ3a\nGwRC+fAy1fNr7IPNocgxru20f/DdlQblfOW7z1Tdzm5FNvwcXpBvA3Ij5fpKb2C1kX403jdpPhrv\nCVIPtWi3MbqEVF6j5T/K/lPddkg+tPd67Au6Jjwwitw9doqfIoedP55IC22/T0LZkhdaMphdeKe1\nqcM3E6nd+yKED88Bvo6cZ34f9k1fOxCfHsbY9bq6EKrPVNnOZiDnNX8G/K0g7xwkfOv6cpkF9yN+\nXo2cFLlA9axE/u3iYCvGRATzz53nF2WMRCKRiDPnI7+tb6m6IpHOMRV5Isi6qisSiUQizxO6kOMg\na4oy1oWq/lXNeGMECSFNQXbtDVdbnUgkEhn3HI1sIFpGZ89eRyKRSCQSiUQikUgkEolEIpFIZPzz\nP2b1RC+L2RphAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$4 H^{4} W^{2} \\left(H^{4} - H^{2} R^{2} - H^{2} W^{2} + H^{2} X^{2} + R^{2} W^{2}\\right) \\geq 0$$"
      ],
      "text/plain": [
       "   4  2 ⎛ 4    2  2    2  2    2  2    2  2⎞    \n",
       "4⋅H ⋅W ⋅⎝H  - H ⋅R  - H ⋅W  + H ⋅X  + R ⋅W ⎠ ≥ 0"
      ]
     },
     "execution_count": 120,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e6 = Poly(Add(e4.lhs,-e4.rhs)*H**2*W**2,x)\n",
    "a = e6.coeffs()[0]\n",
    "b = e6.coeffs()[1]\n",
    "c = e6.coeffs()[2]\n",
    "factor(expand(Ge(b**2 -4*a*c,0)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Solve for points (x,y) which intersect both"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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4zYDz2H1h+dJ8iNCvvf6OeVbb+2t2zHOX+u/md8LTgAcSBlyuJAy0AHyinpqW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\nJEmSJEmSJI3q/wPc24zPz4XkWQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left [ \\left ( - \\frac{W}{H^{2} - W^{2}} \\left(W X - \\sqrt{H^{4} - H^{2} R^{2} - H^{2} W^{2} + H^{2} X^{2} + R^{2} W^{2}}\\right), \\quad \\sqrt{\\frac{H^{2}}{H^{4} - 2 H^{2} W^{2} + W^{4}} \\left(H^{2} R^{2} - H^{2} W^{2} - H^{2} X^{2} - R^{2} W^{2} + W^{4} - W^{2} X^{2} + 2 W X \\sqrt{H^{4} - H^{2} R^{2} - H^{2} W^{2} + H^{2} X^{2} + R^{2} W^{2}}\\right)}\\right ), \\quad \\left ( - \\frac{W}{H^{2} - W^{2}} \\left(W X - \\sqrt{H^{4} - H^{2} R^{2} - H^{2} W^{2} + H^{2} X^{2} + R^{2} W^{2}}\\right), \\quad - \\sqrt{- \\frac{H^{2}}{H^{4} - 2 H^{2} W^{2} + W^{4}} \\left(- H^{2} R^{2} + H^{2} W^{2} + H^{2} X^{2} + R^{2} W^{2} - W^{4} + W^{2} X^{2} - 2 W X \\sqrt{H^{4} - H^{2} R^{2} - H^{2} W^{2} + H^{2} X^{2} + R^{2} W^{2}}\\right)}\\right ), \\quad \\left ( - \\frac{W}{H^{2} - W^{2}} \\left(W X + \\sqrt{H^{4} - H^{2} R^{2} - H^{2} W^{2} + H^{2} X^{2} + R^{2} W^{2}}\\right), \\quad \\sqrt{\\frac{H^{2}}{H^{4} - 2 H^{2} W^{2} + W^{4}} \\left(H^{2} R^{2} - H^{2} W^{2} - H^{2} X^{2} - R^{2} W^{2} + W^{4} - W^{2} X^{2} - 2 W X \\sqrt{H^{4} - H^{2} R^{2} - H^{2} W^{2} + H^{2} X^{2} + R^{2} W^{2}}\\right)}\\right ), \\quad \\left ( - \\frac{W}{H^{2} - W^{2}} \\left(W X + \\sqrt{H^{4} - H^{2} R^{2} - H^{2} W^{2} + H^{2} X^{2} + R^{2} W^{2}}\\right), \\quad - \\sqrt{- \\frac{H^{2}}{H^{4} - 2 H^{2} W^{2} + W^{4}} \\left(- H^{2} R^{2} + H^{2} W^{2} + H^{2} X^{2} + R^{2} W^{2} - W^{4} + W^{2} X^{2} + 2 W X \\sqrt{H^{4} - H^{2} R^{2} - H^{2} W^{2} + H^{2} X^{2} + R^{2} W^{2}}\\right)}\\right )\\right ]$$"
      ],
      "text/plain": [
       "⎡⎛                                                            ________________\n",
       "⎢⎜   ⎛         ____________________________________⎞         ╱    ⎛           \n",
       "⎢⎜   ⎜        ╱  4    2  2    2  2    2  2    2  2 ⎟        ╱   2 ⎜ 2  2    2 \n",
       "⎢⎜-W⋅⎝W⋅X - ╲╱  H  - H ⋅R  - H ⋅W  + H ⋅X  + R ⋅W  ⎠       ╱   H ⋅⎝H ⋅R  - H ⋅\n",
       "⎢⎜───────────────────────────────────────────────────,    ╱    ───────────────\n",
       "⎢⎜                       2    2                          ╱                    \n",
       "⎣⎝                      H  - W                         ╲╱                     \n",
       "\n",
       "______________________________________________________________________________\n",
       "                                           ___________________________________\n",
       " 2    2  2    2  2    4    2  2           ╱  4    2  2    2  2    2  2    2  2\n",
       "W  - H ⋅X  - R ⋅W  + W  - W ⋅X  + 2⋅W⋅X⋅╲╱  H  - H ⋅R  - H ⋅W  + H ⋅X  + R ⋅W \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                         4      2  2    4                                     \n",
       "                        H  - 2⋅H ⋅W  + W                                      \n",
       "\n",
       "___⎞  ⎛                                                             __________\n",
       "_⎞ ⎟  ⎜   ⎛         ____________________________________⎞          ╱     ⎛    \n",
       " ⎟ ⎟  ⎜   ⎜        ╱  4    2  2    2  2    2  2    2  2 ⎟         ╱    2 ⎜   2\n",
       " ⎠ ⎟  ⎜-W⋅⎝W⋅X - ╲╱  H  - H ⋅R  - H ⋅W  + H ⋅X  + R ⋅W  ⎠        ╱   -H ⋅⎝- H \n",
       "── ⎟, ⎜───────────────────────────────────────────────────, -   ╱    ─────────\n",
       "   ⎟  ⎜                       2    2                           ╱              \n",
       "   ⎠  ⎝                      H  - W                          ╲╱               \n",
       "\n",
       "______________________________________________________________________________\n",
       "                                                    __________________________\n",
       "  2    2  2    2  2    2  2    4    2  2           ╱  4    2  2    2  2    2  \n",
       "⋅R  + H ⋅W  + H ⋅X  + R ⋅W  - W  + W ⋅X  - 2⋅W⋅X⋅╲╱  H  - H ⋅R  - H ⋅W  + H ⋅X\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                 4      2  2    4                             \n",
       "                                H  - 2⋅H ⋅W  + W                              \n",
       "\n",
       "_____________⎞  ⎛                                                            _\n",
       "__________⎞  ⎟  ⎜   ⎛         ____________________________________⎞         ╱ \n",
       "2    2  2 ⎟  ⎟  ⎜   ⎜        ╱  4    2  2    2  2    2  2    2  2 ⎟        ╱  \n",
       "  + R ⋅W  ⎠  ⎟  ⎜-W⋅⎝W⋅X + ╲╱  H  - H ⋅R  - H ⋅W  + H ⋅X  + R ⋅W  ⎠       ╱   \n",
       "──────────── ⎟, ⎜───────────────────────────────────────────────────,    ╱    \n",
       "             ⎟  ⎜                       2    2                          ╱     \n",
       "             ⎠  ⎝                      H  - W                         ╲╱      \n",
       "\n",
       "______________________________________________________________________________\n",
       "   ⎛                                                      ____________________\n",
       " 2 ⎜ 2  2    2  2    2  2    2  2    4    2  2           ╱  4    2  2    2  2 \n",
       "H ⋅⎝H ⋅R  - H ⋅W  - H ⋅X  - R ⋅W  + W  - W ⋅X  - 2⋅W⋅X⋅╲╱  H  - H ⋅R  - H ⋅W  \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                        4      2  2    4                      \n",
       "                                       H  - 2⋅H ⋅W  + W                       \n",
       "\n",
       "__________________⎞  ⎛                                                        \n",
       "________________⎞ ⎟  ⎜   ⎛         ____________________________________⎞      \n",
       "   2  2    2  2 ⎟ ⎟  ⎜   ⎜        ╱  4    2  2    2  2    2  2    2  2 ⎟      \n",
       "+ H ⋅X  + R ⋅W  ⎠ ⎟  ⎜-W⋅⎝W⋅X + ╲╱  H  - H ⋅R  - H ⋅W  + H ⋅X  + R ⋅W  ⎠      \n",
       "───────────────── ⎟, ⎜───────────────────────────────────────────────────, -  \n",
       "                  ⎟  ⎜                       2    2                           \n",
       "                  ⎠  ⎝                      H  - W                          ╲╱\n",
       "\n",
       "     _________________________________________________________________________\n",
       "    ╱     ⎛                                                        ___________\n",
       "   ╱    2 ⎜   2  2    2  2    2  2    2  2    4    2  2           ╱  4    2  2\n",
       "  ╱   -H ⋅⎝- H ⋅R  + H ⋅W  + H ⋅X  + R ⋅W  - W  + W ⋅X  + 2⋅W⋅X⋅╲╱  H  - H ⋅R \n",
       " ╱    ────────────────────────────────────────────────────────────────────────\n",
       "╱                                               4      2  2    4              \n",
       "                                               H  - 2⋅H ⋅W  + W               \n",
       "\n",
       "____________________________⎞⎤\n",
       "_________________________⎞  ⎟⎥\n",
       "    2  2    2  2    2  2 ⎟  ⎟⎥\n",
       " - H ⋅W  + H ⋅X  + R ⋅W  ⎠  ⎟⎥\n",
       "─────────────────────────── ⎟⎥\n",
       "                            ⎟⎥\n",
       "                            ⎠⎦"
      ]
     },
     "execution_count": 117,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(solve((e1,e2), (x,y)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see it gets very complicated, very quickly... :("
   ]
  }
 ],
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