bezier_curve.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.special import binom"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2 Point Curve"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x117eebf10>]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "t = np.linspace(0, 1, 20)\n",
    "\n",
    "p0 = np.array([1, 1]).reshape((-1, 1))\n",
    "p1 = np.array([2, 2]).reshape((-1, 1))\n",
    "\n",
    "curve = (1 - t) * p0 + t * p1 \n",
    "xs = curve[0]\n",
    "ys = curve[1]\n",
    "\n",
    "plt.plot(xs, ys)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Recursive Definition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def bezier_curve(points, t):\n",
    "    \"\"\" Returns the bezier curve given contorl points\n",
    "    \n",
    "    Args: \n",
    "        points: A list of 2D points.\n",
    "        t: The sampling density of the curve.\n",
    "    \"\"\"\n",
    "    \n",
    "    if len(points) == 2:\n",
    "        curve = (1 - t) * points[0] + t * points[1] \n",
    "        return curve\n",
    "    \n",
    "    else: \n",
    "        left_influence = (1 - t) * bezier_curve(points[:-1], t) \n",
    "        right_influence = t * bezier_curve(points[1:], t)\n",
    "        curve = left_influence + right_influence\n",
    "        return curve"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 4 Point Curve"
   ]
  },
  {
   "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": [
    "p0 = np.array([1, 0]).reshape((-1, 1))\n",
    "p1 = np.array([2, 2]).reshape((-1, 1))\n",
    "p2 = np.array([3, 0]).reshape((-1, 1))\n",
    "p3 = np.array([4, 2]).reshape((-1, 1))\n",
    "\n",
    "points = [p0, p1, p2, p3]\n",
    "curve = bezier_curve(points, t)\n",
    "xs = curve[0]\n",
    "ys = curve[1]\n",
    "\n",
    "plt.scatter(xs, ys)\n",
    "for point in points: \n",
    "    plt.scatter(point[0], point[1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Gradient Vector Plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.quiver.Quiver at 0x821d40f10>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "diff = curve[:, 1:] - curve[:, :-1]\n",
    "plt.quiver(curve[0], curve[1], diff[0], diff[1], color='b')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Normalize Sampling by Subsampling"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x821e29c50>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "t = np.linspace(0, 1, 10_000)\n",
    "\n",
    "curve = bezier_curve(points, t)\n",
    "\n",
    "diff = curve[:, 1:] - curve[:, :-1]\n",
    "diff_len = np.linalg.norm(diff, axis=0)\n",
    "cum_diff_len = np.cumsum(diff_len)\n",
    "curve_len = diff_len.sum()\n",
    "\n",
    "sub_t = np.linspace(0, curve_len, 30)\n",
    "sub_idx = [np.argmin(abs(cum_diff_len - dist)) for dist in sub_t]\n",
    "sub_curve = curve[:, sub_idx]\n",
    "\n",
    "plt.scatter(sub_curve[0], sub_curve[1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Subsample statistics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The variance in point distances\n",
      "original curve: 1.34e-03\n",
      "subsampled curve: 4.00e-08\n",
      "\n",
      "The gap between smallest and largest length\n",
      "original curve: 1.15e-01\n",
      "subsampled curve: 1.01e-03\n"
     ]
    }
   ],
   "source": [
    "t = np.linspace(0, 1, 30)\n",
    "orig_curve = bezier_curve(points, t)\n",
    "\n",
    "diff1 = orig_curve[:, 1:] - orig_curve[:, :-1]\n",
    "diff2 = sub_curve[:, 1:] - sub_curve[:, :-1]\n",
    "len1 = np.linalg.norm(diff1, axis=0)\n",
    "len2 = np.linalg.norm(diff2, axis=0)\n",
    "\n",
    "print('The variance in point distances')\n",
    "print('original curve: {:.2e}'.format(len1.var()))\n",
    "print('subsampled curve: {:.2e}'.format(len2.var()))\n",
    "print()\n",
    "print('The gap between smallest and largest length')\n",
    "print('original curve: {:.2e}'.format(len1.max() - len1.min()))\n",
    "print('subsampled curve: {:.2e}'.format(len2.max() - len2.min()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Rotate Curve\n",
    "A bezier curve can be rotated by rotating the control points"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x821c65b50>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "t = np.linspace(0, 1, 30)\n",
    "curve = bezier_curve(points, t)\n",
    "\n",
    "theta = np.radians(90)\n",
    "rot_mat = np.array([[np.cos(theta), -np.sin(theta)],\n",
    "                    [np.sin(theta), np.cos(theta)]])\n",
    "\n",
    "center_of_mass = np.array(points).mean(axis=0)\n",
    "rot_points = [center_of_mass + np.matmul(rot_mat, point - center_of_mass) for point in points]\n",
    "rot_curve = bezier_curve(rot_points, t)\n",
    "\n",
    "# Cast lists to numpy arrays to ease scatter plotting\n",
    "rot_points = np.array(rot_points)\n",
    "points = np.array(points)\n",
    "\n",
    "plt.plot(curve[0], curve[1], c='g')\n",
    "plt.scatter(points[:, 0], points[:, 1], c='g')\n",
    "\n",
    "plt.plot(rot_curve[0], rot_curve[1], c='b')\n",
    "plt.scatter(rot_points[:, 0], rot_points[:, 1], color='b')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Explicit defintion\n",
    "\n",
    "$$ B(t) = \\sum^{n}_{i=0} \\binom{n}{i}(1- t)^{n-1} t^i P_{i}$$   \n",
    "    \n",
    "Or \n",
    "    \n",
    "$$ B(t) = \\sum^{n}_{i=0} b_{i,n} P_{i}$$\n",
    "where $b_{i,n}$ is the Bernstein polynomial.\n",
    "$$b_{i,n} = \\binom{n}{i}(1- t)^{n-1} t^i$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def bernstein_polynomail(i, n, t):\n",
    "    binomial = binom(n, i)\n",
    "    return binomial * ((1 - t)**(n - i))  * (t**i)\n",
    "\n",
    "def bezier_curve(points, t):\n",
    "    influences = list()\n",
    "    n = len(points) - 1\n",
    "    \n",
    "    for idx, point in enumerate(points):\n",
    "        b = bernstein_polynomail(idx, n, t)\n",
    "        point_influence = b * point\n",
    "        influences.append(point_influence)\n",
    "    \n",
    "    influences = np.array(influences)\n",
    "    curve = influences.sum(axis=0)\n",
    "    return curve"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "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": [
    "t = np.linspace(0, 1, 20)\n",
    "\n",
    "p0 = np.array([1, 0]).reshape((-1, 1))\n",
    "p1 = np.array([2, 2]).reshape((-1, 1))\n",
    "p2 = np.array([3, 0]).reshape((-1, 1))\n",
    "p3 = np.array([4, 2]).reshape((-1, 1))\n",
    "\n",
    "points = [p0, p1, p2, p3]\n",
    "curve = bezier_curve(points, t)\n",
    "\n",
    "xs = curve[0]\n",
    "ys = curve[1]\n",
    "\n",
    "plt.scatter(xs, ys)\n",
    "for point in points: \n",
    "    plt.scatter(point[0], point[1])"
   ]
  }
 ],
 "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": "3.7.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}