{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( x, \\quad y, \\quad x_{0}, \\quad y_{0}, \\quad x_{a}, \\quad y_{a}, \\quad x_{b}, \\quad y_{b}, \\quad k\\right )$$"
      ],
      "text/plain": [
       "(x, y, x₀, y₀, xₐ, yₐ, x_b, y_b, k)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x,y,x0,y0,xa,ya,xb,yb,k = symbols('x y x_0 y_0 x_a y_a x_b y_b k', real=True)\n",
    "x,y,x0,y0,xa,ya,xb,yb,k"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- \\frac{k}{x - x_{0}} + y - y_{0}$$"
      ],
      "text/plain": [
       "    k            \n",
       "- ────── + y - y₀\n",
       "  x - x₀         "
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e1 = y-y0 - k/(x-x0)\n",
    "e1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- \\frac{k}{- x_{0} + x_{a}} - y_{0} + y_{a}$$"
      ],
      "text/plain": [
       "     k              \n",
       "- ──────── - y₀ + yₐ\n",
       "  -x₀ + xₐ          "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s1 = e1.subs(((x,xa),(y,ya)))\n",
    "s1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{k + y_{a} \\left(x_{0} - x_{a}\\right)}{x_{0} - x_{a}}$$"
      ],
      "text/plain": [
       "k + yₐ⋅(x₀ - xₐ)\n",
       "────────────────\n",
       "    x₀ - xₐ     "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s5 = solveset(s1,y0).args[0]\n",
    "s5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- \\frac{k}{- x_{0} + x_{b}} - y_{0} + y_{b}$$"
      ],
      "text/plain": [
       "      k               \n",
       "- ───────── - y₀ + y_b\n",
       "  -x₀ + x_b           "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s2 = e1.subs(((x,xb),(y,yb)))\n",
    "s2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{k + y_{b} \\left(x_{0} - x_{b}\\right)}{x_{0} - x_{b}}$$"
      ],
      "text/plain": [
       "k + y_b⋅(x₀ - x_b)\n",
       "──────────────────\n",
       "     x₀ - x_b     "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s6 = solveset(s2,y0).args[0]\n",
    "s6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{k + y_{a} \\left(x_{0} - x_{a}\\right)}{x_{0} - x_{a}} - \\frac{k + y_{b} \\left(x_{0} - x_{b}\\right)}{x_{0} - x_{b}}$$"
      ],
      "text/plain": [
       "k + yₐ⋅(x₀ - xₐ)   k + y_b⋅(x₀ - x_b)\n",
       "──────────────── - ──────────────────\n",
       "    x₀ - xₐ             x₀ - x_b     "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s7 = s5-s6\n",
    "s7"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The above expression equates to zero so we can multiply it by effectively anything so we set about removing the denominator expressions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$k x_{a} - k x_{b} + x_{0}^{2} y_{a} - x_{0}^{2} y_{b} - x_{0} x_{a} y_{a} + x_{0} x_{a} y_{b} - x_{0} x_{b} y_{a} + x_{0} x_{b} y_{b} + x_{a} x_{b} y_{a} - x_{a} x_{b} y_{b}$$"
      ],
      "text/plain": [
       "                 2        2                                                   \n",
       "k⋅xₐ - k⋅x_b + x₀ ⋅yₐ - x₀ ⋅y_b - x₀⋅xₐ⋅yₐ + x₀⋅xₐ⋅y_b - x₀⋅x_b⋅yₐ + x₀⋅x_b⋅y_\n",
       "\n",
       "                          \n",
       "b + xₐ⋅x_b⋅yₐ - xₐ⋅x_b⋅y_b"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s8 = cancel(s7*(x0-xb)*(x0-xa))\n",
    "s8"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we solve for $x_0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left\\{\\frac{x_{a}}{2} + \\frac{x_{b}}{2} - \\frac{1}{2 \\left(y_{a} - y_{b}\\right)} \\sqrt{- \\left(x_{a} - x_{b}\\right) \\left(y_{a} - y_{b}\\right) \\left(4 k - x_{a} y_{a} + x_{a} y_{b} + x_{b} y_{a} - x_{b} y_{b}\\right)}, \\frac{x_{a}}{2} + \\frac{x_{b}}{2} + \\frac{1}{2 \\left(y_{a} - y_{b}\\right)} \\sqrt{- \\left(x_{a} - x_{b}\\right) \\left(y_{a} - y_{b}\\right) \\left(4 k - x_{a} y_{a} + x_{a} y_{b} + x_{b} y_{a} - x_{b} y_{b}\\right)}\\right\\}$$"
      ],
      "text/plain": [
       "⎧             ________________________________________________________________\n",
       "⎪xₐ   x_b   ╲╱ -(xₐ - x_b)⋅(yₐ - y_b)⋅(4⋅k - xₐ⋅yₐ + xₐ⋅y_b + x_b⋅yₐ - x_b⋅y_b\n",
       "⎨── + ─── - ──────────────────────────────────────────────────────────────────\n",
       "⎪2     2                                2⋅(yₐ - y_b)                          \n",
       "⎩                                                                             \n",
       "\n",
       "__               _____________________________________________________________\n",
       ")   xₐ   x_b   ╲╱ -(xₐ - x_b)⋅(yₐ - y_b)⋅(4⋅k - xₐ⋅yₐ + xₐ⋅y_b + x_b⋅yₐ - x_b⋅\n",
       "──, ── + ─── + ───────────────────────────────────────────────────────────────\n",
       "    2     2                                2⋅(yₐ - y_b)                       \n",
       "                                                                              \n",
       "\n",
       "_____⎫\n",
       "y_b) ⎪\n",
       "─────⎬\n",
       "     ⎪\n",
       "     ⎭"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s9 = solveset(s8,x0)\n",
    "s9"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\frac{x_{a}}{2} + \\frac{x_{b}}{2} - \\frac{1}{2 \\left(y_{a} - y_{b}\\right)} \\sqrt{- \\left(x_{a} - x_{b}\\right) \\left(y_{a} - y_{b}\\right) \\left(4 k - x_{a} y_{a} + x_{a} y_{b} + x_{b} y_{a} - x_{b} y_{b}\\right)}, \\quad \\frac{x_{a}}{2} + \\frac{x_{b}}{2} + \\frac{1}{2 \\left(y_{a} - y_{b}\\right)} \\sqrt{- \\left(x_{a} - x_{b}\\right) \\left(y_{a} - y_{b}\\right) \\left(4 k - x_{a} y_{a} + x_{a} y_{b} + x_{b} y_{a} - x_{b} y_{b}\\right)}\\right )$$"
      ],
      "text/plain": [
       "⎛             ________________________________________________________________\n",
       "⎜xₐ   x_b   ╲╱ -(xₐ - x_b)⋅(yₐ - y_b)⋅(4⋅k - xₐ⋅yₐ + xₐ⋅y_b + x_b⋅yₐ - x_b⋅y_b\n",
       "⎜── + ─── - ──────────────────────────────────────────────────────────────────\n",
       "⎝2     2                                2⋅(yₐ - y_b)                          \n",
       "\n",
       "__               _____________________________________________________________\n",
       ")   xₐ   x_b   ╲╱ -(xₐ - x_b)⋅(yₐ - y_b)⋅(4⋅k - xₐ⋅yₐ + xₐ⋅y_b + x_b⋅yₐ - x_b⋅\n",
       "──, ── + ─── + ───────────────────────────────────────────────────────────────\n",
       "    2     2                                2⋅(yₐ - y_b)                       \n",
       "\n",
       "_____⎞\n",
       "y_b) ⎟\n",
       "─────⎟\n",
       "     ⎠"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s11 = s9.args[0]\n",
    "s12 = s9.args[1]\n",
    "s11,s12"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "a=(1,5)\n",
    "b=(3,2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from plotlibs import plot_hyperbolae1\n",
    "\n",
    "plot_hyperbolae1(a, b)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from plotlibs import plot_hyperbolae2\n",
    "    \n",
    "plot_hyperbolae2(a,b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( x_{s}, \\quad y_{s}, \\quad x_{d}, \\quad y_{d}, \\quad x^\\prime, \\quad y^\\prime, \\quad x_a^\\prime, \\quad y_a^\\prime, \\quad x_b^\\prime, \\quad y_b^\\prime, \\quad x_d^\\prime, \\quad y_d^\\prime, \\quad x_s^\\prime, \\quad y_s^\\prime\\right )$$"
      ],
      "text/plain": [
       "(xₛ, yₛ, x_d, y_d, x__\\′, y__\\′, x_a__\\′, y_a__\\′, x_b__\\′, y_b__\\′, x_d__\\′, \n",
       "y_d__\\′, x_s__\\′, y_s__\\′)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    " xs,ys, xd,yd = symbols('x_s y_s x_d y_d')\n",
    "xp,yp,xap,yap,xbp,ybp, xdp, ydp, xsp, ysp = symbols('x^\\prime y^\\prime x_a^\\prime y_a^\\prime x_b^\\prime y_b^\\prime x_d^\\prime y_d^\\prime x_s^\\prime y_s^\\prime')\n",
    "xs,ys, xd,yd, xp,yp,xap,yap,xbp,ybp,xdp,ydp, xsp, ysp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{x_{a}}{2} + \\frac{x_{b}}{2} - \\frac{1}{2 y_{d}} \\sqrt{- x_{d} y_{d} \\left(4 k - x_{a} y_{a} + x_{a} y_{b} + x_{b} y_{a} - x_{b} y_{b}\\right)}$$"
      ],
      "text/plain": [
       "             ____________________________________________________\n",
       "xₐ   x_b   ╲╱ -x_d⋅y_d⋅(4⋅k - xₐ⋅yₐ + xₐ⋅y_b + x_b⋅yₐ - x_b⋅y_b) \n",
       "── + ─── - ──────────────────────────────────────────────────────\n",
       "2     2                            2⋅y_d                         "
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s15 = s11.subs(((ya-yb,yd),(xa-xb,xd)))\n",
    "s15"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x_a/2 + x_b/2 - sqrt(-x_d*y_d*(4*k - x_a*y_a + x_a*y_b + x_b*y_a - x_b*y_b))/(2*y_d)\n"
     ]
    }
   ],
   "source": [
    "print (s15)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- \\left(x_{a} - x_{b}\\right) \\left(y_{a} - y_{b}\\right) \\left(4 k - x_{a} y_{a} + x_{a} y_{b} + x_{b} y_{a} - x_{b} y_{b}\\right)$$"
      ],
      "text/plain": [
       "-(xₐ - x_b)⋅(yₐ - y_b)⋅(4⋅k - xₐ⋅yₐ + xₐ⋅y_b + x_b⋅yₐ - x_b⋅y_b)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s13 = s11.args[2].args[1].args[0]\n",
    "s13"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left(x_{a} - x_{b}\\right) \\left(y_{a} - y_{b}\\right) \\left(- 4 k + x_{a} \\left(y_{a} - y_{b}\\right) + x_{b} \\left(- y_{a} + y_{b}\\right)\\right)$$"
      ],
      "text/plain": [
       "(xₐ - x_b)⋅(yₐ - y_b)⋅(-4⋅k + xₐ⋅(yₐ - y_b) + x_b⋅(-yₐ + y_b))"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(s13,(xa,xb,ya-yb))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABUAAAAlBAMAAABfQ78IAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NuzJUme9mZ+xlAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAwklEQVQYGWNgEFJ2VVNgAAPGBPYmjgkQNpsA20dWB6g4A9cCCAtE8isg2Ps3ANlsSgcYGHgE9BmYBBhSuRcwMMQfWM/gzsCw6qAAA4OwktD1AgYG5/8OQJVgwOPArABlMnBs4AWZAAY85aowJpn0fzj4QKYJCG2yt1bDOIwGDPkBUA6vAAPfAyibr4GB+QuUzfmRgfk3lA2kOD4i2PEJCHYzgsl7AcHWRTB5LjC4wHjbGBhUoGyWtrTLDVA2J9CbIDYAN9ouZ4RSpL0AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{x_{s}}{2}$$"
      ],
      "text/plain": [
       "xₛ\n",
       "──\n",
       "2 "
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e2 = xs/2\n",
    "e2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{1}{2 y_{d}} \\sqrt{x_{d} y_{d} \\left(- 4 k + x_{d} y_{d}\\right)}$$"
      ],
      "text/plain": [
       "  __________________________\n",
       "╲╱ x_d⋅y_d⋅(-4⋅k + x_d⋅y_d) \n",
       "────────────────────────────\n",
       "           2⋅y_d            "
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e3 = sqrt(xd*yd*(xd*yd - 4*k))/(2*yd)\n",
    "e3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- \\frac{k}{x_{a} - \\frac{x_{s}}{2} - \\frac{1}{2 y_{d}} \\sqrt{x_{d} y_{d} \\left(- 4 k + x_{d} y_{d}\\right)}} - y_{0} + y_{a}$$"
      ],
      "text/plain": [
       "                    k                             \n",
       "- ────────────────────────────────────── - y₀ + yₐ\n",
       "              __________________________          \n",
       "       xₛ   ╲╱ x_d⋅y_d⋅(-4⋅k + x_d⋅y_d)           \n",
       "  xₐ - ── - ────────────────────────────          \n",
       "       2               2⋅y_d                      "
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s1.subs(((x0,e2+e3),))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- \\frac{k}{\\frac{x_{d}}{2} - \\frac{1}{2 y_{d}} \\sqrt{x_{d} y_{d} \\left(- 4 k + x_{d} y_{d}\\right)}} - y_{0} + \\frac{y_{d}}{2} + \\frac{y_{s}}{2}$$"
      ],
      "text/plain": [
       "                  k                         y_d   yₛ\n",
       "- ────────────────────────────────── - y₀ + ─── + ──\n",
       "          __________________________         2    2 \n",
       "  x_d   ╲╱ x_d⋅y_d⋅(-4⋅k + x_d⋅y_d)                 \n",
       "  ─── - ────────────────────────────                \n",
       "   2               2⋅y_d                            "
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s21 = s1.subs(((x0,e2+e3),(xa,xs/2+xd/2),(ya,ys/2+yd/2)))\n",
    "s21"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAa0AAAA4BAMAAAC8pLpAAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM0yu+/dmVREiWYidqtuSOxyAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAG7UlEQVRoBdVZa4gbVRQ+s8kmO9nMZlpqxepuQgQrWGwUpT8UtkgLhWpdxSdo3VoUi33EVrs/6tpYxIp95Y8Kat1UxJbWxQhisVqcH1ot/dGsKKmPbUMRwUdslkXZtivruXfmztyZO5NNdtKaXAj33O8758w5mTuTc3MAGjWkpf2NctVcftrSzRVPo6KJZRrlqbn8DKnNFU+jonm/UY6azM982PH8ZQlJKhbgK9fN0T0I0XsaHIM0oRy/2GCf7u4ich+cQWprysnnXgOl4gR9rjvGRqS8Tx+1mX8TTsASVI071aOZjRBc7ER9rsMP+nRQs7naXoB/UXub00KCRyFWcKI+16HyXT491Gw+pAbGUXm/YCCPQa+wOQWt+oB49mS2PosZa28EpQ+uHL1ecIAb9HuA4VMC4QP4AuKZnA/7OkxXQLAkrQ6MCSbBEqyASPYWgfAB/AKxVM6HfR2mL8BQPpwO9Akm4ZJ8EYLLZgmED2AZKIOqD/s6THtGt2rtGaUkmEjl4iREz00KRKsAGyGuBT9y+RrxCdsJ5VZJwx5nV0I6D+1a7212GFdnIZ6F1bBJIJoa2GVEF8oHCyCXd21h0TICngb8rdlx9HK9l9n1fc4sfHnUcUMYAXuLKZ/X+D/MzfCdF/cknIpNsnaU557hexJNkoczDFqeK8M5A/cM35NwOmyOtV6eH4HdM8lrdpKMa5sjExKFRANK4gtAL8/XQIz8Tr2bTD6RTN4oxulJGKpTzTDsYZPy/KZxaM/oMNtuVqCGOiNiJlOxO2qyFSnPA5MQytvzEqJkeQlEkwKkPMeqPVS4hHk1tPav8Xsk5TneLzwe0+F5WzyJ6a+zaLOgY+1mc1v7FRyPBSnPpXGIZS9dXvC6kNeXAtJ4AJ8w432Ivj1viydRQ0BiXr/VYOVPhZbnRfP3izsuSgNLUpZvT0JXGd50mlO2zKgk5BVNODTMZTRvivUIOwVlWp4rB9MCAT0QOS+iiLgQino2lAUQHyTqQMgrjLruYw4PLwKY8wAPeMpdmpPyLs+fAbjfqU3XLkQAbiDUG6764vM1210P0QU8sx4X83nAU5YSnpRA4Dd1ThNQBFwJ+r/AIabeUWASmYX7dYBnebmT3zfKSqQmTLrqEz5sqk0rLFThXMpNy40IVALQObqWqVfPS0owPecc0jjkBFZ0WA6xUTWvK5hWTfMG1UPNSZyMJCLwKvQxdXteLzPYmCN5B2Auj5kSCjnMK9JvIlXzasuaetML8gUPHYHo+XR4HywIlJi+La+vN5QZrs8hrLdH/9xr7B3+BFjkFDs1zKst3XFY00GWV/fp4z8WKMRZRtKc5XRiqOSh4U6sUn5g+ra8GGjOPwHMg08Kj+sA36BZbuoAdAPmFc/8PGTcXSMvKde5LpSgepxlp45w5lXEAS/OnXjzqPkvT/W8+gH2wYfq2+g+PM4aNGHSDniKu2KO5LWnW73awIy8ImpkTNYQ23q70dohlh39hlYNk+J1bz0J3Wkkmbzu3mSy5HYJFUGlAKDSLhTuxwpr0EgVpB62zoYdKZLXc0YM1glQYm2qOG8ZINY1ju0Q1VxVRcIsWg337H7Z6lvkjuXQIT5eOEgXCr/nBHnlkQYNSjQvgpIxD0he678r6EuuxIvp0Dbeso68AjlQNOaUnz0JS4nlZSFUkj/uw/kIkWkXCt93aQjTBg2RbPtw98DA34PS+ahZ87D3BmtT7ect5X7itKZx7OD+X10V3YnIs1iHsGHmZUPx5+giaqzCz4vYyx0BafDbDPZtsEFDJQD+vQGwEjor0kTEcGrk1aGeAUWlLSzOso73xsKpqX9YnLbZnfgA/+E2h5mXDUX6SdxwOcxvIliJ5mFuqleFMGnQUMnZRrwPuhbDuCOvoczdcAJoC4uzDKPTSzPWQq9mejbzsqFIn8lCm4ovh/JI8TOAAbiVyNigoZKj4npv6o9gBoojhlfjfvWMdm/JA21hcZZtKfPaDRbWANdLNfOyoXjFeAm2Wxd+CE6TBT5hhhTSyNp9mM8XoY0Wlmk5192mIegevBHGkCxxj7q3RB8pSoUnuZJdmoTls2iDRpcAbHUvc2bMs/g1bWGRs6NhOcyTDZbXufpbB7kx0vbcTNnoBblkqT0SGM/rDRoqIXGnRVaVaAuLnh2ppZSoqu2LpC9qwUMkHb2qX1lsns2WBjVL551TxbzeoKESErZzpaUoSLSFRc+O1JJ3Kuj6BA672iMaznUVAA7p9NBfrmoMxJfkTMbnMzGqzUbOySlRk6ChbHvKPJuFSBnYSuPmV15SxXgJGkoNWWczsiVbajw2NeUSL0Hl8h382cxFqzWha7A84s5mrZmES9S/ywngzmYuGq0JvXWwteL+D4vgvk9DR7FNAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- \\frac{k}{\\frac{x_{d}}{2} + \\frac{1}{2 y_{d}} \\sqrt{x_{d} y_{d} \\left(- 4 k + x_{d} y_{d}\\right)}} - y_{0} + \\frac{y_{d}}{2} + \\frac{y_{s}}{2}$$"
      ],
      "text/plain": [
       "                  k                         y_d   yₛ\n",
       "- ────────────────────────────────── - y₀ + ─── + ──\n",
       "          __________________________         2    2 \n",
       "  x_d   ╲╱ x_d⋅y_d⋅(-4⋅k + x_d⋅y_d)                 \n",
       "  ─── + ────────────────────────────                \n",
       "   2               2⋅y_d                            "
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s22=s1.subs(((x0,e2-e3),(xa,xs/2+xd/2),(ya,ys/2+yd/2)))\n",
    "s22"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAPBAMAAAAv0UM9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEGZURLuZMu/ddqvNiSJwwRR/AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAYElEQVQIHWNgYGBUZgACV+EEBgb2RoaPDAycCgyfGRg0GBg+MTBMY2D/zMDyhYHrCwP37907PjBwHmDgB6ILDPUGDPIGDBkMDPIODL8YGDgcuCYwMLAtePkAaGJWIgMDAFkWFM4t8HnwAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\theta$$"
      ],
      "text/plain": [
       "θ"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "theta = symbols('theta')\n",
    "theta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\cos{\\left (\\theta \\right )} & - \\sin{\\left (\\theta \\right )}\\\\\\sin{\\left (\\theta \\right )} & \\cos{\\left (\\theta \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡cos(θ)  -sin(θ)⎤\n",
       "⎢               ⎥\n",
       "⎣sin(θ)  cos(θ) ⎦"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = Matrix([[cos(theta), -sin(theta)],[sin(theta), cos(theta)]])\n",
    "m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & - \\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡√2  -√2 ⎤\n",
       "⎢──  ────⎥\n",
       "⎢2    2  ⎥\n",
       "⎢        ⎥\n",
       "⎢√2   √2 ⎥\n",
       "⎢──   ── ⎥\n",
       "⎣2    2  ⎦"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mplus = m.subs(theta, pi/4)\n",
    "mplus"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2}}{2}\\\\- \\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ √2   √2⎤\n",
       "⎢ ──   ──⎥\n",
       "⎢ 2    2 ⎥\n",
       "⎢        ⎥\n",
       "⎢-√2   √2⎥\n",
       "⎢────  ──⎥\n",
       "⎣ 2    2 ⎦"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mminus = m.subs(theta, -pi/4)\n",
    "mminus"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\cos{\\left (\\theta \\right )}\\\\\\sin{\\left (\\theta \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡cos(θ)⎤\n",
       "⎢      ⎥\n",
       "⎣sin(θ)⎦"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m*Matrix([1,0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}- \\sin{\\left (\\theta \\right )}\\\\\\cos{\\left (\\theta \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-sin(θ)⎤\n",
       "⎢       ⎥\n",
       "⎣cos(θ) ⎦"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m*Matrix([0,1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{\\sqrt{2} x_d^\\prime}{2} - \\frac{\\sqrt{2} y_d^\\prime}{2}\\\\\\frac{\\sqrt{2} x_d^\\prime}{2} + \\frac{\\sqrt{2} y_d^\\prime}{2}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡√2⋅x_d__\\′   √2⋅y_d__\\′⎤\n",
       "⎢────────── - ──────────⎥\n",
       "⎢    2            2     ⎥\n",
       "⎢                       ⎥\n",
       "⎢√2⋅x_d__\\′   √2⋅y_d__\\′⎥\n",
       "⎢────────── + ──────────⎥\n",
       "⎣    2            2     ⎦"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mp = mplus*Matrix([xdp,ydp])\n",
    "mp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{\\sqrt{2} x_d^\\prime}{2} + \\frac{\\sqrt{2} y_d^\\prime}{2}\\\\- \\frac{\\sqrt{2} x_d^\\prime}{2} + \\frac{\\sqrt{2} y_d^\\prime}{2}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ √2⋅x_d__\\′   √2⋅y_d__\\′ ⎤\n",
       "⎢ ────────── + ────────── ⎥\n",
       "⎢     2            2      ⎥\n",
       "⎢                         ⎥\n",
       "⎢  √2⋅x_d__\\′   √2⋅y_d__\\′⎥\n",
       "⎢- ────────── + ──────────⎥\n",
       "⎣      2            2     ⎦"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mm = mminus*Matrix([xdp,ydp])\n",
    "mm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{\\sqrt{2} x_s^\\prime}{2} + \\frac{\\sqrt{2} y_s^\\prime}{2}\\\\- \\frac{\\sqrt{2} x_s^\\prime}{2} + \\frac{\\sqrt{2} y_s^\\prime}{2}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ √2⋅x_s__\\′   √2⋅y_s__\\′ ⎤\n",
       "⎢ ────────── + ────────── ⎥\n",
       "⎢     2            2      ⎥\n",
       "⎢                         ⎥\n",
       "⎢  √2⋅x_s__\\′   √2⋅y_s__\\′⎥\n",
       "⎢- ────────── + ──────────⎥\n",
       "⎣      2            2     ⎦"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mms = mminus*Matrix([xsp, ysp])\n",
    "mms"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we want to look at the value of x0 after doing our 45 degree rotation to the right"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "import plotlibs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "a=(0,2)\n",
    "b=(2,6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute the space time interval of these two points.  If the result is > 0 they are time-like, if < 0 they are space-like.  \n",
    "\n",
    "* Time-like interval - where $\\Delta S^2 > 0$ , describes events within A's\n",
    "lightcone. These events are causally related to A, and there will be some\n",
    "inertial frame where A and C occur at the same place but at different\n",
    "times.  \n",
    "* Space-like interval - where $\\Delta S^2 < 0$ , describes events outside A's\n",
    "lightcone. These events are not causally related to A, and there will be\n",
    "some inertial frame where A and C occur at the same time but at different\n",
    "places.\n",
    "* Light-like interval - where $\\Delta S^2 = 0$, describes events on A's lightcone.\n",
    "These events are causally related to A, but they can only be linked to A by\n",
    "a light signal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACMAAAAPBAMAAACcmWsnAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM0yVO+riWZ2mbtE3SJy2j1IAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAhElEQVQYGWNgQAaMq3YbIPFZAhkYxBjY/yCExMo+MjAsZGAIRQgxcACFYhkY8h8gxEBC+gIM+UiGgYSA4L4AqioGBp7vCBGwWQwMvAkMDEJKIKDMABVahaQIIsS5AEPIi4HvAUIQZCNzAQMnkhAnUOhteVUuQhFX6qcVDPr///9FCCFYAFrNIGCqjzdWAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$-12$$"
      ],
      "text/plain": [
       "-12"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(a[0]-b[0])**2 - (a[1]-b[1])**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'reload' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-37-68f91ced4367>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mreload\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mplotlibs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m      2\u001b[0m \u001b[0mplotlibs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mplot_hyperbolae3\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mNameError\u001b[0m: name 'reload' is not defined"
     ]
    }
   ],
   "source": [
    "reload(plotlibs)\n",
    "plotlibs.plot_hyperbolae3(a,b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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