Hasenpfote/2d_curve_frenet.ipynb Secret

## 2d_curve_frenet.ipynb
{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "** 曲線の接線、法線ベクトル **\n",
    "Tangent: $T=\\frac{\\dot{r}}{\\|\\dot{r}\\|}$  \n",
    "Normal: $N=\\frac{\\dot{T}}{\\|\\dot{T}\\|}$  \n",
    "Binormal: $T\\times N$  \n",
    "Curvature: $k=\\|\\dot{r}\\|$  \n",
    "Torsion $t=-\\dot{B}\\cdot N$  \n",
    "\n",
    "see also: [Frenet–Serret formulas](https://ja.wikipedia.org/wiki/%E3%83%95%E3%83%AC%E3%83%8D%E3%83%BB%E3%82%BB%E3%83%AC%E3%81%AE%E5%85%AC%E5%BC%8F)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import sympy as sym\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "sym.init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " ** その１ **"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "t = sym.symbols('t')\n",
    "\n",
    "# r\n",
    "rx = t\n",
    "ry = sym.cos(t)\n",
    "\n",
    "# r'\n",
    "p_rx = rx.diff(t)\n",
    "p_ry = ry.diff(t)\n",
    "# |r'|\n",
    "p_rl = sym.sqrt(p_rx**2 + p_ry**2)\n",
    "\n",
    "# t = r' / |r'|\n",
    "tx = p_rx / p_rl\n",
    "ty = p_ry / p_rl\n",
    "\n",
    "# t'\n",
    "p_tx = tx.diff(t)\n",
    "p_ty = ty.diff(t)\n",
    "# |t'|\n",
    "p_tl = sym.sqrt(p_tx**2 + p_ty**2) \n",
    "\n",
    "# n = t' / |t'|\n",
    "nx = p_tx / p_tl\n",
    "ny = p_ty / p_tl\n",
    "\n",
    "np_r = sym.lambdify(t, [rx, ry], modules='numpy')\n",
    "np_tangent = sym.lambdify(t, [tx, ty], modules='numpy')\n",
    "np_normal = sym.lambdify(t, [nx, ny], modules='numpy')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
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1YB0dAbgHhAFNgG3Dhhm6huaQxOREvjpoWwuT1/eqT3GX4oTHhNN0WVM2n99s\n1vNv2LCB5s2b4+rqyuHDh6lSpYpZz6/YhucmACllayllzQx+NgMRQohSAMY/72RyjlvGP+8Am4D6\nzyhvsZTST0rp96yFR5RcrkwZw2Lgly/D6NHUdHTkKFAJ6HzrFj/27ZsjYeiSdfRZ34frD21rYJqd\nxo72ldoD8CjxEd3XdmfO33MynYY7K3744Qd69uxJnTp1OHLkCNWqVTP5nIptMrUJaAswyPh6EJDu\nMkUIUUAI4fb4NdAWOG1iuYqtKFMGvv8ewsLwGjOG/Y6OdAJGbdrEwq+/tmjRumQdvX/rzabzm6jv\nlek1R67VqXKnlNcSyYSdE3jX/10SkxNNOm+t6tV5q00bdn/xhRrglc+ZmgC+BNoIIS4CrY3vEUKU\nFkI8buj1AA4KIU4CR4E/pZTbTSxXsTVeXjBvHgXDwtg0ahQz7ezofeqUxYrTJmnpsa4Hm0MN1yS2\nmADaVWqXrivokuNL6LCqAw/i0/W2fqZbwcEsHTIEevakWffu/BoXh0vDhuYMV7FBajZQxTrCw2HW\nLHSDBjFg5kw++OADGpqpQkpISqDHuh5svWi4BinoWJCoiVE2OSFdk5+bcOjGoXTbq5aoin8/fyoW\nq5jxB/V6CAyErVvZt2YNvS9cIB64CHj4+MA//4BqYs2T1GygSu5XujR88w233Nw4duwYzZo1Y8GC\nBSa3cSckJdB9bfeUyh/Ar7SfRSr/f2P/Nbk55nlSNwM91qRcEzpW6sjhm4fT/n1FRcHatTBoEHh6\nIhs2ZM7nn9PqwgWKAkcAj0KFDCOzVeWvoBKAYmU+FSsSFBRE69atGTlyJIMHD+bRo0fZOld8Yjxd\nV3dl+6W0LYz1S1um+eevy3+lK8vcOlbumG6bg8aB2W1n82atN9MuzXniBEyfDr/+irx7l4HABAyD\nc44C1e3sYP16qF7dojErtkMlAMXqihUrhr+/P9OmTWPFihUMHjw4y+d4lPiIrmu6sjNsZ7p9lmr/\nP3T9EMtPLn/+gSao5VGLMoXKUK1ENQbXGQzA3qt7WRq8NP3BLVoYmn38/BBAU2AW8BtQCGD+fGjT\nxqLxKrZFTfWn5AoajYapU6dSr149ypcvD4BWq8XR0fGZC9ADxOni6LqmK3uu7Mlwv6USwMEbB7kU\neYnI+EiKuRSzSBlCCDpW6kjP6j2p51WPHZd2cDv2Nh/89QEdKnXAq5BXyrH6Y8f4tmtXSoWH048n\nI7ABGD+saHhZAAALRUlEQVQe3n3XIjEqtkvdASi5SseOHalRowZgmJumU6dO3L6d8aIzjwkhWNx5\nMSHDQ6haomqafZ4FPSlTqIzZ44yMj+Ts3bPoknWsOb3G7OdPbWqLqbSp2IYizkVY0GkBANHaaEZu\nHWl4BhAfz/WRI2nl58cH4eFsB6iY6uFw165g4S63im1SCUDJlaSUvPLKK+zdu5eXXnqJjRs3Znqs\nq4MrFYtV5Eb0Dc7fOw88ueqv71X/uXcQ2XH4xuGU15ZuBirtVjrldbeq3ehdozetSkK/IlsICNCw\ny78goy4sIAhYKgS/TJoEp0+DqyvUqQOrVoGd7fWAUixPJQAlVxJCMGrUKIKDg/Hx8aFHjx4MHjyY\nh5msKZCsT2birokAuDm64d/Pn8F1BlvsAfDB6wdTXh+9dTQl8eSE6fWaM8EXPJ1BCLAvqWfMBDg0\nsDRvHz+O+OILcHaGhg3hjz+gYMEci02xLSoBKLla1apV+fvvv/nss8/YvHkzcXEZT5H868lfOX3H\nMMB8YuOJlCxQku/af0cX3y4WievpvvnLT1j2LiC1++Ff4fzUBb2jM8QNtzdc8T/222+GkdiKkgk1\nEEyxGZGRkRQrVgwpJePHj2fw4MF4ep7mctgktAk3iNDChttFWP7mLVwdLLeSlS5ZR+EvC5OQlJCy\nzcvNi2vvX7P4YLOjR48SF9eAjFu1BC1a2Nasp4r5qYFgSp5UrJihp82VK1dYuXIlEybU4dSpwei0\nNxDC0CQywucRMZGbLBrH8dvH01T+ALdibmXaC8kc7t27x9ChQ2nYsCH37mX839bJqZzFylfyJpUA\nFJtToUIFQkNDef/9gtjbJ6XZp0FHWNgnaT+QaN7Ruqnb/1Oz1MPg2NhYatSowbJlyxg3bhx+fovQ\naNLe4Wg0rlSoMNMi5St5l0oAik0qVqwYBQpk/DxAm3AdUjdtDhgAMTFmK/tx+39BR8PDVVcHVyoV\nq8TGcxuJ1kabpYz4+Hh+++03QzkFCzJjxgxOnjzJnDlz8PF5B1/fxTg5lQcETk7l8fVdnGZdZkV5\nESoBKDYrsyaPBxGS7ytWJH6PsUlm1y7o2BFiY00uU0pJjDaGPQP3MOAlQ4XrYu9C6OhQNvTewNWo\nqyadPy4uzljJ+9C7d29OnjwJwNChQ1PGRwB4eAygUaOrtGihp1Gjq6ryV7JFJQDFZlWoMDNdU4hI\ngF0/wXtXrlChVStm1azJnZgYOHjQLElAItn+5nZe83mNIs5FAIhKiEIg6FC5A7U8amXrvLGxscyc\nORNvb28mTJhAzZo1CQgIoHbt2ibFqyjPohKAYrM8PAakawqpWnkR81+eQIC9PdWBj86cwSspifMA\nBw5A586QSVfSF6ERGuw1hhlUHieAZJlMrO6pxJKUBNHPbg5KSEggNDQUADs7O2bNmoWfnx+HDh1i\n165dNG/ePNtxKsqLMGkuICFEL2AaUA2oL6XMsM+mEKI98B1gB/wkpfzSlHIV5TEPjwHpmz9mQfNR\no9g9bhxnf/+dzYCvcdcn+/YRW7UqA9esoe6rr5o0Srioc9GU11EJUbg5uT3ZaWcHvXrBnTvQuLHh\np0kT9F5eBAYGsnz5clavXo2Hhwfnzp3DxcWFy5cvU7x48WzHoyhZZepkcKeBN4BFmR0ghLAD5gNt\ngJtAoBBii5TyrIllK0rmLl6EwECqA6knP74LLL95k3lNmlCyZElatGhBr1696NWrV5aL8OQ0qxuA\nuxNcOdUAx8qzniQjIWDuXGStWnDiBGL+fH4Apmk03NfrcXZwoEebNgwaMyblfKryV3KaSQlASnkO\neN5VVH3gkpQyzHjsGqAboBKAYhnh4bB6NRQubHidqkfQYgzrlm4G9trbs/fvv3F3d6dXr14kJyfz\nzjvvULFiRXx8fPD29sbHxwdPT080mrStpRERqygUt4TCzob3+qTbnD89hPMLFnD9oDNX7tzh9N27\nBEjJAQx3IF5AF72elkDXxEQKb91qaJZq2NBwhzBwIPj45MhfkaJAzkwH7QXcSPX+JtAgB8pV8qvS\npeHnnw2vo6Ph2DE4ejTlp9jNmwwBhty+jWzXDu2MGQDcuHGDXbt28csvv6Q53axZs5gwYQIXLlzg\nzTffBGDSpJMULapLc5y0S+TfqocY+F/D+/JAe+Bx+ulu/EnDyQmqVYMuXcDb2yxfX1Fe1HMTgBBi\nF+CZwa5PpJSbzR2QEGIYxqnMy5VTIxsVExUqBK+9Zvh5LDw8JRmIo0dxHjkSli3D29ubGzduEB8f\nz7Vr17hy5QpXrlyhSZMmgOFBbYkSJQAoUkSXUWl4uEOomxvlihbFuXhxuHXL8BwgNQcHw8PoQYOg\nQwdwdLTIV1eU5zHLXEBCiABgQkYPgYUQjYBpUsp2xveTAKSU//e886q5gJQcodcbeu1koSI+fNgb\nrfZauu1OTuVo1Mi4PTkZKleGK1cM7/38DJV+375gTCSKYm65bS6gQKCyEMJHCOEI9AW25EC5ivJi\nNJosX4VnNAbBMB3DF082bNsGWi189JFhfv7AQBg9WlX+Sq5hajfQ7sD3QEngTyHECSllOyFEaQzd\nPTtKKZOEEKOBHRi6gf4spTxjcuSKYkWPe/uEhX2CVnsdJ6dyVKgwM22X1Nq14fp1tRiLkmup6aAV\nRVHykKw0AeXqBCCEuAukb2h9MSWAe2YMJ6fZevxg+9/B1uMH2/8OKv6sKy+lLPkiB+bqBGAKIUTQ\ni2bB3MjW4wfb/w62Hj/Y/ndQ8VuWmgtIURQln1IJQFEUJZ/KywlgsbUDMJGtxw+2/x1sPX6w/e+g\n4regPPsMQFEURXm2vHwHoCiKojxDnksAQoj2QohQIcQlIcTH1o4nq4QQPwsh7gghTls7luwQQpQV\nQuwVQpwVQpwRQoy1dkxZJYRwFkIcFUKcNH6H/1o7puwQQtgJIYKFEP7WjiU7hBBXhRCnhBAnhBA2\nNyBICFFECLFeCHFeCHHOOC1OrpKnmoCMaw9cINXaA0A/W1p7QAjRDIgFfpVS1rR2PFklhCgFlJJS\nHhdCuAHHgNdt7HcggAJSylghhANwEBgrpTxi5dCyRAgxHvADCkkpO1s7nqwSQlwF/KSUNjkOQAix\nHDggpfzJOA2Oq5QyytpxpZbX7gBS1h6QUuqAx2sP2Awp5X4g0tpxZJeU8raU8rjxdQxwDsOU4DZD\nGjxe49HB+GNTV0pCiDJAJ+Ana8eSHwkhCgPNgKUAUkpdbqv8Ie8lgIzWHrCpyicvEUJ4Ay8D/1g3\nkqwzNp+cAO4AO6WUtvYdvgU+AvTWDsQEEtglhDhmnCbelvhgWIBumbEZ7ichRAFrB/W0vJYAlFxC\nCFEQ2AC8L6V89urouZCUMllKWQcoA9QXQthMc5wQojNwR0p5zNqxmKiJ8XfQARhlbB61FfZAXWCB\nlPJlIA7Idc8k81oCuAWUTfW+jHGbkoOM7eYbgFVSyo3WjscUxtv2vRgW97IVjYGuxjb0NUBLIcRK\n64aUdVLKW8Y/7wCbMDTx2oqbwM1Ud47rMSSEXCWvJQC19oCVGR+gLgXOSSnnWjue7BBClBRCFDG+\ndsHQqeC8daN6cVLKSVLKMlJKbwz/B/ZIKd+0clhZIoQoYOxEgLHppC1gMz3jpJT/AjeEEL7GTa3I\nheug58SawDkmL6w9IIRYDbQASgghbgJTpZRLrRtVljQG3gJOGdvQASZLKbdaMaasKgUsN/Yq0wDr\npJQ22ZXShnkAmwzXE9gD/5NSbrduSFk2BlhlvBgNA4ZYOZ508lQ3UEVRFOXF5bUmIEVRFOUFqQSg\nKIqST6kEoCiKkk+pBKAoipJPqQSgKIqST6kEoCiKkk+pBKAoipJPqQSgKIqST/0/+AhgFXeIvTwA\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1c6976cab70>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# numpy\n",
    "theta = np.pi / 4\n",
    "\n",
    "T = np.linspace(0, 2 * np.pi)\n",
    "T2= T[::5]\n",
    "\n",
    "pv = np_r(T)\n",
    "rv = np_r(T2)\n",
    "\n",
    "tv = np.array([np_tangent(t) for t in T2]).transpose()\n",
    "#temp = np.array([np_tangent(t) for t in T2])\n",
    "#tv = np.concatenate([temp[:, i:i+1] for i in range(temp.shape[1])], axis=0)\n",
    "#tv = tv.reshape((temp.shape[1], temp.shape[0]))\n",
    "\n",
    "nv = np.array([np_normal(t) for t in T2]).transpose()\n",
    "#temp = np.array([np_normal(t) for t in T2])\n",
    "#nv = np.concatenate([temp[:, i:i+1] for i in range(temp.shape[1])], axis=0)\n",
    "#nv = nv.reshape((temp.shape[1], temp.shape[0]))\n",
    "\n",
    "# plot\n",
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(1, 1, 1)\n",
    "ax.plot(pv[0], pv[1], color='k', linestyle='dashed')\n",
    "ax.plot(rv[0], rv[1], 'o', color='y')\n",
    "ax.quiver(rv[0], rv[1], tv[0], tv[1], color='r')\n",
    "ax.quiver(rv[0], rv[1], nv[0], nv[1], color='g')\n",
    "ax.set_aspect('equal', adjustable='box')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " ** その2 **"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def frenet(x, y, z=0, *, symbol):\n",
    "    def norm(v):\n",
    "        return sym.sqrt(sum([c**2 for c in v]))\n",
    "    # Position\n",
    "    r = np.array([x, y, z])\n",
    "    # Tangent = r' / |r'|\n",
    "    dr = np.array([sym.diff(c, symbol) for c in r])\n",
    "    tangent = dr / norm(dr)\n",
    "    # Normal = T' / |T'|\n",
    "    dt = np.array([sym.diff(c, symbol) for c in tangent])\n",
    "    normal = dt / norm(dt) \n",
    "    # Binormal = T x N \n",
    "    binormal = np.cross(tangent, normal)\n",
    "    # Curvature = |T'|\n",
    "    curvature = norm(dt)\n",
    "    # Torsion = dot(-B', N) \n",
    "    db = np.array([sym.diff(c, symbol) for c in binormal])\n",
    "    torsion = np.dot(-db, normal)\n",
    "\n",
    "    return (tangent, normal, binormal, curvature, torsion)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "t = sym.symbols('t')\n",
    "directrix = (t, sym.cos(t), 0)\n",
    "tangent, normal, _, _, _ = frenet(directrix[0], directrix[1], symbol=t)\n",
    "\n",
    "np_position = sym.lambdify(t, [c for c in directrix], modules='numpy')\n",
    "np_tangent = sym.lambdify(t, [c for c in tangent], modules='numpy')\n",
    "np_normal = sym.lambdify(t, [c for c in normal], modules='numpy')\n",
    "#np_binormal = sym.lambdify(t, [c for c in binormal], modules='numpy')\n",
    "#np_curvature = sym.lambdify(t, curvature, modules='numpy')\n",
    "#np_torsion = sym.lambdify(t, torsion, modules='numpy')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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1YB0dAbgHhAFNgG3Dhhm6huaQxOREvjpoWwuT1/eqT3GX4oTHhNN0WVM2n99s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\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1c6975e3cf8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot\n",
    "T = np.linspace(0, 2.0 * np.pi)\n",
    "skip = 5\n",
    "T2= T[::skip]\n",
    "\n",
    "pv = np.array([np_position(t) for t in T]).transpose()\n",
    "rv = pv[:, ::skip]\n",
    "tv = np.array([np_tangent(t) for t in T2]).transpose()\n",
    "nv = np.array([np_normal(t) for t in T2]).transpose()\n",
    "\n",
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(1, 1, 1)\n",
    "ax.plot(pv[0], pv[1], color='k', linestyle='dashed')\n",
    "ax.plot(rv[0], rv[1], 'o', color='y')\n",
    "ax.quiver(rv[0], rv[1], tv[0], tv[1], color='r')\n",
    "ax.quiver(rv[0], rv[1], nv[0], nv[1], color='g')\n",
    "ax.set_aspect('equal', adjustable='box')\n",
    "plt.show()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
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