Vectorized scanlines calculation

scanlines.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "X: [1, 2, 3, 4]\n",
      "Y: [5, 6, 7, 8]\n",
      "Model: [-2 -1  0  1  2]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1d85fe7a748>]"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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uHtmFqaNjaNRA/10iEhiUVuVk5uTz2LsbSVizl25RjfjnnSOJ69LC67JERKpE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\nu4hIEFK4i4gEIYW7iEgQ+v/2phOJRUaBqQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1d85fdda710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Наша кривая\n",
    "x = [1, 2, 3, 4]\n",
    "y = [5, 6, 7, 8]\n",
    "\n",
    "# Модель сканстроки\n",
    "model = np.arange(-2, 3)\n",
    "\n",
    "print('X:', x)\n",
    "print('Y:', y)\n",
    "print('Model:', model)\n",
    "plt.plot(x, y, 'o-')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "UX: [ 0.70710678  0.70710678  0.70710678  0.70710678]\n",
      "UY: [-0.70710678 -0.70710678 -0.70710678 -0.70710678]\n"
     ]
    }
   ],
   "source": [
    "# Вычислим нормали к нашей кривой в каждой точке\n",
    "# Нормали задаются нормированными (единичными) векторами направления\n",
    "dx = np.gradient(x)\n",
    "dy = np.gradient(y)\n",
    "dl = np.sqrt(dx*dx + dy*dy)\n",
    "\n",
    "ux = dx / dl\n",
    "uy = dy / dl\n",
    "\n",
    "ux, uy = uy, -ux\n",
    "\n",
    "print('UX:', ux)\n",
    "print('UY:', uy)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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7OjOsVAHGXR13SCcBe/Ob4dvfhr//e3jbhNMVXXllfR+8pP0Yd3VU287ueNZZ8IMfwPLl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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1d85ddd54a8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Нарисуем на графике нашу кривую и нормали в каждой точке\n",
    "plt.plot(x, y, 'o-')\n",
    "plt.quiver(x, y, ux, uy, color='r')\n",
    "plt.axis('equal');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1d85fe89b38>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Вычислим сканстроку для одной точки кривой\n",
    "# Тут уже используется векторизация через numpy но пока для одной точки\n",
    "sx = x[0] + np.asarray(model) * ux[0]\n",
    "sy = y[0] + np.asarray(model) * uy[0]\n",
    "\n",
    "plt.plot(x, y, 'o-', sx, sy, '.-')\n",
    "plt.quiver(x, y, ux, uy, color='r')\n",
    "plt.axis('equal');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Чтобы сделать то же самое для всех точек кривой мы могли бы написать цикл, но зачем если у нас есть NumPy?\n",
    "Приведём наши данные к такой форме, чтобы вычислить сканстроки для всех точек кривой без циклов, используя NumPy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " [[-2 -1  0  1  2]\n",
      " [-2 -1  0  1  2]\n",
      " [-2 -1  0  1  2]\n",
      " [-2 -1  0  1  2]]\n",
      "\n",
      " [[1 1 1 1 1]\n",
      " [2 2 2 2 2]\n",
      " [3 3 3 3 3]\n",
      " [4 4 4 4 4]]\n",
      "\n",
      " [[5 5 5 5 5]\n",
      " [6 6 6 6 6]\n",
      " [7 7 7 7 7]\n",
      " [8 8 8 8 8]]\n"
     ]
    }
   ],
   "source": [
    "# Приведём наши данные к удобной форме с размерностью 2\n",
    "# Переход к размерности N+1 позволит векторизовано работать с данными из разных пространств \n",
    "# (пространство кривой и пространство модели)\n",
    "x2 = np.array(x, ndmin=2)\n",
    "y2 = np.array(y, ndmin=2)\n",
    "ux2 = np.array(ux, ndmin=2)\n",
    "uy2 = np.array(uy, ndmin=2)\n",
    "model2 = np.array(model, ndmin=2)\n",
    "\n",
    "# Далее клонируем модель для соответствия длине кривой\n",
    "v_model = model2.repeat(len(x), axis=0)\n",
    "\n",
    "print('\\n', v_model)\n",
    "\n",
    "# Далее клонируем координаты точек кривой и векторы направления для соответствия длине модели\n",
    "# Мы транспонируем данные для соответсвия длине модели\n",
    "len_model = len(model)\n",
    "v_x = x2.T.repeat(len_model, axis=1)\n",
    "v_y = y2.T.repeat(len_model, axis=1)\n",
    "\n",
    "v_ux = ux2.T.repeat(len_model, axis=1)\n",
    "v_uy = uy2.T.repeat(len_model, axis=1)\n",
    "\n",
    "print('\\n', v_x)\n",
    "print('\\n', v_y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-0.41421356  0.29289322  1.          1.70710678  2.41421356]\n",
      " [ 0.58578644  1.29289322  2.          2.70710678  3.41421356]\n",
      " [ 1.58578644  2.29289322  3.          3.70710678  4.41421356]\n",
      " [ 2.58578644  3.29289322  4.          4.70710678  5.41421356]]\n",
      "\n",
      " [[ 6.41421356  5.70710678  5.          4.29289322  3.58578644]\n",
      " [ 7.41421356  6.70710678  6.          5.29289322  4.58578644]\n",
      " [ 8.41421356  7.70710678  7.          6.29289322  5.58578644]\n",
      " [ 9.41421356  8.70710678  8.          7.29289322  6.58578644]]\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1d85ea12e80>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Теперь можно считать сканстроки для всех точек кривой\n",
    "sx = v_x + v_model * v_ux\n",
    "sy = v_y + v_model * v_uy\n",
    "\n",
    "print(sx)\n",
    "print('\\n', sy)\n",
    "\n",
    "# Построим график. Тут всё же с циклом :)\n",
    "\n",
    "plt.plot(x, y, 'o-')\n",
    "\n",
    "for sx_, sy_ in zip(sx, sy):\n",
    "    plt.plot(sx_, sy_, '.-')\n",
    "\n",
    "plt.axis('equal');\n"
   ]
  }
 ],
 "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": "3.6.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}