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   "source": [
    "# The Convex Hull Problem"
   ]
  },
  {
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   "source": [
    "Pound a bunch of nails into a board, then stretch a rubber band around them and let the rubber band snap taut, like this:\n",
    "\n",
    "<img src=\"http://www.personal.kent.edu/~rmuhamma/Compgeometry/MyCG/Gifs-CompGeometry/ch2.gif\">\n",
    "\n",
    "The rubber band has traced out the *convex hull* of the set of nails. It turns out this is an important problem with applications in computer graphics, robot motion planning, geographical information systems, ethology, and other areas.\n",
    "More formally, we say that:\n",
    "\n",
    "*Given a finite set, **P**, of points in a plane, the convex hull of **P** is a polygon, **H**, such that:*\n",
    "\n",
    "- *Every point in **P** lies either on or inside of **H**.*\n",
    "- *Every vertex of **H** is a point in **P**.*\n",
    "- <i>**H**</i> *is convex: a line segment joining any two vertexes of **H** either is an edge of **H** or lies inside **H**.*\n",
    "\n",
    "\n",
    "In this notebook we develop an algorithm to find the convex hull (and show examples of how to use `matplotlib` plotting). The first thing to do is decide how we will represent the objects of interest:\n",
    "\n",
    "- **Point**: We'll define a class such that `Point(3, 4)` is a point where `p.x` is 3 and `p.y` is 4.\n",
    "- **Set of Points**: We'll use a Python set: `{Point(0,0), Point(3,4), ...}`\n",
    "- **Polygon**: We'll represent a polygon as an ordered list of vertex points.\n",
    "\n",
    "First, get the necessary imports done:"
   ]
  },
  {
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   "execution_count": 1,
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   "source": [
    "from __future__ import division, print_function\n",
    "\n",
    "%matplotlib inline \n",
    "import matplotlib.pyplot as plt\n",
    "import collections\n",
    "import random\n",
    "import math"
   ]
  },
  {
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   "source": [
    "# Points and Sets of Points\n",
    "\n",
    "I'll define the class `Point` as a named tuple of `x` and `y` coordinates, and `Points(n)` as a function that creates a set of *n* random points. \n",
    "\n",
    "There are two complications to the function `Points(n)`:\n",
    "1. A second optional argument is used to set the random seed.  This way, the same call to `Points` will return the same result each time.  That makes it easier to reproduce tests.  If you want different sets of points, just pass in different values for the seed.\n",
    "2. Since `matplotlib` plots on a 3&times;2 rectangle by default, the points will be uniformly sampled   from a 3&times;2 box (with a small border of 0.05 on each edge to prevent the points from bumping up against the edge of the box)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
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   "source": [
    "Point = collections.namedtuple('Point', 'x, y')\n",
    "\n",
    "def Points(n, seed=42):\n",
    "    \"Generate n random points within a 3 x 2 box.\"\n",
    "    random.seed((n, seed))\n",
    "    b = 0.05 # border\n",
    "    return {Point(random.uniform(b, 3-b), random.uniform(b, 2-b)) \n",
    "            for _ in range(n)}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "{Point(x=0.15172583449638682, y=1.6108693392839208),\n",
       " Point(x=0.968326330695687, y=1.3139550880088586),\n",
       " Point(x=1.3508070075242857, y=0.22290610532132638)}"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Points(3)"
   ]
  },
  {
   "cell_type": "markdown",
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   },
   "source": [
    "# Visualizing Points and Line Segments\n",
    "\n",
    "\n",
    "Now let's see how to visualize points; I'll define a function `plot_points`.  We will want to be able to see:\n",
    "- The **points** themselves. \n",
    "- Optionally, **line segments** between points. An optional `style` parameter allows you to specify whether you want lines or not, and what color they should be. This parameter uses the standard [style format](http://matplotlib.org/1.3.1/api/pyplot_api.html#matplotlib.pyplot.plot) defined by matplotlib; for example, `'r.'` means red colored dots with no lines, `'bs-'` means blue colored squares with lines between them, and `'go:'` means green colored circles with dotted lines between them.  The lines go from point to point in order; if you want the lines to close\n",
    "back from the last point to the first (to form a complete polygon), specify `closed=True`. (For that to work,\n",
    "the collection of points must be a list; with `closed=False` the collection can be any collection.)\n",
    "- Optionally, **labels** on the points that let us distinguish one from another. You get\n",
    "labels (integers from 0 to *n*) if you specify `labels=True`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
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   "source": [
    "def plot_points(points, style='r.', labels=False, closed=False): \n",
    "    \"\"\"Plot a collection of points. Optionally change the line style, label points with numbers, \n",
    "    and/or form a closed polygon by closing the line from the last point to the first.\"\"\"\n",
    "    if labels:\n",
    "        for (i, (x, y)) in enumerate(points):\n",
    "            plt.text(x, y, '  '+str(i))\n",
    "    if closed:\n",
    "        points = points + [points[0]]\n",
    "    plt.plot([p.x for p in points], [p.y for p in points], style, linewidth=2.5)\n",
    "    plt.axis('scaled'); plt.axis('off')"
   ]
  },
  {
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   },
   "source": [
    "Here's an example:"
   ]
  },
  {
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   "outputs": [
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1097305f8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_points(Points(200))"
   ]
  },
  {
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    "deletable": true,
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   },
   "source": [
    "# Convexity\n",
    "\n",
    "\n",
    "We want to make a *convex* hull, so we better have some way of determining whether a polygon is *convex*. Let's examine one that is:"
   ]
  },
  {
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   "outputs": [
    {
     "data": {
      "image/png": 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j26o+TSAwy0jXzK42sy1mttfMFrdbmvTIABNEt2Z2s5k9bGabzOxOMzs0WIUz1LcJZFaR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\n/wPYPtj1V6+qUQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109767e48>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "octagon = [Point(-10, 0), Point(-7, -7), Point(0, -10), Point(+7, -7), \n",
    "           Point(+10, 0), Point(+7, +7), Point(0, +10), Point(-7, 7)]\n",
    "plot_points(octagon, 'bs-', labels=True, closed=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "If you start at point 0 at the left and proceed in order counterclockwise around the octagon, following edges from point to point, you can see that at every vertex you are making a **left** turn.\n",
    "\n",
    "Now let's consider a non-convex polygon:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1097c7b00>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pacman = octagon[:4] + [Point(0, 0)] + octagon[5:]\n",
    "plot_points(pacman, 'ys-', labels=True, closed=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "The `pacman` polygon is non-convex; you can see that a line from point 3 to point 5 passes *outside* the polygon. You can also see that as you move counterclockwise from 3 to 4 to 5 you turn **right** at 4.  That leads to the idea: **a polygon is convex if there are no right turns** as we go around the polygon counterclockwise.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "# Turn Directions\n",
    "\n",
    "\n",
    "Now how do we determine if a turn from point A to B to C is a left turn at B or a right turn (or straight)? Consider this diagram:\n",
    "\n",
    "<img src=\"http://norvig.com/convexhull.jpg\">\n",
    "\n",
    "It is a left turn at B if angle &beta;  is bigger than angle &alpha;; in other words, if &beta;'s opposite-over-adjacent ratio is bigger than &alpha;'s: \n",
    "\n",
    "    (C.y - B.y) / (C.x - B.x)   >   (B.y - A.y) / (B.x - A.x)\n",
    "    \n",
    "But if we did that computation, we'd need special cases for when each denominator is zero.  So multiply each side by the denominators:\n",
    "\n",
    "    (B.x - A.x) * (C.y - B.y)   >   (B.y - A.y) * (C.x - B.x) \n",
    "    \n",
    "(*Note:* This step should make you very nervous! In general, multiplying both sides of an inequality by a negative number reverses the inequality, and here the denominators might be negative. In this case it works out; basically because we are doing two multiplications so that negatives cancel out, but [the math proof](https://en.wikipedia.org/wiki/Cross_product) is tricky, involving some concepts in vector algebra, so I won't duplicate it here; instead I will provide good test coverage below.)\n",
    "    \n",
    "That leads to the function definition: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [],
   "source": [
    "def turn(A, B, C):\n",
    "    \"Is the turn from A->B->C a 'right', 'left', or 'straight' turn?\"\n",
    "    diff = (B.x - A.x) * (C.y - B.y)  -  (B.y - A.y) * (C.x - B.x) \n",
    "    return ('right' if diff < 0 else\n",
    "            'left'  if diff > 0 else\n",
    "            'straight')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "# Sketch of Convex Hull Algorithm\n",
    "\n",
    "\n",
    "Now we have the first part of a strategy to find the convex hull:\n",
    "\n",
    "> *Travel a path along the points in some order. (It is not yet clear exactly what order.) Any point along the way that does not mark a left-hand turn is not part of the hull.*\n",
    "\n",
    "What's a good order? Let's see what happens if we start at the leftmost point and work our way to the rightmost.  We can achieve that ordering by calling the built-in function `sorted` on the points (since points are tuples, `sorted` sorts them lexicographically: first by their first component, `x`, and if there are ties, next by their `y` component). We start with 11 random points, and I will define a function to help plot the partial hull as we go:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [],
   "source": [
    "def plot_partial_hull(points, hull_indexes=()):\n",
    "    \"Plot the points, labeled, with a blue line for the points named by indexes.\"\n",
    "    plot_points(points, labels=True)\n",
    "    plot_points([points[i] for i in hull_indexes], 'bs-')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "run_control": {}
   },
   "source": [
    "Here are the points without any hull:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXcAAADsCAYAAACPFubKAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nS1JDxl2SGjLuktTQ/wNv4YPlAsECOQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109717588>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "Now I will start building up the hull by following the points in order from point 0 to 1 to 2 to 3:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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GlvtN4dtt51Ms9QBeTMwyc7j3uQgGKfYCXZFOfVpin4wldVI3NuoYVwT3Av+Wmof3+Tr6\nlpnD3YjgIYoJTk9S/E5cVJMRHz0N96SxHMFs4I96+LlmIzjcDRhaBOvPUnNziglOG2YsqRNyhPvl\nwMPp2F0zlo3D3YZEMAB8NTVfT7HgWJX1PNwjeBK4KDX3l3hRrz7bbDiHu412EsWwPoCTJA7PWUyb\nurXF3mQaXTPrA3/S4882AxzuNkq68jyE5nT6syVembGkduTolgG4Dvh5Ol5Qs+GlVhEOd1tHBCsp\nrjiDYg/SJRKzshY1PVnCPQ27bFy9vwr6bntDKwGHu40pgu8Bp6bmDsDXKngF2pih2usrd4DzYGhs\n/YIMn299zuFuEzkV+F46PgT4i4y1TEeubpnG8g7fT80jajDyyCrG4W7jShOcjgTuSqf+QWLPjCW1\nLO0TOzM1c1y5Q3M5gk2BAzLV0Pck/VDSTenrHkmX5a6pFxzuNqEIHqS4al9DMfpjscQWeatqyfAd\nkXKF+2XA6nS8IFMNfS8i9oqIuRExF/gRsCR3Tb3gcLdJRXADcGJqvgi4WGKDjCW1opfL/Y4pgseB\nS1LzbRKb5ajDCpJmAfvS3Cy+1hzu1qqzaXYz7Emx4FiZZQ/3ZCA9zgCOyFiHFVsgXh0RvZ73kIXD\n3VqShvedANySTv2lxCEZS5pMWcL9GpqLsnk5grwOp7ljVu053K1lETxBscBYY+2UcyR2yFjSRHqy\nxd5kRo15n1vTHa9KT9ILgN0p9rvtCw53m5IIfkFzG7nnAP8qjdjxqCzKcuUOcO6wY1+953EI8O2I\n6OqmLWXicLcpi+DbwKdTc0dgYQknOPVyi70JpRm/S1PzyArcjK4WaSHSUqQrkMbbKnI+fdQlAw53\nm76PA1en48NpjqYpizJduUOza2Zz4C05C6mh7YG9gbcCZ431hIjYJyKu6mlVmTncbVoieIZi9Mfd\n6dQXJV6fsaTRyhbul9Ksw10znfV4elwGHJ+zkDJxuNu0RXA/RV/mWoqhfpdIbJ63qiGlCvcIHqU5\neeZAiefnrKdmjgAWA/sRsSp3MWXhcLe2RHA98IHUnEOxRd+MjCU1lCrck0bXzIbAYTkLqZWIVUTM\nd7CP5HC3TvhnYFE63gf4VMZaGhrhPgg8lbOQYZYCv0rHC/KVYf3A4W5tS2O5jwd+kk59ROIdGUuC\nYStCpvqySwuxNYZF7i6xY856rN4c7tYRETxGMcHpkXTqXIltM5aUbbnfSXxj2LFvrFrXONytYyL4\nOc3uhlkUOzjNHP8VXZVr/9QJRXAncG1qHimxfs56rL4c7tZREVwGfC41dwL+JdMEp7JeuUNzMbEX\nA2/KWIfVmMPduuFjNGdkvos8Y49zbrE3mUuAJ9Pxgox1WI053K3jIniaYqjfvenUlyVe2+MySnvl\nHsHDwL+m5jslxpsybzZtDnfrigh+AxwKPE0xrvtSiRf0sITShnvSuLH6LGBezkLqSNJpkm6XdJuk\n9+WuJweHu3VNBNcCH0zNrYALengDsezh/n3gnnTsUTMdJOkYYE5E7BAROwIX5a4pB4e7ddsZwMXp\n+M3AJ3r0uaUO97Q2z3mpuYfE9jnrqZk/BU5tNCLi/oy1ZONwt65KE4jeA/w0nfobibf14KNLHe7J\n8DHvR2Wron5eDhwmaZmkKyTlnG+RjcPdui4tmnUwzTHn50ts3a3PS0MvG+PrSxvuEfwMuD41j5L8\n/2OHbAQ8ERG7A18Fvp65niz8y2Q9EcFPgXen5myKCU4bd+njNoahsfWlDfekcfW+FcW6PNa+u2mO\nRvom9OfWhg5365kILgFOT825wD926aNKsX9qiy4CGlu/+cZqZ3wT2Dcd7w3cnrGWbBzu1msfBq5J\nx8dKHNuFzyjNFnuTieAh4PLUPFhik5z1VMLk2+r9PXCwpOXAaRT3fPqOw916KoK1FPtZ/iad+orE\nrh3+mDKu5T6RgfQ4k2LzE5vYhNvqRcTDEfH2iNgpIv4gIm7teYUl4HC3novg/ygC/hmKm19LOrwz\nUdXC/SrgvnS8IGMdVeFt9VrgcLcsIvgB8JHUfBlwXgdHi1Qq3NNyDY3NTvaS2CZnPRXgbfVa4HC3\nnL5Ac1TDAcBfdeh9KxXuice8t8rb6rXE4W7ZpAlOxwB3pFN/K/GWDrx15cI9gluBH6emx7xb2/wL\nZFlF8AhwEEU/qijWn3lpm29buXBPBtLj1sCeGeuwGnC4W3YR/AR4b2o+H7hEYqM23rKq4X4hsDYd\ne8y7tcXhbqUQwQXAP6Xm7sCX2ni7Kk1iGhLBA8B3UvNQacR/h9mUONytTP4SuC4d/6k07RuLjVAc\nBJ5qu6reatxYfQ5Fd5XZtDjcrTQiWEOxwccD6dRZEjtP462GtthLN22r5AqgsUStu2Zs2hzuVioR\n3E2xRd8gxS5FS6axDV0VlvsdU5rBe0Fq7ivxkpz1WHU53K10Ivg+8Nep+XLgG1McGljZcE8aXTOi\n2GDcbMoc7lZWnwW+nY4PBD40hddWPdxvBpan46PT+vRmU+Jwt1KKYJBipuYv0qnTJN7Y4ssrHe7p\nPkHj6n074PUZy7GKcrhbaUWwimIHpycpflcvlNiyhZdWOtyTRRQLq4FvrNo0ONyt1CK4hWLDY4DN\nKCY4bTjJyyof7hH8BrgyNQ/r4q5VVlMOdyu9CL5Bc93u1wGfn+QllQ/3ZCA9zgLeMdmTJX1N0s2S\nlku6TNJzu1qdlZrD3ariA8AN6fh9EodP8NxGuFdmduo4vgM8lI4XtPD8D0TELhGxE7ACeF+3CrPy\nc7hbJUTwJMUuRb9Lp86WeNU4Tx+axNT1wroogqdojnl/s8SciZ8fqwEkiWKT8Acmen4nSBqQtELS\nTemrLzejLiOHu1VGBHdRbNQQFFvSLZGYNfw5adjgzNSsdLgnjVEz6wFHTvZkSecA9wI7AWd3sa6G\nAD4YEXPT1/JJX2E94XC3Song34BPpub2wDmjxoFvDEPtOoT7DcBP0/GkY94j4hjgxRTj5Du1+clk\nPA6/hBzuVkV/R3MkyUHAycO+N+lyv5JOlHSnpEFJndy7teOKMe8LHi/+Pfvk78ExN0jzlhZfuw2M\n/ZoYBC6iWF2zFz4j6RZJX5Q02Ugm65EZuQswm6oIBiWOBG6k2H/1sxLLIvghra3lfg3F7Nel3ayz\nc9auaf6xwq7N8/NGPEvSthFxZ+pzPxC4qQfFfTQi7kuhvhD4MPCpHnyuTcJX7lZJEfyO4gbrU8D6\nwGKJLWgh3CPi5oi4q/tVdsraNZM9IwX6gKTlwC0Um558utuVRcR96XENcA7w2m5/prXG4W6VFcGN\nwImp+UJgMYxYQbIOfe4ticIbImKn9PXuiHii7TeWFiItRboCaZ3VOSVtkR4F/DFwa9ufaR3hcLdK\ni+Bs4Oup+QZGTnDqm3Dvou2BvYG30pxINtz56a+F5RR/LfxdD2uzCbjP3ergRGAufHIuxQzW5K7P\nS4+thhUrI25YkKm2qns8PS4Djh/9zYhodTE36zGHu1VeBE9IHAKDt8Opw3+n083HeWO+LqnAML4V\nK8f+b1ixsgcffgTFFfvxRKzqwedZhzjcrRYiWCHd+zMYd9bqEEknAadQ9NMvl/TdiDiu2zVOV9a/\nOopAn5/t823aHO5WIw8/2MqzIuIM4IwuF2OWlW+ompnVkMPdzKyG3C1jNZL1xqNZqSgictdgZmYd\n5m4ZM7MacribmdWQw93MrIYc7mZmNeRwNzOrIYe7mVkNOdzNzGrI4W5mVkMOdzOzGnK4m5nVkMPd\nzKyGHO5mZjXkcDczqyGHu5lZDTnczcxqyOFuZlZDDnczsxpyuJuZ1ZDD3cyshhzuZmY15HA3M6sh\nh7uZWQ053M3MasjhbmZWQw53M7MacribmdWQw93MrIYc7mZmbZK0r6QbJd0qaUDS+rlrcribmbVB\n0nrAADA/Il4N3AUcnbUoHO5mZu3aFFgTEXem9tXAwRnrARzuZmbtegCYIek1qX0IsFXGegCHu5lZ\nWyIigMOA0yVdDzwCPJO3KpiRuwAzs6qLiOuAvQAk7Qdsl7ciX7mbmbVN0mbpcSPgQ8C/5K3I4W5m\nNjlpIdJSpCuQZo/xjFMk3QbcAnwrIpb2tsB1qeguMjOzcUlLgb1TazER8zNW0xJfuZuZTe7x9LgM\nOD5nIa3ylbuZ2WSKrpizgOOJWJW7nFY43M3MasjdMmZmNeRwNzOrIYe7mVkNOdzNzGrI4W5mVkMO\ndzOzGnK4m5nVkMPdzKyGHO5mZjXkcDczqyGHu5lZDTnczcxqyOFuZlZDDnczsxpyuJuZ1ZDD3cys\nhhzuZmY15HA3M6shh7uZWQ053M3MasjhbmZWQw53M7MacribmdXQ/wM9dMLn+D4sogAAAABJRU5E\nrkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109b09c18>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [0, 1, 2, 3])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "We see that we made a valid left turn at point 1, but a right turn at 2. So we remove point 2 from the hull:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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i+8AB5fX+MJh7/Hrmrq6KYA3gF8DTy6E3ZPLVipHUbyJmAjvQbNIxbdlnuEfE+jRPKd2G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\nzF1xAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10976d9b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [0, 1, 3])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "We move on to points 4 and 5:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x109c5fa90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [0, 1, 3, 4, 5])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "Point 4 is a right turn, so we remove it:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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jPWAw9/j1zF1dFcGywM+BZ5dDb8zklIqR1G8ipgKb0mzScfCC93CPiEk0dyldn2YT97dk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\npBb6f8UkB+QYCHlkAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109e24f28>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [0, 1, 3, 5])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "But now we see point 3 is also a right turn.  The addition of one new point (5) can remove multiple points (4 and 3) from the hull. We remove 3 and move on to 6, 7, and 8:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x109ea9a90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [0, 1, 5, 6, 7, 8])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "Point 7 is a right turn so we remove 7 and move on to 9:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x109800860>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [0, 1, 5, 6, 8, 9])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "Point 8 is a right turn, so we remove 8.  But then 6 and 5 are also right turns, so they too are removed. We proceed on to 10:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x109f58ac8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [0, 1, 9, 10])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "Now what do we do? We got all the way to the end of our set of 11 points, but we only got half the hull (the lower half).  Well, if looking at all the points in left-to-right order gives us the lower half of the hull, maybe looking at all the points in right-to-left order will give us the upper half.  Let's try. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXcAAADsCAYAAACPFubKAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nIMNdkhr0/9GWHng179/7AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1099b2470>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [10, 9, 8])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "Point 9 is a right turn; remove it and move on:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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R+2fmXc3/XF4JfG/ua9BUGO7S8Hioed0ArC5TwgP3j5u4JCKWAkHnvuzv6H9N6sVwl4bH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\nf7se1IjE6GWnAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109b09518>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [10, 8, 7, 6, 5])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "Adding 5 reveals 6, and then 7, to be right turns; remove them and move on to 4:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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53M1qq8vb7n2a6klSgPPS0sPWQQ53s3prhPvrJbbsxAkktgDOSd1FwJWdOM/q59QFkh6U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\nmVmBHO5mZgX6f4EBFStKeGtdAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109f72eb8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [10, 8, 5, 4])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "Remove 5 and continue on to 3 and then 2:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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D5XET4PCKdbRSRJwUETdExPUR8aHa9dRguEt1XATcUfbtmplHEXEssG1m7paZewBn1a6p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\nYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109ad7d68>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [10, 8, 4, 3, 2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "Now 3 is a right turn; remove it and continue on to 1 and finally 0:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10619ac50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [10, 8, 4, 2, 1 ,0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "Adding 0 makes,  1, and then  2 be right turns, so they are removed:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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1yHCXpAYZ7pLUIMNdkhpkuEtSgwx3SWqQ4S5JDTLcJalBhrskNchwl6QGGe6StEJJzkty\nQ5JbklyV5Oi+azLcJWkFkhwFXAVsraqXAHcBb+21KAx3SVqpk4DHq+r27vmXgV/ssR7AcJeklbof\nWJfkld3zNwHP7rEewHCXpBWpqgLeDPx5kl3Aw8CT/VYF6/ouQJLWuqq6Bng1QJLXAWf1W5Etd0la\nsSSbusf1wO8DH++3IsNdkpaWbCfZSbKDZMM8n/i9JF8HbgY+X1U7x1vgXBl0F0mSFpTsBM7tnl1N\n1dYeqxmKLXdJWtr+7vE6YFufhQzLlrskLWXQFXMlsI2qfX2XMwzDXZIaZLeMJDXIcJekBhnuktQg\nw12SGmS4S1KDDHdJapDhLkkNMtwlqUGGuyQ1yHCXpAYZ7pLUIMNdkhpkuEtSgwx3SWqQ4S5JDTLc\nJalBhrskNchwl6QGGe6S1CDDXZIaZLhLUoMMd0lqkOEuSQ36fxAPnXjKQDLwAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a122dd8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [10, 8, 4, 0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "Let's bring back the lower hull and concatenate it with the upper hull:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a12af98>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_partial_hull(sorted(Points(11)), [0, 1, 9, 10] + [10, 8, 4, 0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "That's all there is to the basic idea of the algorithm, but there are a few edge cases to worry about: \n",
    "\n",
    "* **Degenerate polygons**: What happens when there are only 1 or 2 (or zero) points? Such a set of points should be considered convex because there is no way to draw a line segment that goes outside the points.\n",
    "\n",
    "* **Colinear points:** if three or more points are colinear, we should keep only the two \"outside\" ones.  The rationale for not keeping them all is that we want the convex hull to be the minimal possible set of points. We need to keep the outside ones because they mark true corners in the hull. We can achieve this by rejecting a point when it is a \"straight\" turn as well as when it is a \"right\" turn.\n",
    "\n",
    "* **First and last points:** An astute reader might have noticed that our algorithm only rejects the middle point, point B, in the A->B->C turn. That means that the first and last point in sorted order will never be a candidate for rejection, and thus will always end up on the hull.  Is that correct?  Yes it is. The first point is the leftmost point, the one with lowest `x` value (and if there are ties, it is the lowest-leftmost point). That is an extreme corner, so it should always be on the hull. A similar argument holds for the last point in sorted order.\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "# Implementation of Convex Hull Algorithm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [],
   "source": [
    "def convex_hull(points):\n",
    "    \"Find the convex hull of a set of points.\"\n",
    "    if len(points) <= 3:\n",
    "        return points\n",
    "    # Find the two half-hulls and append them, but don't repeat first and last points\n",
    "    upper = half_hull(sorted(points))\n",
    "    lower = half_hull(reversed(sorted(points)))\n",
    "    return upper + lower[1:-1]\n",
    "\n",
    "def half_hull(sorted_points):\n",
    "    \"Return the half-hull from following points in sorted order.\"\n",
    "    # Add each point C in order; remove previous point B if A->B-C is not a left turn.\n",
    "    hull = []\n",
    "    for C in sorted_points:\n",
    "        # if A->B->C is not a left turn ...\n",
    "        while len(hull) >= 2 and turn(hull[-2], hull[-1], C) != 'left':\n",
    "            hull.pop() # ... then remove B from hull.\n",
    "        hull.append(C)\n",
    "    return hull"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can try it out on our 11 random points, but it is not easy to tell at a glance whether the answer is correct:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Point(x=0.3253748207631174, y=1.7900592822602743),\n",
       " Point(x=1.3968712854329428, y=0.4086086096198411),\n",
       " Point(x=2.7310024878562857, y=0.05565070635109892),\n",
       " Point(x=2.835445111495586, y=1.375183795456248),\n",
       " Point(x=2.7309330192147097, y=1.4818235191572668),\n",
       " Point(x=1.8813296288048804, y=1.666404092312656)]"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "convex_hull(Points(11))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "run_control": {}
   },
   "source": [
    "# Visualization of Results\n",
    "\n",
    "To visualize the results of the algorithm, I'll define a function to call `convex_hull` and plot the results: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [],
   "source": [
    "def plot_convex_hull(points):\n",
    "    \"Find the convex hull of these points, and show a plot.\"\n",
    "    hull = convex_hull(points)\n",
    "    plot_points(points)\n",
    "    plot_points(hull, 'bs-', closed=True)\n",
    "    print(len(hull), 'of', len(points), 'points on hull')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "6 of 11 points on hull\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a251828>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_convex_hull(Points(11))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now the octagon and pacman shapes:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "8 of 8 points on hull\n"
     ]
    },
    {
     "data": {
      "image/png": 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15MpcL8ZnUGTOtWSV60U6BkUhprqWuF7kwaAo0BTWEteLvBgUBatxLXG9yJNBUYHS1xLX\ni/wZFJUpaS1xvSiHQVGpnNcS14vyGBSVy2Utaa0XB4GldXe7XmTOoJiQFGvJbL34aeBtuF4Uy6CY\noDHWEteLuhgUE9b3WuJ6US+DQsBia4nrRf0MCq2x+Vpy8DE4+zicO9O861lXw77r4DnXwq/ubH2s\n60VlDAptaOO15PDsz3rPvN/1olI7Uh9AeYqRCzHyQIy8GvhO4ANwpZ8q574MfB/w8hj5bUOiPgaF\nOsXIQzFyJ3zmoxt/xH/8c4z8pdcg6uXqIamTE4WkTgaFpE4GhaROBoWkTgaFpE4GhaROBoWkTgaF\npE4GhaROBoWkTgaFpE4GhaROBoWkTgaFpE4GhaROBoWkTgaFpE4GhaROBoWkTv8PLG2oD7QJbi4A\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a2518d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_convex_hull(octagon)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "7 of 8 points on hull\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPEAAAEACAYAAABicpABAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nWMqcIZYyZ4ilzP0fPv/IpjzqE7IAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a122550>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_convex_hull(pacman)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "How about 100 random points?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "15 of 100 points on hull\n"
     ]
    },
    {
     "data": {
      "image/png": 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6t8zIMjh1GszcJtxfsxJeXBu2Z6XXov0i6Q3SZajuxZw0O9zd75xrxky6lfHn3flUirZM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\ndxGRBlK4i4g0kMJdRKSBFO4iIg30/wEIQzOQvuK0XQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109749eb8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_convex_hull(Points(100))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "Will 10,000 points be slow? "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 loops, best of 3: 396 ms per loop\n"
     ]
    }
   ],
   "source": [
    "P10K = Points(10000)\n",
    "\n",
    "%timeit convex_hull(P10K)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "No problem! Still well under a second! Here's what it looks like:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "27 of 10000 points on hull\n"
     ]
    },
    {
     "data": {
      "image/png": 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vm9MPEXakcuZAZSgsdTPyJsZZp/ZGUaJLFGLLDGNc1iE2BXav2dYEgPSe90u1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VQJLcwZ1RLKQ4T31zSNkxFnLBhpbQClv/3IoX/uo5hBc4irfZzTfzBaqhP0P4o8raGzWFiLTJFj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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1097d1438>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_convex_hull(P10K)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "How about a non-random set? Here is a set of coordinates of 80 US cities:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "9 of 80 points on hull\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x109942b38>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "P = Point\n",
    "\n",
    "USA = {\n",
    " P(-621, 289), P(-614, 297), P(-613, 319), P(-613, 342), P(-612, 263), \n",
    " P(-612, 332), P(-603, 247), P(-599, 277), P(-592, 238), P(-591, 323), \n",
    " P(-586, 229), P(-581, 289), P(-581, 305), P(-576, 253), P(-568, 260), \n",
    " P(-563, 322), P(-560, 234), P(-560, 285), P(-559, 292), P(-558, 246),\n",
    " P(-557, 259), P(-555, 225), P(-549, 271), P(-543, 321), P(-535, 313), \n",
    " P(-530, 249), P(-524, 278), P(-524, 288), P(-515, 308), P(-505, 206), \n",
    " P(-504, 327), P(-492, 207), P(-488, 194), P(-488, 248), P(-487, 264), \n",
    " P(-484, 305), P(-484, 328), P(-482, 297), P(-480, 289), P(-477, 210), \n",
    " P(-470, 319), P(-468, 291), P(-462, 247), P(-461, 328), P(-452, 271), \n",
    " P(-450, 210), P(-450, 226), P(-450, 245), P(-441, 311), P(-440, 301), \n",
    " P(-438, 233), P(-438, 293), P(-431, 278), P(-425, 266), P(-423, 273),\n",
    " P(-422, 213), P(-422, 236), P(-420, 251), P(-415, 297), P(-413, 196), \n",
    " P(-409, 214), P(-409, 290), P(-401, 181), P(-401, 253), P(-400, 230), \n",
    " P(-400, 282), P(-394, 251), P(-394, 301), P(-387, 263), P(-385, 272), \n",
    " P(-371, 285), P(-370, 285), P(-369, 299), P(-363, 309), P(-357, 292), \n",
    " P(-355, 297), P(-352, 306), P(-344, 314), P(-340, 328), P(-608, 270)\n",
    " }\n",
    "\n",
    "plot_convex_hull(USA)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "run_control": {}
   },
   "source": [
    "A decidedly non-random set of points:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4 of 100 points on hull\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQEAAAEACAYAAACzsMNYAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nRADCiQCEEwEIJwIQTgQg3B+zYwJ3yU0vQgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109f89ba8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "grid = {Point(x+0.5, y+0.5) \n",
    "        for x in range(10) for y in range(10)}\n",
    "\n",
    "plot_convex_hull(grid)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "run_control": {}
   },
   "source": [
    "A variant with some noise thrown in:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "13 of 100 points on hull\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQEAAAEACAYAAACzsMNYAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nAUdeJYlnKFNNAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10994eda0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def noisy(points, d=0.3, seed=42): \n",
    "    \"Add some uniform noise to each of the points.\"\n",
    "    random.seed(seed)\n",
    "    def noise(): return random.uniform(-d, +d)\n",
    "    return {Point(x + noise(), y + noise())\n",
    "            for (x, y) in points}\n",
    "\n",
    "plot_convex_hull(noisy(grid))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Circles and donuts:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "28 of 790 points on hull\n"
     ]
    },
    {
     "data": {
      "image/png": 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A5khaUyam85pNB4TymMnMfEaV43R0s3+1779c5RWvJqbn7CuzsrnLY+siWbA6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\nbFBTwNAAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a235470>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "square = {Point(random.uniform(-1, 1), random.uniform(-1, 1)) for _ in range(1000)}\n",
    "circle = {p for p in square if p.x ** 2 + p.y ** 2 < 1}\n",
    "plot_convex_hull(circle)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "28 of 628 points on hull\n"
     ]
    },
    {
     "data": {
      "image/png": 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zcifA2d73Z68/OSNbvbEn5bY81fd5nEOYwbpBZmhz+qra504UT7YZ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\nFIITHg4ODoXghIeDg0MhOOHh4OBQCP8f+brBN4kaw2oAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109f4e7f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "donut = {p for p in square if 0.2 < (p.x ** 2 + p.y ** 2) < 1}\n",
    "plot_convex_hull(donut)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "19 of 720 points on hull\n"
     ]
    },
    {
     "data": {
      "image/png": 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e5Wi2B/dQ+SLwHf/6bOBOXy+3KWj5lEvn+C7wvvFs69nKxJE+gGwOKmQWo1Xf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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109d67d68>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def rad(degrees): return degrees * math.pi / 180.0\n",
    "\n",
    "sine = {Point(rad(d), 5 * math.sin(rad(d))) for d in range(720)}\n",
    "plot_convex_hull(noisy(sine))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "38 of 360 points on hull\n"
     ]
    },
    {
     "data": {
      "image/png": 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5jK83wAUa9QgwGpMCzO0bsQSTDlyGCEL3MWr/A4DCXD9FJLh+lOKrwL3W4o1asyiFPe0W\nmTgcmIslAC7luU0adSexKcDsmG9vnQAJBhK6h1H7r0LrK7FmBEhw/dTRb9Jwdh0BULT/UCl6p7Sv\n3SCjhIBSDMF4BgLseYhrqzF/kF08dA2mAKXb2i9TgMLx0On104/D437Kf/QC0GSNxBSnDRQZJQQw\nsQFDrPdfH8beQ9b79ZhpwbeBh3CSRIBMAQrHR1fXT8M1PMqJlNsa6PeUYkDqutc1GWMTUIoLgf+1\nFp9pI6sqCz0BE/k13ZrCqcJYdQGWWeqcICQPa9pwNNufeZ/RD1lrf6g1P0pnt9xkhBCwxllvAWOB\nw8AEjXoYt7uvMeDMwzEMLsUkjRCEnsXxI+iFKYlWqtCHMEOGMzHhyidqTXX6OumQKcOBOzACAOD7\nWrMd/0KhtgA4CFyX0h4KUcL2I3AnH2kHvm9t7w98J019iyO0ocROHcHefeFEq3rwkTr43zPNfZ+w\nUOgB4EwZ/wtJxJ223B2f8g/45h7oPwz0N5Xach60NJtN6UtKGlohkKCOYH/YaoKF4gtPeOPCBaHn\niE0+exPwG4wfQUc9Cq3RSh3cBb8ahtHCXdGs6UtKGmIhcJR0XZpaEI4Hd+WiBYntTfW1qepQdwmx\nTSA7OgJMCAOh9TcJsRAYfmK6eyAILkLrbxLKp6kpA1UwMt39EIQOOhtuuuwF0/jI1FdT2a9uEDoh\nYAUIWa7B84Fta6HxsLNHeYVUCBICRoe94GS26nwrKel6Juk6ChW0t0PVrrT1Lt1JDo+2gf6knTQU\n9H1WosfYZKDmvb3TE+nus7SINycR6V7Xdan/TOka1+KP0tW/UHkMKkU+8G+guC/1R97jpLUj2H0I\nGIBTB24JxhljFsYz6xWgFNEGhHThZBwahateYTtqaTbto4CpGI/WMVqT8tmDsBkGb8GqAnMbC3aM\nYPe5xFaGsS2zczHlpPtjeWylvquCYGFnHDLZiG02ZKGvA35sLRcAX0l53whR7IBSjAQ2A32L2dqw\nifGHc2kdirnx52BSPDmOQE72lzWE0GIrZCCxyUjmoXWNUmQBbwCnAfuBIq053MlZepwwaQI/AfoC\nLOTGPpYA2IG5wSuJLwu2F6MNyM0vBAN3MhLHi7AdRxsYAnwp1d0KhSagFFOBVwHOZ+WelVwwjK6e\n8FI0RAgC3ZipUopsTK6LEqAKKNaaI6nqYuA1AUtd+p212JxP0yz8nDLc1YSN2hVaDy4ho+gyfZ3W\ntAE/sxZHAtempmuGQGoCToQgwKBhMHy8eb9zo9b3n+ba0S1lvTMENyIBQ0K66co2ZV3DH+OkccuZ\nMwRye0FrE2x+DdCpiC4MqLOQb4Qg8GnvzewO2rCdLcyTXwKGhGDQVfRqCTDzMCNxTAPkA+ebt8mP\nLgz8cCCW9raOt0aC2lrBekxYZih9t4UMxlWQJMEeDQCHyalLYa9iCJkQiKEEk7kF69WbQFQQwsBc\nYMkGCt9MVwfCLATchr+dSO0AIcjEG65tFgAzplI9LV1dC7MQcEI3HU8smQkQgop7lmCzSxiUACNy\n0GmzzwXUMFheAde6cgfuegcO7jXrLdyGP6XmAmsxFWIfQymJHBSChq251gFDcbTWBoAiKmjgwvrX\nGbKxlVPONrtWV8Rc80kikFOEAEpxKaamIMD5WrOqiwPKEOcgIag4QUQDMfEsZsrQ8ABwBmZYeyib\n1gntZI8B/kdrZie7a0EeDnzA9X5nN/YX5yAhuDhBRKW4Z7FsV2KoxPi5zDqHl2wNfVIquhZkITDK\n9b6q411iA0to0zsJESLxlGHHQ+wtTnvEel+cipJlYRAC+7XuqCQMidwwu56PFYRgYnxeBmAc3ubU\nUviKa+vEZH98kIWAPRzwDgXcan9jAq1AEIJLvDZbghkKjMBMGW5w7X16srsTZCFgawLve9a7pwaL\nEP8AIXx4tVmvPasSOjIMJd0uEAYhEKsJxKr9YgwUwoh93e7DXOc5wFIse5ZJS9ihDURTE7ByCQ6x\nFt93bbDVqO0otRrz4y1DjIFCuLC12c2YYcDFQIsrK9aia3jEKrCrT1Mquf48gRQCmJhqG7cmYKtR\no3F+vGYRAEKoiM85aDQCl43gw/zL0oRVPnBSMrsTVCGQyEfAVqPs8ZL3xxOEMOHVCDpsBKfH2AaT\naxcIqhBw+wi4DYP2j3a69dqO8+M9mLLeCUJPEK8RdGTLPpl3nwRth84n1S4QOLdhk1Xo7BkwxEoj\n/u6L0Nbqm2FFqWqccOKlJKwEKwgBxnEpjkk8ohQbgVOB57RmVrI+PoABRMVFcF+xa4WVMszKsBKb\nUmwD8GFMUpHrUtdHQehBfLJgmYfhzKEmRUbbR5R6t8xs6fl0YwEUAl3iTim2DDufoBgHhbAR+0Db\ni/F7MVmJmVMEvzrB2jOPjmu+59ONhVEIuH0D5snNL4QY9wNtHybEGFLs+BZUw2AcubTbAksChYRM\nwR721gIbrfcpd3wLjRAYz6ESINZjMHFEoSCEge3WawFGEKTl4RbA4cD2HTAfgNFUtBVTkd2GatvI\nwZU+O7vVqYVIinEhXLidhYYSW7A0ZQRQCLxyB3ANwNf51rZv8dA4IBvjE+BFYgeEMGPXJHCXLLce\nZuUVxgg4YSbAcHY0V1O7poWKip7uRACFAMPsN0PYb6tFiW7yrgo7CEJwsacGTZUicIXHa2iAdZ9U\ntO8A1a+euj816/5JKV0eRJtAhxC4l699m87GSZJIRMgM5gJbMYlyryAmzFjVAdTTPy9ZHx4oj0Gl\nJm+BkpHQq7dZ03AItIby/VqvG9edCq+CEEpiE+WClYhUoV/FXPNLtE6OzStgw4HiIfB4b9cKK7/a\nHHtZDIFCpmLbt9ZjkopcZ2bAsMuT9UvWBwdxONAZYggUMhXb/+UtTDzMY9a0ty0EklZiL2CaQJeI\nIVDITBwjYRUm1yCYyNikC4FQaAJ9ae2LUoWu0MsFKFWFUtUotUIchYQMYpDrfS5Qb72PthCw6rS5\n/alLMNJyECa7kCQZFTKFBtf7ZqI3HCjfD6UK+gwAyKNR9+aIGswWgBM6tIHYH2o9Yh8QMoe1mAeb\nHR7/A2t90oRAoKYIbZRiHzDkU/xlz18oHebaZGoMGvX/AUAhkYRCmPFOexsWAjcqJt8D02fC4CKz\n+u0XAN3TOQUCpgl0sAUYsoVxua51zhPf3PSSRUjIBGKnvY3Ny6q2XVoE9xa59j3fvPRsToGg2gTe\nA9jA6U2YfOzLgAtjnvgSQShkBmmf9g6yJoAma6RCX6N1jA3AVqFKMSGYII5DQnhJ+7R3oDUBi3E+\n20twBMABxDAohBVz49cCy7xa7QRqS1LRhUBrAhbjgDc9223N4ABwphgGhZDjtgusRan3gYZeXNw3\nFR8eVCHQoQncxs9/gfru9Ri1aQHmB2vG2AlkZkDIBNxpxvZhCYQC6o/YCXay2PluO7W7zW7lFT35\n4YEUAlpToxT7gSEHGFSM+ZEWYsKMbYm5RASAkCFsx5TWKwDGWOvqxnJdTRnXjwbd3o46V2uqk/Hh\nQbUJgDUkeM+UYbMtp2m3pApCEnBXIJqO0Qb6v8bU0Wa1WpMsAQABdRYCUIqHgc8NZ1fbLka+gvmh\nbsIMCSSASMgcXBWIFJPvmcTAK/JRg17hHGuHA5XwUllPFx2xCeRwwGILwG5GZDfQ+9w+NAIssDwG\nF6GUJBcRMgN3BSJVWvQGSwZ59hgDpUXJ+vggDwc6jINbzCyhU4EYJhCTgkkQhGMl8JoAwCpmvDiR\nt8DJyLrLehXbgCAcJ0HWBDqEwFf5/dPEG0+kCpEg9ACBFQJacxA6LKLjiC0/VhmTZVjiCAThmAny\ncACMXWAwcJJf+WYXkoBUCCdxocSTK+D70yA3H5pqofwNs2PPOgi5CYMQOBv/+AE34j8ghA+fQLiV\n9FEz+XE+wEAO/OqAHnRXsrsRWCGg1JTFMH2GUQQYpdSmF6C9PUFChbRHYgnCMeAOhDsCjFrFjDPs\njcu5bDq8kvROBFYIQHGRJ6HCDPPik1Ch86GCIAQVW4NtBRqBc5/jowCMZvuR8Wyam+jAniSwhkFB\niABzMf4vOUDhfgbzkuUleJCBfyzUNSnRakMnBKaz/2yUWo1S261XmREQwonRYNdaS+sf4Lp17WQD\nUE//J1PVjQAPB/zJo70XjtOQFWAhMwJCaJn7Uca9/jynN7bQXmJCh9tbYdOdSpVvS1a8gJvQCYE2\nVBuQjYm9LkBmBIQwo3XNP1RpJSxxFyPNAc6H0pRE9wVYCJRXGCNg9ekwuhCgFw1tL9H/wEQuZgd7\nVx5kQzsyIyCEiQBW1g6sELDVIKU+swoWnwdwxGgAQ98CoHQo+o0L0tU/QThG4lKJTeKi095IY4cC\nKwQc2tr81ubSHoK+C0Ic9rRgHaaM3tgBtKSxO6EQAv6M55CTidVRsYoxqZoOERBVSxA8zAU2A0Pt\nFTUUtJLGezG0QmATAza7Ft0qlswYCMFF6xqUWovJhbEeqGwh5wo7oegprDvwDr3NiDeJ8QJuQisE\nWshqdS3a2VrbMb4PtcBtKe+UIHSPWDd39XYrTGAKa1jDnS+g9ZWp7EwIhIA9S3DwDBg1wKxrOgxH\nxitVWgblFdrJ1mo7PxVgchGKJiCkh0SzALHrUYpxMCEb4CM8XwXMS3lftdahaHD1KtA6vs0p07Dc\nWqixXl/TUJjuPkuLcIMy10X6RIL1W7/H3VvsxXNZNT0dfQ2R23C77yyBhZ1w5HQk45AQDBKFt7vX\n73qB88cCDGdX/Yucl/yQQR9CJAQ6QesaTKah2IxDgpA+3JmwavzWb2d004uWB3w9/RYbRSH1hMAm\nIAihZAGmYtZjKOUuodfQh0k1jZy0LIuBIzU/AqCe7ecp9ebiVMQKeAm9EJhAbQlKlREgN0xBID7l\nXUcJvXEM3fcWS4a2x+4/CUprU9lBmxAJAXuWIDsHSqaDyoLD+8exJQvJLygED69N4DF7eRMDmnE5\nC6Wb0AgBt5qkFA8A87JpG3IvT9gZiSWaUAgSti9AI6aCdkcl7RaylqWzY17Cahj8A0Ab2TzAdYMx\n+dlqgHsk9bgQCGxjNRRhNNWLgeYgDldDKQS0Zs3JbKoFWMQNuoWcXpgf+TK85cmkJoGQXnymCrMD\npYEHqjNHQw2F3wfu28VI9TRXcBVL12C0gYuJHRpITQIhnSTIhD3fet3xJtQdNO9TEyvgJbClybtC\nKfoAO4HCM1i/Zz2Tx1ubYn9wU8B0Fiah42YkwlBIBy534V405TXT60LMNTlSa1o7PzjJXQurEABQ\ninuAm63Fk7Vms89Odu33UTi5CZdY4zVB6Dk6yxpkprFn7mcww9ij28lWwB+05itp6auLUNoEXPzB\n9f7Lvns4Bhp3QVOZRRCSgT30nIXJGuS2RTUAPMi8CksAADyRpn7GEGpNAECpW6tgwAiToXXzy6ZK\nEcRVKnI0AslJKCQHZ+i5BjMl6Gie5sGzMI8jw1vIOx/YBYzWms5iYlJCaA2DDvUH4dcjMN9lhrPe\nU6lIqhQJyccxArqcg/owqa2Rk5ZBTi78xFQXoaYdVv8JUu8m7CUDhEDN/i53CWCGVyFDiL+2rrbW\ndwgEIwBiUooDjILSolR2NRFhtwkA3Yq8co/VFia3O0LE8L+2nMjWwD9wMkAT6ARHSk+w1sQaBUVD\nEI4f/7wBrmsrl9k56c0n3DmZLQRiHYWagDmeG10ciYTjJYEzkHNtjSCndns6etZNMkAI2NGFAKNO\ngoKR5n1uHsaD0Caf+LyDthTfB4yyrLuiEQjdJ7HBuQHgFaZt3E7JKant1NER+ilCN0rRH9gIfBCo\nm82T059kzj+BEZhiD68ApS5vQnEkEpKDUoVbKX6ohM1ntHOXlQZ/50aorXZ28kxjp4mMEgIASnEp\n8Jy1+OxBCucWUusu9hB/kzvzu/GCQhAS0YlNSSl6A/8Cplmrvq01v0x9J7smE2YHYtCafwAPWYuz\n7uVrS8Eq+m6KPfh5C87FDAn6YwKQZAZB6A6+MwNKoYA/4QiAB4Bfpbx33STjNAEApRgE398JuflZ\ntOppvKbyaKaG3P0bOPiMrwoW6+0l2YqFrklwzSjFHcCd1l4rgUu0pjk9neyajBQCAErdsBEWnRq/\npXSl1ks31o2SAAAMl0lEQVQu8DlA3IqFo8O5ZhoxyUMaRrN9yfuMftDaYyswTWuqE5whEGTA7EAi\nuuFJaOOpCiMIQNd+JPbMgBUh+BpnsZ8hswAU7Yc0WZcHXQBABtoEjhHxKBT8iL0uEmepKt7BB/gE\nT9FEb7Jp5WE+/7rWbEpPt48OEQKGRNVihGjjvS58HxY1FLz/cf7GbkYAcBe3V17Df38yxX09ZiIo\nBAoG+6xMVC1GiDbe6yLuYaEUWZ9kWfEbnAHAHJaU/wc/mxSm6yiDDYNTFkNxkVnKzYO6s2GQMvkG\nWl6GNiulUzAcNoSAY+wDE4BioAwYCTQMoPbdOgbcApDHkf9rptdH050u7GjJWMOg98ZW6sub4L9O\nxmg/tncguXwqY38DoUcpwbluZgN5i7mWOgbMsta900yv2WETABCp4cCB3X5rx3OoJNU9EUKJe+Yo\nbxXn8SXut9XoauAKrUlLGbHjJfJPwXcYtO2oD5IQ5GgQ+z/fBLwMjCjnRK5kKa3kKqAFuFJrtqax\np8dFhDQBfxQnnrVKzVh1lIVJZEoxU3FPAxobgP0/LwAm1FDQfDl/p5ohAAxn1y1asyp9HT5+Ii8E\nWuilZvHseQu54em4jeaCqEKpapRa4RIUMqWYubgFfLG1bh8wqpncxy9jef07Vo6a2/g5uxjpTRsW\nOjJ2dsBL7GyBYTh9p+7mg70dN2++D/xUaytlmVJVYE3+GsqBHZhMsoeBebjqziNDg/ATGw8wB/P/\njgLO/Tq/5V6+DsAnWMZfmb0mm/bQTytHRgj4olTh/Vz/9A0smgSqn7X2ceCLWtOIUk1AL2t9DfBv\nHAvxPmAtMADJRZA5+MQDADl/4MsXf8Uqc5FN68a9nLB5EAe/GHYBAFEXAhZK8SHgb8CJ1qo1wCc1\naiMw0Fr3DGb4ZOcd6G+t34XRFiT6MJNwaYGP8+lXruHRaVbRkN3AVK3Zkdb+9SCRtwkAaM1GYCom\n7BPgrCHs2/YyZ9uVYtYD1+B4kL1irV8DTEe8DYPNsVWm7gWwiZP5EvdPtgRAE/CJTBIAAGitpVkN\ndB7ohaA1aJ1Pg36Qa/drKIzZFwo1PBG3XlowG5Rp+0+FJ7p5zIr9DNKjqWxyDtVXp/27JKGlvQNB\na6DV7dz572xaXNeNvht0Vrr7Ju0YGyy3/sjXuiu4n+OSoaexYa/rGvhh2r9HklraOxDIBoWPMrdM\n0VbjugiWgu7XxXGLrKfOctESAtSOUnNrQy26kr/ucv33j4NWaf8eSWpiGOwEpSjBGAxPBhjLlroJ\nvH3e3/TH30xwQBlOHQOZKUg3R+HZ6Z5CHkmfaVUU5QP058CROn43UGsaU9DjtBB5t+HO0JrNSnH2\ndF4qf5lzBm5lXP99DH1NKS7SmtU+h4gTUbA4iuIyxUV2vcAq19pGPrMhkwUAyOxAl2hNTRkXvPoN\nfg3AIQp6Ac8rxRd9dpe8BKnAsfZvR6nVnVj9j1sot9KW0QIARBPoFnm0fObXfHPhY8x9YQ/DfwPz\nc4E/KlUzH6rKAT2B2pJxFGc9Rfn6hCeSwKOewv2Etwp7+D7p92Kcupzf2ec/UIo8GDkuqT0OMCIE\nuoOVUHI3oBQboO15uCsX+IDVeBsYygVAuR1U5Kd6Su3DnsF+wtcCBSR+0hdhis7YtSSuBkq+wJiZ\nFRRxgLzNTerjug8nDW6gJtvn+EggQuAo0ZrVSm1bB5ydYJdW4O4E28Rm0DPMBSoBBRwB4t13zRP/\nNGvJXXSmoYIiVlIGTlUqYH4SuxtsRAgcE81H/Na+y8nUsS6nP/XPYOohGswFeTmmKOoe4qsjC0eD\n1jUopTFaABiX7g969ioBBlnvt9u/dw0Fc7czdBvgY0OYcwjU67Hryit6qttBRYRAD7KbEZzGW3o8\nm259LnZTCbHRiN7qyAal3rH2awbOQuvKpHU2DHRuQ7Er+hwGZvgc59YC5plxP3Og5laY7yMA5gOl\nr/sWpslwZHagh6mkSP2Djy5RigVKdUQgulNTOappvE/7CMzTbSiw6SgTnWQinSVvOQsT1n2qj7Ds\n0AKqGFGl0F8GtgGPAmcmtcchRDSBY6K8Akp91vfqDUzEqP3fhu98SandW3KZfWQCtfsB3qagsYWK\nezR8gXhDobteXT62MSuKswqJx/QGc+N7hwA2De8xjh9xx95HueYjmKGYzV6obsAYDQVECBwTnaUo\nV4pTgEeAydC7EB6a0gJscHYZAqV2RlqvobAA2IQRAG7jYaywUGoiYR82dC3Y3GP6yu4IPqsa8Pl5\nNOkWctFkneDavBH4NfA4vPxfUOrzm2X++N+XdPstZ2IDnQv6TvihOwipo03iomrLn71Qw1YNqzvi\nDfz83L0BMFDjOuH2TvsTtHgGpz/VnUb2xX7nxQm/AyxqpNfKX3PLG8Vs2e/zey8HfXEm+/4f9/Wa\n7g5kcoMvrPMTAh9gnt7AaU9rrbsX5uoVDLDX2r9Nw/Od3tzdDaNNlbCI7Y923eRVlmBYYS2vttaN\nSfQdQA/6FgvKR/J+zCl70aiv4KmtoCek+xoIQ0t7BzK5wZwyPyFgNIT2NtBLn+SqV9qPMszVujGa\nunlzdy+M9lhi7rvbYgXMCusz1mlYagk4r2DYG9MXz3cAXQL696APuw8bSPWRH/EDvZMR6wOh9YSk\npb0Dmdw6FwLO8hi21Qxlz80zOeW9M7mwZhIXVecye7U5fk4ZTF4cd36/m9vvad7dMNpjiLl3HRv7\nufHL7pt8aVx/nM/WlnBYEdMXKGxDPTGety8H/RTodvfvV0R5zTB2feXPlA7r1neVFtMklDiJ+GU4\nNhw6BM/tw3i+5dtrs/iBbuduFb9/6cq4+WsnIeaN2EYzbyizcavt3qyC3/m6S/znDnMtb8XkaRyE\nsfJfGHd+89kPAhq4zlq7ELhRoRswUzG3glX10+FZjLHv/4wsEY4FmR1IIl0VOlWK24Drgf8HjG4n\nx0cAAPTppxS5WtPiOnkN8Q5H9mxDHebGG4WTCTk+ViHeQt/V9kTCwTvL8ZhruRkYay37W/nNuitj\nP5qbMFV/voop/mnTBDwM3KM17yToj3A0pFsVkaYBnQP6KrittpPhQyPol0H/7k5u/9caJr/WTM5y\nj1pd6BlPV7nG2M4MhLN/53aA7hktF2m3Ec/pxxPWq63qH7LU/DhVHSYvnsAlVWdyYc14rqhR3FwF\nt7d5hk17QN8Oemi6/69Ma2nvgDTXn9FNG4Ld+nFIW3nwfgn6atBj21Dusf0Y62Zc7REMtr2gK7vC\nirjt2ty0tr3iTC6smclMPZOZ+lLGbfOxB3gF0xOg+4AuBn0O6Nlw0+ZOvveboOeBzk/3/5OpTYYD\noWD3JuCfGFfZSVh2hHr68xYThwLftPfMobXmDF7fcyr/7nUBZc+cx+q949jScB0fpMI4yY04QN7m\nNnVpa28u7HsS5Y2Ps80d0OR2TFqKGeN77AROFh538oQPcdnAlxky5RADztjNcNZz5srfcfO/LmZF\nazN5VFDUVMmYj2L8/V2cgD873gQmGdkkJAsRAoEikTtyeYXWpv6VUuRO5dWzP8Yzv/o53/l3A30n\nYlyVcwA0WYXrmcx6Jg97hM8DnHoCe3Qrd3OAe+0TdoTQNnMV/8Ul/32T4j4g/0f8YPAADnGIAYer\nGTz5MH1b/synf1xvvPHygV4w5lS/3m9kasE5zHcb7yYCE//JJfZyPi5DaNfUHRQBkHxECASIrgyJ\nZh9aYNoqmDb1DmudUuQDpwNTMNrCFEX7qdqKD9vLMAWDfc/3FhO5ifnnYhkQ7+Aue1NfqwF8Jfao\nvhwF9ZiqPbsx1Zp83r/3e0wRFyENiBDIALSmCXjVagD8Rn1z1BgqH17LlA9UUFTyV5p0s0nCcVTk\n0KJbyakHdQRjmW+C5hH4SIIs9pS3w3dx3eBaU9/VZyjV0tzVPkLyECGQoXxD/2YncOFV1vx/HrPd\n04UdZLPtzTa4FpOhp8lui/hS/hdY/NtcWm/AM62n1JYyHLtBB+1U79Cavxx9bxMPg47+XMLRIkIg\n07H8CVpUaZnf5jYaD2rNG/Fb7gfu97sz6embtjvDICF5iBCIDD1348pNm1mI27AgRBxJLyYIEUeE\ngCBEHBECghBxRAgIQsQRISAIEUeEgCBEHBECghBxRAgIQsQRISAIEUeEgCBEHBECghBxRAgIQsQR\nISAIEUeEgCBEHBECghBxRAgIQsQRISAIEUeEgCBEHBECghBxRAgIQsQRISAIEUeEgCBEHBECghBx\nRAgIQsQRISAIEUeEgCBEHBECghBxRAgIQsQRISAIEUeEgCBEHBECghBx/j+BGmmo+vdcMQAAAABJ\nRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109aa7860>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "donut2 = noisy({Point(5 * math.sin(rad(d)), 5 * math.cos(rad(d))) for d in range(360)})\n",
    "\n",
    "plot_convex_hull(donut2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Tests\n",
    "\n",
    "So far, everything looks good! But I would gain even more confidence if we could pass a test suite:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'tests pass'"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def tests():\n",
    "    # Tests of `turn`\n",
    "    assert turn(octagon[0], octagon[1], octagon[2]) == 'left'\n",
    "    assert turn(octagon[2], octagon[3], octagon[4]) == 'left'\n",
    "    assert turn(octagon[1], octagon[0], octagon[7]) == 'right'\n",
    "    assert turn(octagon[5], octagon[6], octagon[7]) == 'left'\n",
    "    assert turn(octagon[2], octagon[1], octagon[0]) == 'right'\n",
    "    assert turn(pacman[1], pacman[2], pacman[3]) == 'left'\n",
    "    assert turn(pacman[3], pacman[4], pacman[5]) == 'right'\n",
    "    assert turn(Point(0, 0), Point(0, 1), Point(0, 2)) == 'straight'\n",
    "    assert turn(Point(2, 1), Point(3, 1), Point(4, 1)) == 'straight'\n",
    "    assert turn(Point(2, 1), Point(4, 1), Point(3, 1)) == 'straight'\n",
    "    assert turn(Point(0, 0), Point(1, 1), Point(2, 2)) == 'straight'\n",
    "    assert turn(Point(0, 0), Point(-1, -1), Point(2, 2)) == 'straight'\n",
    "    # More tests of `turn`, covering negative denominator\n",
    "    A, B = Point(-2, -2), Point(0, 0)\n",
    "    assert turn(A, B, Point(1, 3)) == 'left'\n",
    "    assert turn(A, B, Point(2, 2)) == 'straight'\n",
    "    assert turn(A, B, Point(3, 1)) == 'right'\n",
    "    assert turn(A, B, Point(-1, 1)) == 'left'\n",
    "    assert turn(A, B, Point(-1, -4)) == 'right'\n",
    "    assert turn(A, B, Point(-1, -1)) == 'straight'\n",
    "    assert turn(B, A, Point(-3, -4)) == 'left'\n",
    "    assert turn(B, A, Point(-4, -3)) == 'right'\n",
    "    assert turn(B, A, Point(-1, -1)) == 'straight'\n",
    "    assert turn(B, A, Point(-3, -3)) == 'straight'\n",
    "    \n",
    "    # Tests of convex_hull\n",
    "    assert convex_hull(octagon)== octagon\n",
    "    assert convex_hull(circle) == convex_hull(donut)\n",
    "    assert convex_hull(circle) == convex_hull(convex_hull(circle))\n",
    "    for n in (0, 1, 2, 3):\n",
    "        assert convex_hull(Points(n)) == Points(n)\n",
    "    collinear = {Point(x, 0) for x in range(100)}\n",
    "    assert convex_hull(collinear) == [min(collinear), max(collinear)]\n",
    "    P = Point(5, 5)\n",
    "    assert convex_hull(collinear | {P}) == [min(collinear), max(collinear), P]\n",
    "    grid1 = {Point(x, y) for x in range(10) for y in range(10)}\n",
    "    assert convex_hull(grid1) == [Point(0, 0), Point(9, 0), Point(9, 9), Point(0, 9)]\n",
    "\n",
    "    return 'tests pass'\n",
    "        \n",
    "tests()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "## How Many Points on the Hull?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "The number of points on the hull for `Points(N)` seems to increase slowly as `N` increases. \n",
    "How slowly? Let's try to find out.  We'll average the number of points on the hull for `Points(N)` over, say, 60 random trials:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [],
   "source": [
    "def average_hull_size(N, trials=60):\n",
    "    \"\"\"Compute the average hull size of N random points\n",
    "    (averaged over the given number of random trials).\"\"\"\n",
    "    return sum(len(convex_hull(Points(N, seed=trials+i)))\n",
    "               for i in range(trials)) / trials"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We'll do this for several values of *N*, taken as powers of 2:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "   N   Hull Size\n",
      "   1:  1.0\n",
      "   2:  2.0\n",
      "   4:  3.7\n",
      "   8:  5.1\n",
      "  16:  7.1\n",
      "  32:  8.6\n",
      "  64: 11.0\n",
      " 128: 12.6\n",
      " 256: 14.6\n",
      " 512: 16.4\n",
      "1024: 18.1\n",
      "2048: 19.8\n",
      "4096: 21.6\n",
      "8192: 23.2\n"
     ]
    }
   ],
   "source": [
    "hull_sizes = [average_hull_size(2**e) \n",
    "              for e in range(14)]\n",
    "\n",
    "print('   N   Hull Size')\n",
    "for e in range(14): \n",
    "    print('{:4}: {:4.1f}'.format(2**e, hull_sizes[e]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then we'll plot the results, with *N* on a log<sub>2</sub> scale:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
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7AG5ogXhEMpo7PPkknHce/PrXoWxzQUHcUYm0DlUDlbw1Zw6cey4sWQLDhsEhh8Qdkci6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\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109f66048>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def plot_hull_sizes(hull_sizes):\n",
    "    plt.plot(hull_sizes, 'bo-')\n",
    "    plt.ylabel('Hull size')\n",
    "    plt.xlabel('log_2(number of points)')\n",
    "\n",
    "plot_hull_sizes(hull_sizes)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That sure looks like a straight line! \n",
    "\n",
    "That means we can define `estimated_hull_size` by computing a slope and intercept of the line. (I won't bother doing linear regression; I'll just draw a straight line from the first to the last point in `hull_sizes`.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "button": false,
    "collapsed": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x109af6cc0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def estimated_hull_size(N):\n",
    "    \"Estimated hull size for N random points, (inter/extra)polating from hull_sizes.\"\n",
    "    slope = (hull_sizes[-1] - hull_sizes[0]) / (len(hull_sizes) - 1)\n",
    "    return hull_sizes[0] + slope * math.log(N, 2)\n",
    "\n",
    "# Plot actual average hull sizes in blue, and estimated hull sizes in red\n",
    "plot_hull_sizes(hull_sizes)\n",
    "plt.plot([estimated_hull_size(2**e) \n",
    "          for e in range(len(hull_sizes))], \n",
    "         'r--');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's an estimate of the number of points on the hull of a quadrillion random points:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "86.15634904778565"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "estimated_hull_size(10**15)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "deletable": true,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    }
   },
   "source": [
    "# Concluding Remarks and Further Reading\n",
    "\n",
    "The convex hull problem is an interesting exercise in algorithm design.\n",
    "The algorithm covered here is called  [Andrew's Monotone Chain](https://en.wikibooks.org/wiki/Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain).\n",
    "It is a variant of the [Graham Scan](https://en.wikipedia.org/wiki/Graham_scan).\n",
    "You can read more from [Tamassia](http://cs.brown.edu/courses/cs016/docs/old_lectures/ConvexHull-Notes.pdf) or [Wikipedia](https://en.wikipedia.org/wiki/Convex_hull)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "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": "3.5.0"
  }
 },
 "nbformat": 4,
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}