Graham Scan.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Point functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "#based on S. Halim's code\n",
    "import math\n",
    "import numpy as np\n",
    "EPS = 1e-6\n",
    "\n",
    "#the representation on a point\n",
    "class Point:\n",
    "    def __init__(self, _x=0, _y=0):\n",
    "        self.x = _x\n",
    "        self.y = _y\n",
    "    \n",
    "    #operator overloaded (==)\n",
    "    def __eq__(self, other_point):\n",
    "        if abs(self.x - other_point.x) < EPS and abs(self.y - other_point.y) < EPS:\n",
    "            return True\n",
    "        return False\n",
    "    \n",
    "    def get_position(self):\n",
    "        return (self.x,self.y)\n",
    "\n",
    "#the Euclidean distance\n",
    "def dist(x, y):\n",
    "    return ((x.x-y.x)**2 + (x.y-y.y)**2)**0.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p1 = Point(3,4)\n",
    "p2 = Point(3,2)\n",
    "p1.x == p2.x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Vector functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "#the representation of a vector\n",
    "class vec:\n",
    "    def __init__(self, _x, _y):\n",
    "        self.x = _x\n",
    "        self.y = _y\n",
    "    \n",
    "#multiply vector by scalar\n",
    "def scale(v, s):\n",
    "    return vec(s*v.x, s*v.y)\n",
    "        \n",
    "#point to vector\n",
    "def toVec(p1, p2):\n",
    "    return vec(p2.x-p1.x, p2.y-p1.y)\n",
    "\n",
    "#translate a point \n",
    "def translate(p, v):\n",
    "    return point(p.x+v.x, p.y+v.y)\n",
    "\n",
    "#return the dot product\n",
    "def dot(v1, v2):\n",
    "    return v1.x*v2.x + v1.y*v2.y\n",
    "\n",
    "#the square of norm\n",
    "def norm_sq(v):\n",
    "    return (v.x*v.x + v.y*v.y)\n",
    "\n",
    "#norm\n",
    "def norm(v):\n",
    "    return (v.x*v.x + v.y*v.y)**0.5\n",
    "\n",
    "#distance from p to the line ab\n",
    "#a parameter c can be added to store the closest point\n",
    "#a and b must be diferent\n",
    "def distToLine(p, a, b):\n",
    "    ap = toVec(a, p)\n",
    "    ab = toVec(a, b)\n",
    "    u = dot(ap, ab)/norm_sq(ab)\n",
    "    c = translate(a, scale(ab, u))\n",
    "    return dist(p, c)\n",
    "\n",
    "#distance form p to ab\n",
    "#a and b can be the same point\n",
    "def distToLineSegment(p, a, b):\n",
    "    ap = toVec(a, p)\n",
    "    ab = toVec(a, b)\n",
    "    #the projection of p on ab\n",
    "    u = dot(ap, ab)/norm_sq(ab)\n",
    "    if u < 0.0:\n",
    "        #c = point(a.x, a.y) used to store the closest point\n",
    "        return dist(p, a)\n",
    "    \n",
    "    if u > 1.0:\n",
    "        c = point(b.x, b.y)\n",
    "        return dist(p, b)\n",
    "    \n",
    "    return distToLine(p, a, b)\n",
    "    \n",
    "#cross product\n",
    "def cross(v1, v2):\n",
    "    return v1.x*v2.y - v2.x*v1.y\n",
    "\n",
    "#return true if the point r is on the left side of pq\n",
    "def ccw(p, q, r):\n",
    "    return cross(toVec(p, q), toVec(p, r))\n",
    "\n",
    "def angle(v1,v2):\n",
    "    v1_norm = norm(v1)\n",
    "    v2_norm = norm(v2)\n",
    "    cos = dot(v1,v2)/(v1_norm*v2_norm)\n",
    "    return np.arccos(cos)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(1, 2), (2, 1)]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = [-1,-2,3,5,0]\n",
    "sorted(a, key =lambda x: x**2, reverse=True)\n",
    "\n",
    "p0 = Point(0,0)\n",
    "p1 = Point(2,1)\n",
    "p2 = Point(1,2)\n",
    "points2 = [p1,p2]\n",
    "ccw(p0,p1,p2)\n",
    "v1 = toVec(p0,p1)\n",
    "v2 = toVec(p0,p2)\n",
    "angle(v1,v2)\n",
    "\n",
    "points2[0], points2[1] = points2[1], points2[0]\n",
    "[p.get_position() for p in points2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<function sorted(iterable, /, *, key=None, reverse=False)>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sorted"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "anim = {}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "def graham_scan(points):\n",
    "    \n",
    "    \n",
    "    #Find the lowest y-coordinate and leftmost point, called p0\n",
    "    y_min = 99999\n",
    "    mini = 0\n",
    "    for i, point in enumerate(points):\n",
    "        if point.y < y_min:\n",
    "            y_min = point.y\n",
    "            mini = i\n",
    "        elif point.y == y_min:\n",
    "            if point.x < points[mini].x:\n",
    "                mini = i\n",
    "    p0 = points[mini]\n",
    "    \n",
    "    \n",
    "    #Swap p[0] and p[i]\n",
    "    points[0], points[mini] = points[mini], points[0]\n",
    "    \n",
    "    #Define the parallel vector\n",
    "    p_para = Point(p0.x, p0.y)\n",
    "    p_para.x = p_para.x +1\n",
    "    v_para = toVec(p0, p_para)\n",
    "    \n",
    "    #Sort the points by polar angle with P0\n",
    "    def sort(p):\n",
    "        \n",
    "        v_p = toVec(p0, p)\n",
    "        return angle(v_para, v_p)\n",
    "    \n",
    "    sorted_pts = sorted(points[1:], key=sort, reverse=False)\n",
    "    \n",
    "    \n",
    "    #if several points have the same polar angle then only keep the farthest\n",
    "    for i in range(len(sorted_pts)-1):\n",
    "         remove = []\n",
    "         #If the polar angle is the same\n",
    "         if sort(sorted_pts[i]) - sort(sorted_pts[i+1]) == 0:\n",
    "\n",
    "            if sorted_pts[i].x < sort(sorted_pts[i+1]):\n",
    "                remove += i\n",
    "\n",
    "            else:\n",
    "                 remove += (i+1)\n",
    "    sorted_pts = [i for j, i in enumerate(sorted_pts) if j not in remove]\n",
    "        \n",
    "    #If there are less than or eq to 2 points then the convex hull is empty\n",
    "    if len(sorted_pts) <= 2:\n",
    "        print('Convex hull is empty')\n",
    "    \n",
    "    else:\n",
    "            \n",
    "            stack = [p0]\n",
    "\n",
    "            it = 0\n",
    "            for p in sorted_pts:\n",
    "             # pop the last point from the stack if we turn clockwise to reach this point\n",
    "                while len(stack) > 1 and ccw(stack[len(stack)-2],stack[len(stack)-1], p) <0:\n",
    "                    \n",
    "                    #clear the current plot\n",
    "                    \n",
    "                    stack.pop()\n",
    "                    anim[it] = stack.copy()\n",
    "                    it+=1\n",
    "        \n",
    "                stack.append(p)\n",
    "                anim[it] = stack.copy()\n",
    "                it+=1\n",
    "                \n",
    "                \n",
    "    stack.append(p0)\n",
    "    anim[it] = stack.copy()\n",
    "    it+=1\n",
    "    return stack"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(20, 2)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import random\n",
    "N = 20\n",
    "points = np.array([[random.randint(0, 100), random.randint(0, 100)] for i in range(N)])\n",
    "points.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "points1 = [Point(points[i][0],points[i][1]) for i in range(len(points))]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "hull = graham_scan(points1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import random\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot(points):\n",
    "    \n",
    "    X = []\n",
    "    Y = []\n",
    "    for p in points:\n",
    "        X.append(p.x)\n",
    "        Y.append(p.y)\n",
    "    \n",
    "    return X, Y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "from matplotlib.animation import FuncAnimation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "from matplotlib import animation, rc\n",
    "from IPython.display import HTML, Image\n",
    "\n",
    "# equivalent to rcParams['animation.html'] = 'html5'\n",
    "rc('animation', html='html5')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# First set up the figure, the axis, and the plot element we want to animate\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(*plot(points1), 'o')\n",
    "line, = ax.plot(*plot(anim[0]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "# initialization function: plot the background of each frame\n",
    "def init():\n",
    "    \n",
    "    X, Y = plot(points1)\n",
    "    line.set_data(X, Y)\n",
    "    return line,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "# animation function. This is called sequentially\n",
    "import time\n",
    "def animate(i):\n",
    "    X, Y = plot(anim[i % len(anim)])\n",
    "    line.set_xdata(X)\n",
    "    line.set_ydata(Y)\n",
    "    time.sleep(1)\n",
    "    return (line,)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "# call the animator. blit=True means only re-draw the parts that \n",
    "# have changed.\n",
    "ani = animation.FuncAnimation(fig, animate, init_func=init,\n",
    "                               frames=100, interval=500, blit=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
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     "execution_count": 34,
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   "outputs": [],
   "source": []
  }
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