samples/PaperEqnVisualisations.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import torch\n",
    "import numpy as np\n",
    "from dsketch.raster.disttrans import point_edt2, line_edt2, curve_edt2_bruteforce\n",
    "from dsketch.raster.raster import exp\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib \n",
    "\n",
    "sz = 10\n",
    "r = torch.linspace(-1, 1, sz)\n",
    "c = torch.linspace(-1, 1, sz)\n",
    "grid = torch.meshgrid(r, c)\n",
    "grid = torch.stack(grid, dim=2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "matplotlib.rcParams.update({'font.size': 42})\n",
    "matplotlib.rcParams['text.usetex'] = True\n",
    "matplotlib.rcParams['text.latex.preamble'] = r'\\usepackage{bm}'\n",
    "\n",
    "line = torch.tensor([[0.5, 0.5], [0, 0]])\n",
    "ldt2 = line_edt2(line, grid)\n",
    "\n",
    "fr = plt.imshow(ldt2, cmap=matplotlib.cm.gray, interpolation='nearest')\n",
    "fr.axes.xaxis.set_ticklabels([])\n",
    "fr.axes.yaxis.set_ticklabels([])\n",
    "plt.xticks(np.arange(0.5, 10, step=1))\n",
    "plt.yticks(np.arange(0.5, 10, step=1))\n",
    "fr.axes.set_ylabel(r'$\\bm{n}_{2}$')\n",
    "fr.axes.set_xlabel(r'$\\bm{n}_{1}$')\n",
    "\n",
    "plt.axline((0, 9), (sz-1, 0), color='white', linewidth=2)\n",
    "plt.axline((5, 9), (sz-1, 5), color='white', linewidth=2)\n",
    "\n",
    "plt.scatter([4.5,7],[4.5,7], color='cyan') \n",
    "plt.plot([4.5,7],[4.5,7], color='cyan', linewidth=2) \n",
    "plt.annotate(r\"$\\bm s$\", (5, 4.8), color='cyan')\n",
    "plt.annotate(r\"$\\bm e$\", (6.5, 6.5), color='cyan')\n",
    "\n",
    "plt.annotate(r\"$\\scriptstyle t<0$\", (2,2.5), color='white')\n",
    "plt.annotate(r\"$\\scriptstyle t>0$\", (7,9), color='white')\n",
    "\n",
    "plt.savefig('eqn4.pdf', bbox_inches='tight')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# fr = plt.imshow(ldt2, cmap=cmap_map(lambda x: x/2 + 0.5, matplotlib.cm.gray), interpolation='nearest')\n",
    "fr = plt.imshow(ldt2, cmap=matplotlib.cm.gray, interpolation='nearest')\n",
    "fr.axes.xaxis.set_ticklabels([])\n",
    "fr.axes.yaxis.set_ticklabels([])\n",
    "plt.xticks(np.arange(0.5, 10, step=1))\n",
    "plt.yticks(np.arange(0.5, 10, step=1))\n",
    "fr.axes.set_ylabel(r'$\\bm{n}_{2}$')\n",
    "fr.axes.set_xlabel(r'$\\bm{n}_{1}$')\n",
    "\n",
    "plt.savefig('eqn4b.pdf', bbox_inches='tight')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fr = plt.imshow(ldt2 < 0.01, cmap=matplotlib.cm.gray, interpolation='nearest')\n",
    "fr.axes.xaxis.set_ticklabels([])\n",
    "fr.axes.yaxis.set_ticklabels([])\n",
    "plt.xticks(np.arange(0.5, 10, step=1))\n",
    "plt.yticks(np.arange(0.5, 10, step=1))\n",
    "fr.axes.set_ylabel(r'$\\bm{n}_{2}$')\n",
    "fr.axes.set_xlabel(r'$\\bm{n}_{1}$')\n",
    "\n",
    "plt.savefig('eqn5.pdf', bbox_inches='tight')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fr = plt.imshow(exp(ldt2, 0.04), cmap=matplotlib.cm.gray, interpolation='nearest')\n",
    "fr.axes.xaxis.set_ticklabels([])\n",
    "fr.axes.yaxis.set_ticklabels([])\n",
    "plt.xticks(np.arange(0.5, 10, step=1))\n",
    "plt.yticks(np.arange(0.5, 10, step=1))\n",
    "fr.axes.set_ylabel(r'$\\bm{n}_{2}$')\n",
    "fr.axes.set_xlabel(r'$\\bm{n}_{1}$')\n",
    "\n",
    "plt.savefig('eqn6.pdf', bbox_inches='tight')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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QESJUEhKMPZfcW9bGCAkZIUKlXz+Ii4O9e8GiJS3CkYSMEKEycKDxunu3vfUIMQkZIULFHTJ79thbjxCTkBEiVMaMMV4//9zWaoSahIwQoZCSAuPGQVMTLF9ud21CSkJGiFC49FJjHZlPP4UjR+yuTUhJyAgRClddZbwuW2ZvPWwQ1bsVJCQkUGLBeIQjFv1PVFlZaXqZdXV1ppcJ0GDRiFWrVtTXWltSrl8yM+G664zv/QiZuLg4S6oRH2/PP/eoa8l47lZg11+qEC388pdG0Lz/PmzfbndtQi7kIaOUyvfzvNhZBFVEr8xMI2QAHnrI1qrYJSQho5TKV0r9n1KqEfhKKdWolNqtlJrcwWX9lVKPKqXeVEq9oZR6JBR1FcJUd91lrIL3/vvw8cd218YWloeMUupa4CvgUkABFa7XAcBCV9j09b5Oa71Fa32f1voGYAQw0+q6CmGqkSPhvvuM73/7W1urYidLQ0YplQksAUqAS7XWDq11N621A+gP/AHIAYqVUhd3UFSxlfUUwnSZmfDGG8YiVU89BevW2V0j21jdkpkLbNZaD9Baf+D5hta6RGs9U2vdFSNs3ldKXWNxfYQIjeefh4IC2LgRZsZ2I9zqkBkBXO/rJK31TGAk8IIEjYhoDgcsWADXXw8VFXDjjeB02l0rW1n9jDdLa73XnxO11puVUv0wWjTdtNYvWFs1IUwWFwcvvQQ/+hGcPAk33ADFcqdvdchUBHKy1rocKFRKvaeUUlrr562plhAmS0+Hv/wFrrkGqqrgyivhn/+0u1ZhwerbpY3+jovxpLX+PjBAKXW3+VUSwmSjRsEXXxgBU1ZmzFOSgGlmachoracBj3kOrFNKjfUnPLTW9wEVEjQibHXpArNnw9q1Rifv5s1wwQXwr3/ZXbOwEorBeLcDzyul7naFzRJgrmv8TIe01ouALUChxXUUwn8Oh9HvsmsX/PrXoBTMmQPnnw87d9pdu7Bj+eQerXUFcINS6hLgfmAjUA687+f1HyilCoE3LaukEP5Qyuhr+f3vYehQ49iOHTBtGnz0kb11C2Mhm0HoGifzgc8T2762GGnNCLt06wY/+QlMnQpnnGEc27cPHnwQ/vpXYyEq0S6ZpixEWxITYexYmDgRJkwwFpwCKC2FJ56A554Di5bRiDa2hYxS6mygWGtt/iIqQgQjKwuuuAKuvhouu8x4LA2nlsx87jlYsUJaLgGysyWzGdBKqQXAfRI2IuTi4oy+lQsvNFauGzPm1DayYDwtWroUXn7Z2CtJBMXOkFmE8eRpGjBVKTVFRvkKS/XoYTwB+o//MF4LCyE19dT7DQ3GkgxLlxor2JWW2lfXKGJbyGitpyql7gWmArMwln3I0lo/bledRPTQSUkwfLgRJuedZ7zm57c+cc8e+OQTWLnSuBUqLw91VaOe6SHj6muZivE06FuM26JVWuvV3ue6Hm8/ppRaCGxyff++1voLs+slopMGY3/pYcOMr6FDjdeBA43bIU+VlcZuARs2GMHyr3/B8eN2VDummBoySql7gEfdP2L8DowD7lVKgTE25jmt9Tue12mty5VSIzBCaRZwk5n1EtFBp6XBkCGtAyUzs/XJDQ1QVGQEySefGMHy5ZfSaWsD00JGKVWAsX6Mm8YIGk/jgHGuwFkMLHS3cFxBsxCY0Ml6NO9WEBcXx+HDhztTXJvKyspMLxOgtrbW9DKtWv3fKkoptFLQv3/rMOnfv+2Ljhwx5g5t2wZbtxpfX36J8njE7HA4jMF0Ju8EYMVi9WlpaaaXCZCRYc+y2Wb+DU11vbrDxd2S8eQZOhOACUqpcuANjNuqcUBWZyqhtV4ILARISEiwcR8M4beUFGOpylGj0KNGGR2zXbu2Pq+uzhhhu3Vri0BRMbZZWqQxM2TGuV4rMKYOdAP60TI0PP/RuwOnK6cCSmGsByyiWV6eMXP5gguM17PPNiYbejp0yNgz2rN1smsXyqL9noR1zAyZfhgBUejq0AWa1/kdh9HPcp3H+W0FjgYWmFgnYTeHw7jdGTXqVLB4P+VpbDTGpKxbZ/SdrFuH2r/fluoK85kZMlnAAs+AgeYnSG+5vlBKXY/RZzLOuwCMEcB/MLFOItTS043Hxe5Wyvnnnxo561ZRYXTGrltnfH36KVRXA0afjIguZoZMOUZrpkNa6yXAElcL50ZcnbQYT57mmFgfESoDBsC11xpfhYWtO1eLi08Fyvr1xi6KEdYhLYJnZshsBM7192RXC6e5k1ZEmKFDjVC57rpTyx6AsWj2Z58ZYeIOlW++sa+ewnZmhswSYL5SargMpotCShlPgNwtFveSB2CMkn33XXj7bXjvPaipsa2aIvyYFjJa64VKqbkYc5K+a1a5wma9esFPf2p89fXY6PPoUfj73+Gtt2D1aqivt62KIryZPZLoduBNpdSftNY/M7lsESrx8cYKcLffbix54HCt0nrggNFaefttY1/nxkZ76ykigqkho7VeopS6D3jUtXeSTA+IJKedBj//udFqOf1041hdHbzzjrEj4urVoGV8owiMFbOw3YPpJiilzgVmAh/IejFhrH9/uOcemDTJWBEOjJG1ixYZy0vKJELRCWZPkHwO45bJPbVgAEaHMEqpzRiPqduckS1scM45xj7N119vPHZuaoIlS+DJJ42nQkKYwMwJkpdwasyL5tSIXvfoqhEYj7g9Z2SvAt7XWn9uVj2EH4YPh0cfNfpbwHjs/OKLMG8e7N5tb91E1LFygqT2+BlOTZp0G+f6co/yXAXM1Vp/aGKdhKe8PGM7j1tvNTpzq6uNdWuffNKYKySEBczc3M09TWALxoxq5fWlvb683/8+8L5S6i4T6yTAWGLyoYeMVeAmTTLWWnniCeOR9D33SMAIS5kZMlnAYq11oevLAfTHaOEswZid3VHo4Do+zcQ6xTQNxn5B//43/OY3xpIKixfDd74Dd90F335rdxVFDDDzdmkLXnOXtNYlGIPzFkHzwlbjgEtpvXaMO2i6mVinmKX79kUvXAjjXA3MTz4xgkU6dEWImdmSmQOMUEp9qpQa3tYJWusSrfUirfUNWutuGC2dmRidwO6WzkwT6xRzNKCnTkV/8YURMEePws03GwtBScAIG5g5rWCJUmoa8BywWSlVjLHE5pvtPT1ytXTmub5EJ+m+fdGLFsEllxgHFi+GGTOMoBHCJma2ZNxLX/YHnne93oexC4GwmP7pT9FbtxoBc/Qo6sYbcdx0kwSMsJ3pI35drRP3nko34scaMyJ4OiEB/fTTcNttxoE330TdcQfq2DF7KyaEi2Wbu3msFxNSnrsVKKWorDR/NoMVuwpAEDsL9OxpzII+7zw4eRKmTYOXX27xuK6L99q5JklISLCkXCtW/wfr/h4S3dMwTJSVlWV6mQCnu+ejmayoqKjD9+3cptYSnrsVxMXFRe9svgsvhDffNLZe3bvXWONlyxa7ayVEK6b2yYgQ+cUv4IMPjIBZtcpY8lICRoQpCZlIopQxBeCpp4w1X+bOhcsvl1nSIqxF3e1S1HI4YOFCmDzZmNB4663wxht210oInyRkIkGXLsa6LjfeaKyfe801xlq6QkQACZlwl5RkDKq78kpjv6IrrzSWvhQiQkjIhLO0NFi2DC6+GI4dg//8T2OnRSEiiIRMuEpIOBUwhw4Z85C+/NLuWgkRMAmZcKQUvPTSqYD53veMXRiFiEDyCDsczZkDEydCVRWMHy8BIyKahEy4mTHDWNy7vt7YAvYL2YxTRDYJmXDywx/CH/9ofH/bbcZoXiEinIRMmNDnnw+vvmoMuvv1r+Hll+2ukhCmkJAJA7p7d/Sbb0JysjGq9/e/t7tKQphGQsZmWin0iy8ayzZ89BH8TLYQF9FFQsZuv/iF8QTp22+NtXhlE3sRZSRkbKQLC9Fz5gCgJk+GAwdsrpEQ5pOQsYnOyEC/+qoxsvfpp1HLltldJSEsISFjAw3o556D/v1h82bUvffaXSUhLCMhY4f/+i9j2YbqatTNN6OcTrtrJIRlom7ukvdC4uFGd+2KnmdsM6V+8QvUnj0210gIa0VdyHguJO5wOLTTglZCwLsKeHr4YcjJgTVr0H/+M54rnVuxon5mZqbpZVpZbnJysiXlWrGrAFhT3+zsbNPLBOjVq5cl5a7yMTJdbpdC6bvfhSlTjHlJMh5GxAgJmVBxOGD+fOP18cdlbRgRMyRkQmX6dDj3XCgthdmz7a6NECEjIRMKPXqcmo90553GYuBCxAgJmVCYNw8yM+Ef/4ClS+2ujRAhJSFjteHDjXExtbVwxx1210aIkJOQscrEiVBScmr72NWrjT2rhYgxEjJWmDgRFi2C/HxjUXCAMWOM40LEGAkZKzzyCKSmtjyWmmocFyLGSMhYoU+fwI4LEcUkZKxQWhrYcSGimISMFe6/H06caHnsxAnjuBAxRkLGCq+9BrffbjxNamoyXm+/3TguRIyJulnYYeO11yRUhEBaMkIIi0nICCEsJSEThZrGjqVRxuSE1Je/+x21p59udzXCkoRMlHEHTJw8yQqpgvnzKXrySQmaNkjHr9UKCiArC/r1M74AysuhuBg++KDDS5uuvRZVXIz6/HO/PkqffTaNzz5L/Pnnd6rKInBJX3/NgHnzKHrySc6+7Tbiq6vtrlLYkJAxW0EBXH+9sRvBiBG+z1+8GObMOTWR0qXx2Wdpuv12HIsWETdjhs9idGYmDa+9RtyMGaiKimBrH9Ma0tI4du21VJ53XvOxlJ07Oe3FF/0KjaxNm8hes4adv/sdQ371KyurGlGiLmS8dyuwagFpb42TJ9Nw112nWiueiotRJSWo8nJUSQkAuqCApuuugwkTYMIE4v7wB5Jnz0ZnZnJi2TKazj4bgISiIhL8WKy6ZtEiuvzznyRv2AAe52dlZfm8tnbwYOqGDiXzjTf8+rMCFBQU+H1uILp27WpJuSkpKR2+v7ewkPWTJjFw7Voufe655uO7xoxh0/LljFi8mEFr17a6Li0trcXPZ3/yCS9///vE/+xnDFm/Pqi65uTkBHWdL31smtYSdSHjuVtBXFyc9nF6pzWNHUv9M8+0DpfiYuKef564d95pDpZWbrnFCKeHH6bx7rs5MWIEjeecY9xeucSvWeOzDg0XXUTDD39IehC/RI3p6RxwLaQVV1FB2sqVAZcR6XaNGcOGSZO46Nlnyd+4scV7g9aupfvevbz7299SlZtL4eLFPsu7YNkyVk6ezBlbtpB48qRV1Y4Y0vHbCc7ly6lfsaJlwJSVET9jBomDBxP/xBPtB4xL3AsvkHDBBVBWRuPFF7cIGMrLcfixBk3tQw+R8NRTQd0mVY8f3/y9o7Iy4OsjXVVODhsmTaLvZ5+1Chi37H37GLJiBUXjx3O8b1+fZQ74/HNySktZf9VVZlc3IknIBEFnZlK3Ywf6kktaHHe89RaJp59O3AsvBFSeKimhyy23tDre5Z13fF5bf/XVNJ19NolPPBHQZ7rVjBrV/H1KkM37SLZxwgQAhi1f3uF5Q//xDwDW+LmVzYhVq9g+ejQVFu2hFEkkZALUOHw4zp07W90eOZ5/vs2g8Jdj9WriPvywxbF4r5/bUverXxH34YdBd/aevOACABKLioK6PpLVJSezb+RIEk6cIHvfvg7PTTx5kuy9e6nKzfW7NQOw3SPEY5WETADcnbJ4dU463nqLLj//eafL7/L3v7f42Vd/TFN+Pk1nn03Ciy8G9XmN6ek0uXaCTI7BVszuiy4CINvPZVHd5+1yXefLWR9/zL9dHfixTEImACeWLWvZZwJQXNypFoynOI/H2KqkxGfrxPmTnwD+dQ635aTnrdK6dUGVEckOnXUWAOlHj/p1fvqRIwDsHTnSr/N779pFZU4ORyzaHjZSSMj46eRTTzU/VvZkRgvGLe6LL5q/9yc46q++Gsfnnwd9q+TZH5O4bVtQZUSy4/n5wKnw8MUdRs7UVOr8GFYgt0wGCRk/NOXnU+9qNXhSmzfjWL3aks/01R+jMzPRBQVBt2IAalz9MfGlpcRVVQVdTiSqS07G6VqH2e+WjMd51bm5fl2TU1rK/kGDAq9gFIm6cTJWOPnUU20ej//1ry37TF/h0eDqF4jbvNmv8oo3bmzuf2lVVp8+/HvPnhbHTrvjjqgeM+P0GkQXqKqcHJ+dxQC5paVsHz2auuTkmB0zIyHjQ+Pw4cb4FW9lZZa0YjLaCQJvDa46ed5idSTv1ltp8uhPqhk1ivIpUwDIfuwxErdvb37PUV5O0o4dftY4Mjk9RgAn+LltsOd5dd67UbQjZ/9+APYPGtR8+xRrJGR8cLZxmwTGEyU7NZ5zjlEPP5+MeIdG9eWXN3+f8frrsXe75GdItMfpY5qCW+bx4wAc6dMnZkNG+mR8qL/mmjaPx739dohr0lJTfr4xmztIsdwfA/6HRGdlHDsGQEX37iH5vHAkLZkONA4f3vqRNVh2qxSQrCwcQf7P2JieToNrnpMVo3w/fOstGtLTTS/XU5fqaq644w6/b3W8BdOSSfCYiV3nZ5+OuyVTGcMjfyVkOuC+JfGm/OxstYr2s9+mPXVDhzZ/b8VI3zNeeIGTQS7elJiU5Nd5aYcPBx0wwepMZ3GltGREW5raWc7A4bX2S6hp14hjVVYW1PVWz1fqtWJF0NdatdSDt0TvfbECvV4WpfKb9Ml0oMk1WMub2rQptBXx4m7JdHa+kqOigi6upx+xprOtoECuT6ypoS5EfUDhSEImCL6WbwhG4+TJOCdNCqweQbZk6oYMAWJzvlJbrO4E7myrKdJJyHSgvX/EZi9vqQsKaHj2Wep++cvArgvi1sL9VAlic76Sm+c/fH87gT3DKJDg6Ozj8kgnIdMBR3stlm+/NfVzGu66C8Dv2dTukAumAzjW149x8xyt629LpspjWUx/pyIA1KWkkBjiTupwIh2/HWhvNK0uKPB7BwFfdEEBTbfdBkDCSy/5dY27hRVMSyYU/TEHxo8PydOl/j52e/Al4cQJnKmpVPk5D8mzReLPlAJhkJDpQPyaNcaAN6+xMk1jxwY9RsVb/SuvGJ8VwC4Dnbld89UfUzt4MJUTJ5L7wANBf8aeyZNDMk6m94YNnerAPX3HDvaNHOn37Yw7jPxdf8aTe1BeLIq6kPHcrSA+Pp5cP/+Xak+XhQs5fu+9LY41zZxJ7t/+1qlyAY7Nn48+91wSP/qI3HffJTGAulZXVEBBgc8/X48ePU5dM3Bg8/fdS0pavOe285e/pPfSpWS18Z6nnj17tvvej+68s8NrOzJ8+PBATvb71PQ2Qi+pspLngJpBgxg9erTPMv7l+ryLjhxpPj8jI6PDa0506QJAf6W48MIL/aprZ39n22PVDhO+RF2fjNZ6oda6UGtdGBcX1+nyui5aRHxpacvPyMykfNasoMtsysjgyCuvcPKKK+iybRu5QSx61WX//nZnVben+swzm7/PamPR7Nq8PGrz8tp8LxoNPnQIgKMZGdQkJPg8/0tXsBYG0JL5ynVLmxvDT5iiLmSs0HPSJBxetyhV06dTOW1awGXVjh7NN8uXU/e975H40UecdsUVQdXJfbtT37t3UNcnuf6Bedo5ezYD5s4NqrxIlOJ0ctnWrQB85uN/+Y2uMVMX7txJTgBzvY64bsUGmPywIJJIyPihy/799L7mmlYtmor77+eb5cup9aOpXTN+PN8sX87RV1+lsW9fMh95JKgWjFuSayW7OtcSkv5I27mz3fd2zp5N1saNpO3eHXSdItH4L74gpa6Ot3wsqbli+HBS6uq4/rPPAiq/2NWSGern6nvRKOr6ZKzSZf9+8i+5hGP33NO8DgtA/dChHH31VeOcbduILy1tDqOmjAycw4ZR7zFXKHn5crLmzGkVWIFKdo1xqR061O/FpdJ27yb7ww85fvHFlBcW0n3NGqoHDmTv9OkkHTpE/vz5napTJEpxOnlg6VJmX30188eOZXobE19fHjWKY+np3PW//0uK0xlQ+V9160aP6mpS6+vNqnLEUVpbvsmibZKSknTvIG8nOnIyMZGaK6+k+pZbWgRIe+L27SNlxQoynn22ww3UAt1Sd+8HHxBXWUnvdpajANrs3N07fTrHL7qI+oyM5nAJtB/mjDPOCOh8fwXU8RuAtjp+PdUkJDB/7FhOJCYysriYwYcOcTQ9nRWu+kxfvbrN2yRfHb83TJjAuK++YkoAk2ojreO3b9++m7TWhe29Ly2ZIDgqK0l79VXSXC0Y55AhNPTpQ1NWFk2uXzpHZSXxpaUkbN1q2c6Mma+/zvF776UxPT2gNWHy58+PyVZLR1KcTu5auZKahAQ+KyhgR14eKU5nu+Hij62usDj/wAEzqxpxJGRMkFBURIINm6OlrVzJ8Xvv5eSoUVG9Hm8opTidjNm1y5SyPunVi1Snk2Ex3B8D0vEb0brs309iUREVN91kd1VEGzb07s0lxcV2V8N2EjIRruuCBZwcNSroR9nCGp/07MmJhAQulZCRkIl0aStXEl9aKq2ZMPPOd77DVTt30iOGB+G5SchEgdwHHqB8yhQaLZ4vJPyzNTeXkq5duaaDcUmxREImCqSsX0/yunWt5lgJeywaMYIpGzfG9NgYTxIyUeK0O+6g+vLLWyxKJUJv6aBBFJSVMc6C1RMjlYRMlIirqiLv1ls5Mnu23DbZpDgri/f79WOqzWtAhxsJmSiStGMHuQ88wKGXX7a7KjHncGoqCwoL+fU//ym3SV4kZKJMyvr1ZM+bx7F77rG7KjHllaFD+dWGDfI0qQ0y4jcKpaxfH9Pr99rhV598YncVwpa0ZIQQlpKQEUJYSkJGCGEpCRkhhKWietEqpdRRwIoNcroDVuxxEUnlRlJdI63cSKorwCCtdbuDs6L66ZLWOsf3WYFTSm3saCWwWCg3kuoaaeVGUl3d5Xb0vtwuCSEsJSEjhLCUhExwFkq5EVXXSCs3kurqs9yo7vgVQthPWjJCCEtJyAghLCUhI4SwlISMEMJSEjJCCEtJyAghLCUhI4SwlISMEMJSEjJCCEtJyAghLCUhI4SwlISMEMJSEjJCCEtJyAghLCUhI4SwlISM6DSl1PVKqU1KKa2UKlNKLVZKjWvn3ClKqVWu89znb1JKzVVK9Qt13YX1ZNEq0SlKqSnAAmAzUAxc7/H2Qq31VNd51wOLgCzXeZ7ne4bLTK31Y9bXXISKhIwImqvl8RWwWWs9wnVsE3Cux2kTMEJkLkaw3K613uxRRhZQghE+biM8zxGRTW6XRGfMdL3e3sE5izECZqHWulV4aK3LAe8tNaaaVkNhO2nJiKC4WiBlQLHWur/H8bZ+oZZorSd0UJZ366e5ZSQin7RkRLBucL0ucR9op+O2uKOAcfG+7ttAKqKUOlcptTiQa0ToSMiIYLlbL294HGvridLMNo41c7WIsrwO+9Ufo5Tqp5SaC2yidVCJMCEhI4KitZ6ptVZefSyXtnHq+z6KaiuYVnV0gfuROUZfzyqg3MdnCBtJn4wwjVKqjJatEp99K67bHM/H3mitVRCfWyz9OOFJWjLCFK7+mCyvw75aMeAVMH5eIyKIhIwwS1C3PW0cXmBOdUS4kJARZmnVH6O19tUqaTUeRmu9pK0TReSSkBFm8W7JdBgwrqdK3tcs8TpngWt+07mIiCUhIzqtnf6YDm+VODXOxlPzrZIrhKa4fiwOtm7CfhIywgxt9a0EeqtU7nV75Q6Yha6pByJCScgIM7TVHxPoBMc33d+4WjGzXD92OJhPhD8JGWGGQq+f/XkM7X3OJmgOmE0Yt19TpRUT+SRkhBmyvH6e68c1c2jZ1zJTKbUAY9JlP4x1ZRaaUz1hJwkZYYapGEP7izHCwWdLRmtd7pq9PdN1XT+MzuAlGOvJyMJVUUKmFYiIJ9MKwpu0ZIQQlpKQEUJYSkJGCGEpCRkRDbJo/YRLhAkJGRFxXMttrnLv9eQ63M9jD6dVsodT+JCnS0IIS0lLRghhKQkZIYSlJGSEEJaSkBFCWEpCRghhKQkZIYSlJGSEEJaSkBFCWOr/A2WmQpJskWovAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from matplotlib.path import Path\n",
    "import matplotlib.patches as patches\n",
    "\n",
    "sz = 10\n",
    "r = torch.linspace(0, sz-1, sz)\n",
    "c = torch.linspace(0, sz-1, sz)\n",
    "grid = torch.meshgrid(r, c)\n",
    "grid = torch.stack(grid, dim=2)\n",
    "\n",
    "line = torch.tensor([[[2, 1], [6, 3], [3, 8], [7, 7]]])\n",
    "ldt2 = curve_edt2_bruteforce(line, grid, iters=10, slices=10)[0]\n",
    "\n",
    "verts = [\n",
    "   (1, 2),  # P0\n",
    "   (3, 6),  # P1\n",
    "   (8, 3),  # P2\n",
    "   (7, 7),  # P3\n",
    "]\n",
    "\n",
    "codes = [\n",
    "    Path.MOVETO,\n",
    "    Path.CURVE4,\n",
    "    Path.CURVE4,\n",
    "    Path.CURVE4,\n",
    "]\n",
    "\n",
    "path = Path(verts, codes)\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "ax.set_xlim(-0.5, 9)\n",
    "ax.set_ylim(-0.5, 9)\n",
    "\n",
    "fr = ax.imshow(ldt2, cmap=matplotlib.cm.gray, interpolation='nearest')\n",
    "fr.axes.xaxis.set_ticklabels([])\n",
    "fr.axes.yaxis.set_ticklabels([])\n",
    "plt.xticks(np.arange(0.5, 10, step=1))\n",
    "plt.yticks(np.arange(0.5, 10, step=1))\n",
    "fr.axes.set_ylabel(r'$\\bm{n}_{2}$')\n",
    "fr.axes.set_xlabel(r'$\\bm{n}_{1}$')\n",
    "\n",
    "plt.scatter([1,7],[2,7], color='cyan') \n",
    "plt.annotate(r\"$\\bm C(t=0)$\", (0.2, 0.2), color='cyan')\n",
    "plt.annotate(r\"$\\bm C(t=1)$\", (2.5, 7.3), color='cyan')\n",
    "\n",
    "patch = patches.PathPatch(path, facecolor='none', lw=2, edgecolor='cyan')\n",
    "ax.add_patch(patch)\n",
    "\n",
    "plt.savefig('eqn7.pdf', bbox_inches='tight')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "ax.set_xlim(-0.5, 9)\n",
    "ax.set_ylim(-0.5, 9)\n",
    "\n",
    "fr = ax.imshow(ldt2, cmap=matplotlib.cm.gray, interpolation='nearest')\n",
    "fr.axes.xaxis.set_ticklabels([])\n",
    "fr.axes.yaxis.set_ticklabels([])\n",
    "plt.xticks(np.arange(0.5, 10, step=1))\n",
    "plt.yticks(np.arange(0.5, 10, step=1))\n",
    "fr.axes.set_ylabel(r'$\\bm{n}_{2}$')\n",
    "fr.axes.set_xlabel(r'$\\bm{n}_{1}$')\n",
    "\n",
    "plt.savefig('eqn7b.pdf', bbox_inches='tight')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAARgAAAEYCAYAAACHjumMAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjMuMiwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy8vihELAAAACXBIWXMAAAsTAAALEwEAmpwYAAAMR0lEQVR4nO3dTWwc5R3H8d/f4uXWOG65IV7sSDkWxa5EOBEauJPGUe9VXo5IpTj0wgUBDpwpccK9EPeKSh16A4k2dnpOZTsNV5Q4ucAh8O9hZp31Zr12nOfn8U6+H8lyvZ59/FSFb2dmZ56JzBQAOIw0PQEA7UVgANgQGAA2BAaADYEBYENgANg81vQESouIU5JO1T9ONjkX4BHxfWY+1e8X0ebrYCKivf/lgL1jMTOn+v2CQyQANgQGgA2BAWBDYADYEBgANgQGgA2BAWBDYADYEBgANgQGgA2BAWBDYADYEBgANgQGgA2BAWBDYADYEBgANgQGgA2BAWBDYADYtP2pAoAk6bHHPP+oP/PMM8XHvHHjRvExJenu3buWcQdpXWAyc07SnMRTBYCmcYgEwIbAALAhMABsCAwAGwIDwIbAALAhMABsCAwAGwIDwIbAALAhMABsCAwAGwIDwIbAALAhMABsCAwAGwIDwIbAALAhMABsCAwAm9Yt+s1TBdCPY/V/SVpeXi4+5sGDB4uPKUnXrl2zjDtI6wLDUwWAvYNDJAA2BAaADYEBYENgANgQGAA2BAaADYEBYENgANgQGAA2BAaADYEBYENgANgQGAA2BAaADYEBYENgANgQGAA2BAaADYEBYNO6NXlZ9Hu4jYx4/j/vzp07lnF//PHH4mOOjY0VH7MprQsMi34DeweHSABsCAwAGwIDwIbAALAhMABsCAwAGwIDwIbAALAhMABsCAwAGwIDwIbAALAhMABsCAwAGwIDwIbAALAhMABsCAwAGwIDwIbAALBp3aLfw/xUgccff7z4mM8++2zxMSXppZdesoz74osvWsY9cuSIZdybN28WH/P69evFx2xK6wLDUwWAvWPXD5Ei4rltbvcL81QAmO1KYCLiuYj4MiJ+krQcET9FxLWI+MOAt01ExAcR8XlEfBYR7+3GXAGUYw9MRByTtCzpVUkh6Xb9/YCkuTo0950oyMyrmXk2M09ImpQ0454rgLKsgYmIfZLmJa1KejUzRzJzLDNHJE1I+kjSU5JWImLQWbgV5zwBeLj3YGYlLWXmgcz8qvsXmbmamTOZuV9VaC5HxOvm+QDYRe7ATEo6vtVGmTkj6TeSPiUyQHu4P6Yezczr29kwM5ciYlzVnsxYZn7qnRoAN3dgbj/Ixpm5JmkqIv4REZGZFz3TArAb3IdIV7Z73Uu3zHxN0oGIeLP8lADsFmtgMvOMpHPdF81FxCvbCUdmnpV0m8gAw2s3LrQ7KeliRLxZh2Ze0mx9fcxAmXlB0lVJU+Y5AjCw34uUmbclnYiI30r6s6QrktYkXd7m+7+KiClJn9smCcBi1252rK+D+WrLDfu/d0XsxQBDh/VgANg0FpiIeIE7poF2a3IPZknSrYj4mNAA7dRkYC6ouqv6jKrQDFq6AcAQaiwwmXla0n5JZyXdUbV0wx+bmg+A8ooHpj638peI+He9yNT7EfFKv20z83ZmnpP0vKTrqi7K+3XpOQFoRtHARMSfJC2qWnR7UtJRSW9JWqhXsfuy393S9T1Ik6oOmd4uOScAzYnMMutiR8Tzqlau60hVwVDPax2XJM1l5j+7xvhE0nRm/vIh5tH9VIHJnY4zyL59+xzD6p133ik+5htvvFF8TElaXV21jPvNN99Yxv36668t437xxRfFx/zuu++KjylJpf5d72MxM/tep1byQrvT9fdOWEIbgyJtDM60pOmIWJP0mapPlY5KGn2YSfBUAWDvKBmYo/X326puBxiTNK6Nwej+F74Tm/26F6fQxr0gAEOsZGDGVcVhqr7/SNL6urxHJf1e0u+6tu8Xm5R0vuCcADSo5EneUUnz3XGR1j8p+ltmTteLfZ9QdaNjdH11rGTmRwXnBKBBJQOzpmovZqDMnK8XlNqv6iK7pfrrnLihEWiVkodIVyQd2u7G9Z7O+glZAO1Tcg9mXtI4F8oB6CgWmPrj4Tuq7jECgOK3CpxU9VSAjwuPC2AIFQ1MZs6runnxTET8teTYAIaP427qzoVy0/WD7V9nvRfg0VT6ZsdPVC3O3bld4ICqk7+36rurN72zGkD7FPuYun5qQOcmw9S9K3U7F9JNqvoY+62IkKqL7RYkXc7M/5SaB4C9o+QeTO/Njur6uTs2na+jkmYlLdZLOfw9Io4UnA+AhpUMTOdmx6uqrsyNnq/s+er9/WuqHnzPqnZAS5S+F+lSZk7VXyOSJlTt2cyrust6UHCke2v0AmiBkoG5qp57kTJzNTMvZOaJzBzT9oIzVnBOABpUMjDvS5qMiH9tdrvAJsGZUXXCtxOcmYJzAtCgkrcKzKs6vJmStBQR/42I9yLihQHvWc3MDzPztcwcy8yRzLxYak4AmlX6St45VXslF+vvZ1UtAg7gEVT8St56r6TzzKMzkj4s/TcADIeS68Fs0LXey67qeaqAnnjiieJ/49133y0+piQdO3as+Jgvv/xy8TElaWlpyTLuDz/8YBn3559/toxrXKm/FWyBaUr3UwVGRkb4Xx9oUJPPpgbQcgQGgA2BAWBDYADYEBgANgQGgA2BAWBDYADYEBgANgQGgA2BAWBDYADYEBgANgQGgA2BAWBDYADYEBgANgQGgA2BAWDTujV5exf9BtCcaPOq6E8++WQ+/fTTxcddXl4uPqYkHT58uPiY3377bfExJVbTxwaLmTnV7xccIgGwITAAbAgMABsCA8CGwACwITAAbAgMABsCA8CGwACwITAAbAgMABsCA8CGwACwITAAbAgMABsCA8CGwACwITAAbAgMABsCA8Cm9U8VuHHjRvG/MTExUXxMyTNXFudGk1oXmMyckzQnSRHBv11AgzhEAmBDYADYEBgANgQGgA2BAWBDYADYEBgANgQGgA2BAWBDYADYEBgANgQGgA2BAWBDYADYEBgANgQGgA2BAWBDYADYEBgANgQGgE3rFv3ufarA3bt3i/+NlZWV4mMCbRRtfqwFTxUAdsViZk71+wWHSABsCAwAGwIDwIbAALAhMABsCAwAGwIDwIbAALAhMABsCAwAGwIDwIbAALAhMABsCAwAGwIDwIbAALAhMABsCAwAGwIDwIbAALBp3VMFenwv6X+GcX9Vj/0ojztMcx22cYdprpJ0cLNftDowmfmUY9yIuLLZKuqPyrjDNNdhG3eY5toZd7PfcYgEwIbAALAhMDszx7hDNddhG3eY5jpw3FY/2RFAs9iDAWBDYADYEBgANgQGgA2BAWBDYADYEBgANgQGgA2BAWBDYADYEBgANgQGgA2BAWBDYADYEBgANgQGDy0ijkfEYkRkRNyKiEsRcXSTbU9FxEK9XWf7xYiYjYjx3Z47vFhwCg8lIk5JOi9pSdKKpONdv57LzNP1dsclXZA0Wm/XvX13WGYy85x/5tgNBAY7Vu9xLEtayszJ+rVFSYe6NptWFZBZVVE5mZlLXWOMSlpVFZ6Oye5tMLw4RMLDmKm/nxywzSVVcZnLzPvCkZlrknofe3G62AzRKPZgsCP1nsctSSuZOdH1er9/oOYzc3rAWL17Pet7RBhu7MFgp07U3+c7L2xyknZlUFxqve+7+SATiYhDEXHpQd6D3UFgsFOdvZbPul7r98nRTJ/X1tV7QqM9L2/r/EtEjEfErKRF3R8p7AEEBjuSmTOZGT3nVF7ts+nlLYbqF6WFQW/ofCyu6tzOgqS1Lf4GGsI5GBQTEbe0cW9ky3Mp9aFN90fbyszYwd9d4bzN3sMeDIqoz7+M9ry81d6L1BOXbb4HQ4LAoJQdHer0efl8melgLyAwKOW+8y+ZudXeyH3Xu2TmfL8NMZwIDErp3YMZGJf606Pe98z3bHO+vl/pkDCUCAwe2ibnXwYeHunedTTd1g+P6gCdqn9c2enc0CwCgxL6nUt50MOjtZ5Dqk5c5urbCTCECAxK6Hf+5UFvVvy88x/qvZe36x8HXqiHvY3AoISpnp+381Fz7zaL0npcFlUdcp1m72W4ERiUMNrz8+w23vO+Np5bmYmI86puoBxXtS7MXJnpoSkEBiWcVnW5/oqqMGy5B5OZa/Vd2DP1+8ZVnfidV7UeDItOtQC3CmDocavA3sUeDAAbAgPAhsAAsCEwaINR3f9JFvYAAoOhUy+RudB5FlP98njXM5YWeMbS3sCnSABs2IMBYENgANgQGAA2BAaADYEBYENgANgQGAA2BAaAzf8BJNAvQF4vW6gAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "ax.set_xlim(-0.5, 9)\n",
    "ax.set_ylim(-0.5, 9)\n",
    "\n",
    "fr = ax.imshow(exp(ldt2, 0.5), cmap=matplotlib.cm.gray, interpolation='nearest')\n",
    "fr.axes.xaxis.set_ticklabels([])\n",
    "fr.axes.yaxis.set_ticklabels([])\n",
    "plt.xticks(np.arange(0.5, 10, step=1))\n",
    "plt.yticks(np.arange(0.5, 10, step=1))\n",
    "fr.axes.set_ylabel(r'$\\bm{n}_{2}$')\n",
    "fr.axes.set_xlabel(r'$\\bm{n}_{1}$')\n",
    "\n",
    "plt.savefig('eqn8.pdf', bbox_inches='tight')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "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": "3.7.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}