source-code/gis/shapely.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Shapely: computing with geometric objects"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Preparation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Load the required modules."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "from shapely.geometry import Point, MultiPoint\n",
    "from shapely.geometry import LineString, MultiLineString, LinearRing\n",
    "from shapely.geometry import Polygon, MultiPolygon\n",
    "from shapely.geometry import CAP_STYLE, JOIN_STYLE\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Shapes and spatial analysis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create a simple polygon, represented as a `Polygon`.  This shape could also be represented as a `LinearRing`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "polygon1 = Polygon([(1.0, 1.0), (1.0, 3.0), (3.0, 2.0), (3.0, 1.0)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"100.0\" height=\"100.0\" viewBox=\"0.92 0.92 2.16 2.16\" preserveAspectRatio=\"xMinYMin meet\"><g transform=\"matrix(1,0,0,-1,0,4.0)\"><path fill-rule=\"evenodd\" fill=\"#66cc99\" stroke=\"#555555\" stroke-width=\"0.0432\" opacity=\"0.6\" d=\"M 1.0,1.0 L 1.0,3.0 L 3.0,2.0 L 3.0,1.0 L 1.0,1.0 z\" /></g></svg>"
      ],
      "text/plain": [
       "<shapely.geometry.polygon.Polygon at 0x7fa5ac99feb8>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "polygon1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'POLYGON ((1 1, 1 3, 3 2, 3 1, 1 1))'"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "polygon1.wkt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create a second polygon with a hole."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "polygon2 = Polygon([(1.5, 1.5), (1.5, 4.0), (4.5, 3.0), (4.5, 1.0)],\n",
    "                   [[(2.0, 2.0), (2.5, 2.5), (2.8, 2.0)]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"100.0\" height=\"100.0\" viewBox=\"1.38 0.88 3.24 3.24\" preserveAspectRatio=\"xMinYMin meet\"><g transform=\"matrix(1,0,0,-1,0,5.0)\"><path fill-rule=\"evenodd\" fill=\"#66cc99\" stroke=\"#555555\" stroke-width=\"0.06480000000000001\" opacity=\"0.6\" d=\"M 1.5,1.5 L 1.5,4.0 L 4.5,3.0 L 4.5,1.0 L 1.5,1.5 z M 2.0,2.0 L 2.5,2.5 L 2.8,2.0 L 2.0,2.0 z\" /></g></svg>"
      ],
      "text/plain": [
       "<shapely.geometry.polygon.Polygon at 0x7fa5ac9326a0>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "polygon2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute the area of the polygon."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6.55"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "polygon2.area"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute the intersection between the two polygons."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "intersection = polygon1.intersection(polygon2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"100.0\" height=\"100.0\" viewBox=\"1.44 1.19 1.62 1.62\" preserveAspectRatio=\"xMinYMin meet\"><g transform=\"matrix(1,0,0,-1,0,4.0)\"><path fill-rule=\"evenodd\" fill=\"#66cc99\" stroke=\"#555555\" stroke-width=\"0.032400000000000005\" opacity=\"0.6\" d=\"M 1.5,2.75 L 2.3333333333333335,2.3333333333333335 L 2.0,2.0 L 2.8,2.0 L 2.714285714285714,2.142857142857143 L 3.0,2.0 L 3.0,1.25 L 1.5,1.5 L 1.5,2.75 z\" /></g></svg>"
      ],
      "text/plain": [
       "<shapely.geometry.polygon.Polygon at 0x7fa5ac9320f0>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "intersection"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'POLYGON ((1.5 2.75, 2.333333333333333 2.333333333333333, 2 2, 2.8 2, 2.714285714285714 2.142857142857143, 3 2, 3 1.25, 1.5 1.5, 1.5 2.75))'"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "intersection.wkt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute the union of the two polygons."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "union = polygon1.union(polygon2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"100.0\" height=\"100.0\" viewBox=\"0.86 0.86 3.78 3.28\" preserveAspectRatio=\"xMinYMin meet\"><g transform=\"matrix(1,0,0,-1,0,5.0)\"><path fill-rule=\"evenodd\" fill=\"#66cc99\" stroke=\"#555555\" stroke-width=\"0.0756\" opacity=\"0.6\" d=\"M 1.0,1.0 L 1.0,3.0 L 1.5,2.75 L 1.5,4.0 L 4.5,3.0 L 4.5,1.0 L 3.0,1.25 L 3.0,1.0 L 1.0,1.0 z M 2.3333333333333335,2.3333333333333335 L 2.714285714285714,2.142857142857143 L 2.5,2.5 L 2.3333333333333335,2.3333333333333335 z\" /></g></svg>"
      ],
      "text/plain": [
       "<shapely.geometry.polygon.Polygon at 0x7fa5ac932240>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "union"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'POLYGON ((1 1, 1 3, 1.5 2.75, 1.5 4, 4.5 3, 4.5 1, 3 1.25, 3 1, 1 1), (2.333333333333333 2.333333333333333, 2.714285714285714 2.142857142857143, 2.5 2.5, 2.333333333333333 2.333333333333333))'"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "union.wkt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Check whether the area of the union is the sum of the areas of the polygons minus the area of their intersection."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8.202380952380953"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "polygon1.area + polygon2.area - intersection.area"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8.202380952380953"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "union.area"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create a point."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "point1 = Point((2.1, 2.1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Check whether the point lies within the first polygon."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "point1.within(polygon1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For the second polygon, the point is within the hole."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "point1.within(polygon2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create a horizontal line."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "line1 = LineString([(0.5, 2.1), (5.0, 2.1)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute the intersection between the line and the second polygon."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"100.0\" height=\"100.0\" viewBox=\"1.38 1.98 3.24 0.2400000000000002\" preserveAspectRatio=\"xMinYMin meet\"><g transform=\"matrix(1,0,0,-1,0,4.2)\"><g><polyline fill=\"none\" stroke=\"#66cc99\" stroke-width=\"0.06480000000000001\" points=\"1.5,2.1 2.1,2.1\" opacity=\"0.8\" /><polyline fill=\"none\" stroke=\"#66cc99\" stroke-width=\"0.06480000000000001\" points=\"2.7399999999999998,2.1 4.5,2.1\" opacity=\"0.8\" /></g></g></svg>"
      ],
      "text/plain": [
       "<shapely.geometry.multilinestring.MultiLineString at 0x7fa5ac932978>"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "line2 = polygon2.intersection(line1)\n",
    "line2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array('d', [1.5, 2.1]), array('d', [2.1, 2.1]))"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "line2[0].coords.xy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array('d', [2.7399999999999998, 4.5]), array('d', [2.1, 2.1]))"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "line2[1].coords.xy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "shapely.geometry.multilinestring.MultiLineString"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(line2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Check whether the line intersects with the first polygon."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "line1.intersects(polygon1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A line can be turned into an object with a finite width by creating a buffer around it of the specifie size. The result is a `Polygon`."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Collections of objects"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create two lines, and a `MultiLineString` out of them."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"100.0\" height=\"100.0\" viewBox=\"-0.08 -0.08 2.16 1.1600000000000001\" preserveAspectRatio=\"xMinYMin meet\"><g transform=\"matrix(1,0,0,-1,0,1.0)\"><polyline fill=\"none\" stroke=\"#66cc99\" stroke-width=\"0.0432\" points=\"0.0,0.0 1.0,1.0 2.0,0.5\" opacity=\"0.8\" /></g></svg>"
      ],
      "text/plain": [
       "<shapely.geometry.linestring.LineString at 0x7fa5ac932f98>"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "line3 = LineString([(0.0, 0.0), (1.0, 1.0), (2.0, 0.5)])\n",
    "line3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"100.0\" height=\"100.0\" viewBox=\"-0.58 -0.08 2.16 0.6599999999999999\" preserveAspectRatio=\"xMinYMin meet\"><g transform=\"matrix(1,0,0,-1,0,0.49999999999999994)\"><polyline fill=\"none\" stroke=\"#66cc99\" stroke-width=\"0.0432\" points=\"1.5,0.5 -0.5,0.5 -0.5,0.0\" opacity=\"0.8\" /></g></svg>"
      ],
      "text/plain": [
       "<shapely.geometry.linestring.LineString at 0x7fa5ac943160>"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "line4 = LineString([(1.5, 0.5), (-0.5, 0.5), (-0.5, 0.0)])\n",
    "line4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"100.0\" height=\"100.0\" viewBox=\"-0.6 -0.1 2.7 1.2000000000000002\" preserveAspectRatio=\"xMinYMin meet\"><g transform=\"matrix(1,0,0,-1,0,1.0)\"><g><polyline fill=\"none\" stroke=\"#66cc99\" stroke-width=\"0.054000000000000006\" points=\"0.0,0.0 1.0,1.0 2.0,0.5\" opacity=\"0.8\" /><polyline fill=\"none\" stroke=\"#66cc99\" stroke-width=\"0.054000000000000006\" points=\"1.5,0.5 -0.5,0.5 -0.5,0.0\" opacity=\"0.8\" /></g></g></svg>"
      ],
      "text/plain": [
       "<shapely.geometry.multilinestring.MultiLineString at 0x7fa5ac932e48>"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mline = MultiLineString([line3, line4])\n",
    "mline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create two polygons, and a new `MultiPolygon` containing them."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"100.0\" height=\"100.0\" viewBox=\"0.92 0.92 1.1600000000000001 2.16\" preserveAspectRatio=\"xMinYMin meet\"><g transform=\"matrix(1,0,0,-1,0,4.0)\"><path fill-rule=\"evenodd\" fill=\"#66cc99\" stroke=\"#555555\" stroke-width=\"0.0432\" opacity=\"0.6\" d=\"M 1.0,1.0 L 1.0,3.0 L 2.0,2.0 L 2.0,1.0 L 1.0,1.0 z M 1.4,1.4 L 1.4,2.4 L 1.6,1.6 L 1.6,1.4 L 1.4,1.4 z\" /></g></svg>"
      ],
      "text/plain": [
       "<shapely.geometry.polygon.Polygon at 0x7fa5ac943668>"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "polygon3 = Polygon([(1.0, 1.0), (1.0, 3.0), (2.0, 2.0), (2.0, 1.0)],\n",
    "                   [[(1.4, 1.4), (1.4, 2.4), (1.6, 1.6), (1.6, 1.4)]])\n",
    "polygon3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "line4_outer = LinearRing([(3.0, 2.0), (3.0, 5.0), (6.0, 5.0), (6.0, 2.0)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "line4_inner1 = LinearRing([(3.5, 2.5), (3.5, 4.5), (4.5, 4.5), (4.5, 2.5)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "line4_inner2 = LinearRing([(5.0, 2.5), (5.0, 4.5), (5.5, 4.5), (5.5, 2.5)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "polygon4 = Polygon(line4_outer, [line4_inner1, line4_inner2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"100.0\" height=\"100.0\" viewBox=\"2.88 1.88 3.24 3.24\" preserveAspectRatio=\"xMinYMin meet\"><g transform=\"matrix(1,0,0,-1,0,7.0)\"><path fill-rule=\"evenodd\" fill=\"#66cc99\" stroke=\"#555555\" stroke-width=\"0.06480000000000001\" opacity=\"0.6\" d=\"M 3.0,2.0 L 3.0,5.0 L 6.0,5.0 L 6.0,2.0 L 3.0,2.0 z M 3.5,2.5 L 3.5,4.5 L 4.5,4.5 L 4.5,2.5 L 3.5,2.5 z M 5.0,2.5 L 5.0,4.5 L 5.5,4.5 L 5.5,2.5 L 5.0,2.5 z\" /></g></svg>"
      ],
      "text/plain": [
       "<shapely.geometry.polygon.Polygon at 0x7fa5ac9431d0>"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "polygon4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "multi_polygon = MultiPolygon([polygon3, polygon4])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"100.0\" height=\"100.0\" viewBox=\"0.8 0.8 5.4 4.4\" preserveAspectRatio=\"xMinYMin meet\"><g transform=\"matrix(1,0,0,-1,0,6.0)\"><g><path fill-rule=\"evenodd\" fill=\"#66cc99\" stroke=\"#555555\" stroke-width=\"0.10800000000000001\" opacity=\"0.6\" d=\"M 1.0,1.0 L 1.0,3.0 L 2.0,2.0 L 2.0,1.0 L 1.0,1.0 z M 1.4,1.4 L 1.4,2.4 L 1.6,1.6 L 1.6,1.4 L 1.4,1.4 z\" /><path fill-rule=\"evenodd\" fill=\"#66cc99\" stroke=\"#555555\" stroke-width=\"0.10800000000000001\" opacity=\"0.6\" d=\"M 3.0,2.0 L 3.0,5.0 L 6.0,5.0 L 6.0,2.0 L 3.0,2.0 z M 3.5,2.5 L 3.5,4.5 L 4.5,4.5 L 4.5,2.5 L 3.5,2.5 z M 5.0,2.5 L 5.0,4.5 L 5.5,4.5 L 5.5,2.5 L 5.0,2.5 z\" /></g></g></svg>"
      ],
      "text/plain": [
       "<shapely.geometry.multipolygon.MultiPolygon at 0x7fa5ac9434a8>"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "multi_polygon"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Predicates"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create an invalid `LinearRing`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "ring = LinearRing([(0.0, 0.0), (1.0, 1.0), (2.0, 0.5), (0.0, 0.5)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Ring Self-intersection at or near point 0.5 0.5\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ring.is_valid"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create a `Polygon` with a hole and check the semantics of contains."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"100.0\" height=\"100.0\" viewBox=\"-3.24 -3.24 6.48 6.48\" preserveAspectRatio=\"xMinYMin meet\"><g transform=\"matrix(1,0,0,-1,0,0.0)\"><path fill-rule=\"evenodd\" fill=\"#66cc99\" stroke=\"#555555\" stroke-width=\"0.12960000000000002\" opacity=\"0.6\" d=\"M -3.0,-3.0 L -3.0,3.0 L 3.0,3.0 L 3.0,-3.0 L -3.0,-3.0 z M -1.0,-1.0 L -1.0,1.0 L 1.0,1.0 L 1.0,-1.0 L -1.0,-1.0 z\" /></g></svg>"
      ],
      "text/plain": [
       "<shapely.geometry.polygon.Polygon at 0x7fa5ac943e10>"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "large_polygon = Polygon([(-3.0, -3.0), (-3.0, 3.0), (3.0, 3.0), (3.0, -3.0)],\n",
    "                        [[(-1.0, -1.0), (-1.0, 1.0), (1.0, 1.0), (1.0, -1.0)]])\n",
    "large_polygon"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"100.0\" height=\"100.0\" viewBox=\"-2.16 -2.16 4.32 4.32\" preserveAspectRatio=\"xMinYMin meet\"><g transform=\"matrix(1,0,0,-1,0,0.0)\"><path fill-rule=\"evenodd\" fill=\"#66cc99\" stroke=\"#555555\" stroke-width=\"0.0864\" opacity=\"0.6\" d=\"M -2.0,-2.0 L -2.0,2.0 L 2.0,2.0 L 2.0,-2.0 L -2.0,-2.0 z\" /></g></svg>"
      ],
      "text/plain": [
       "<shapely.geometry.polygon.Polygon at 0x7fa5ac943a90>"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "small_polygon = Polygon([(-2.0, -2.0), (-2.0, 2.0), (2.0, 2.0), (2.0, -2.0)])\n",
    "small_polygon"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "large_polygon.contains(small_polygon)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "small_polygon.within(large_polygon)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "small_polygon.crosses(large_polygon)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"100.0\" height=\"100.0\" viewBox=\"-1.08 -1.08 2.16 2.16\" preserveAspectRatio=\"xMinYMin meet\"><g transform=\"matrix(1,0,0,-1,0,0.0)\"><path fill-rule=\"evenodd\" fill=\"#66cc99\" stroke=\"#555555\" stroke-width=\"0.0432\" opacity=\"0.6\" d=\"M -1.0,-1.0 L -1.0,1.0 L 1.0,1.0 L 1.0,-1.0 L -1.0,-1.0 z\" /></g></svg>"
      ],
      "text/plain": [
       "<shapely.geometry.polygon.Polygon at 0x7fa5ac943c88>"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smaller_polygon = Polygon([(-1.0, -1.0), (-1.0, 1.0), (1.0, 1.0), (1.0, -1.0)])\n",
    "smaller_polygon"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "small_polygon.contains(smaller_polygon)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smaller_polygon.within(small_polygon)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smaller_polygon.touches(large_polygon)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Constructive methods"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Buffers"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Buffers make 2D objects from `Point` or `LineString`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "line = LineString([(0.0, 1.0), (3.0, 1.0)])\n",
    "buffer1 = line.buffer(0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"100.0\" height=\"100.0\" viewBox=\"-0.228 0.772 3.4560000000000004 0.4560000000000002\" preserveAspectRatio=\"xMinYMin meet\"><g transform=\"matrix(1,0,0,-1,0,2.0)\"><path fill-rule=\"evenodd\" fill=\"#66cc99\" stroke=\"#555555\" stroke-width=\"0.06912000000000001\" opacity=\"0.6\" d=\"M 3.0,1.1 L 3.0098017140329563,1.0995184726672196 L 3.019509032201613,1.0980785280403231 L 3.029028467725446,1.0956940335732208 L 3.038268343236509,1.0923879532511287 L 3.0471396736826,1.0881921264348355 L 3.05555702330196,1.0831469612302544 L 3.0634393284163646,1.0773010453362737 L 3.0707106781186546,1.0707106781186546 L 3.0773010453362737,1.0634393284163646 L 3.0831469612302547,1.0555570233019602 L 3.0881921264348353,1.0471396736825997 L 3.0923879532511287,1.038268343236509 L 3.095694033573221,1.0290284677254462 L 3.098078528040323,1.0195090322016127 L 3.09951847266722,1.009801714032956 L 3.1,1.0 L 3.09951847266722,0.990198285967044 L 3.098078528040323,0.9804909677983872 L 3.095694033573221,0.9709715322745538 L 3.0923879532511287,0.961731656763491 L 3.0881921264348353,0.9528603263174003 L 3.0831469612302547,0.9444429766980398 L 3.0773010453362737,0.9365606715836354 L 3.0707106781186546,0.9292893218813453 L 3.0634393284163646,0.9226989546637263 L 3.05555702330196,0.9168530387697454 L 3.0471396736826,0.9118078735651645 L 3.038268343236509,0.9076120467488713 L 3.029028467725446,0.9043059664267791 L 3.0195090322016127,0.901921471959677 L 3.009801714032956,0.9004815273327803 L 3.0,0.9 L 0.0,0.9 L -0.009801714032956222,0.9004815273327803 L -0.019509032201612955,0.901921471959677 L -0.029028467725446414,0.9043059664267792 L -0.03826834323650912,0.9076120467488714 L -0.04713967368259987,0.9118078735651646 L -0.055557023301960363,0.9168530387697456 L -0.06343932841636467,0.9226989546637264 L -0.07071067811865482,0.9292893218813453 L -0.07730104533627374,0.9365606715836355 L -0.08314696123025457,0.9444429766980399 L -0.08819212643483555,0.9528603263174003 L -0.0923879532511287,0.961731656763491 L -0.0956940335732209,0.9709715322745538 L -0.09807852804032306,0.9804909677983872 L -0.09951847266721969,0.990198285967044 L -0.1,1.0 L -0.09951847266721969,1.009801714032956 L -0.09807852804032305,1.0195090322016127 L -0.09569403357322089,1.0290284677254462 L -0.0923879532511287,1.038268343236509 L -0.08819212643483554,1.0471396736825997 L -0.08314696123025458,1.0555570233019602 L -0.07730104533627379,1.0634393284163643 L -0.07071067811865488,1.0707106781186546 L -0.06343932841636471,1.0773010453362737 L -0.05555702330196042,1.0831469612302544 L -0.047139673682600004,1.0881921264348353 L -0.03826834323650926,1.0923879532511285 L -0.029028467725446563,1.0956940335732208 L -0.019509032201613194,1.098078528040323 L -0.009801714032956464,1.0995184726672196 L -4.379659758543259e-16,1.1 L 3.0,1.1 z\" /></g></svg>"
      ],
      "text/plain": [
       "<shapely.geometry.polygon.Polygon at 0x7fa5ac943e48>"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "buffer1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'POLYGON ((3 1.1, 3.009801714032956 1.09951847266722, 3.019509032201613 1.098078528040323, 3.029028467725446 1.095694033573221, 3.038268343236509 1.092387953251129, 3.0471396736826 1.088192126434836, 3.05555702330196 1.083146961230254, 3.063439328416365 1.077301045336274, 3.070710678118655 1.070710678118655, 3.077301045336274 1.063439328416365, 3.083146961230255 1.05555702330196, 3.088192126434835 1.0471396736826, 3.092387953251129 1.038268343236509, 3.095694033573221 1.029028467725446, 3.098078528040323 1.019509032201613, 3.09951847266722 1.009801714032956, 3.1 1, 3.09951847266722 0.990198285967044, 3.098078528040323 0.9804909677983872, 3.095694033573221 0.9709715322745538, 3.092387953251129 0.961731656763491, 3.088192126434835 0.9528603263174003, 3.083146961230255 0.9444429766980398, 3.077301045336274 0.9365606715836354, 3.070710678118655 0.9292893218813453, 3.063439328416365 0.9226989546637263, 3.05555702330196 0.9168530387697454, 3.0471396736826 0.9118078735651645, 3.038268343236509 0.9076120467488713, 3.029028467725446 0.9043059664267791, 3.019509032201613 0.901921471959677, 3.009801714032956 0.9004815273327803, 3 0.9, 0 0.9, -0.009801714032956222 0.9004815273327803, -0.01950903220161295 0.901921471959677, -0.02902846772544641 0.9043059664267792, -0.03826834323650912 0.9076120467488714, -0.04713967368259987 0.9118078735651646, -0.05555702330196036 0.9168530387697456, -0.06343932841636467 0.9226989546637264, -0.07071067811865482 0.9292893218813453, -0.07730104533627374 0.9365606715836355, -0.08314696123025457 0.9444429766980399, -0.08819212643483555 0.9528603263174003, -0.0923879532511287 0.961731656763491, -0.0956940335732209 0.9709715322745538, -0.09807852804032306 0.9804909677983872, -0.09951847266721969 0.990198285967044, -0.1 1, -0.09951847266721969 1.009801714032956, -0.09807852804032305 1.019509032201613, -0.09569403357322089 1.029028467725446, -0.0923879532511287 1.038268343236509, -0.08819212643483554 1.0471396736826, -0.08314696123025458 1.05555702330196, -0.07730104533627379 1.063439328416364, -0.07071067811865488 1.070710678118655, -0.06343932841636471 1.077301045336274, -0.05555702330196042 1.083146961230254, -0.0471396736826 1.088192126434835, -0.03826834323650926 1.092387953251128, -0.02902846772544656 1.095694033573221, -0.01950903220161319 1.098078528040323, -0.009801714032956464 1.09951847266722, -4.379659758543259e-16 1.1, 3 1.1))'"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "buffer1.wkt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create a `MultilineString` to illustrate a buffer operation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"100.0\" height=\"100.0\" viewBox=\"-1.6400000000000001 0.36 3.7800000000000002 1.7800000000000002\" preserveAspectRatio=\"xMinYMin meet\"><g transform=\"matrix(1,0,0,-1,0,2.5)\"><g><polyline fill=\"none\" stroke=\"#66cc99\" stroke-width=\"0.0756\" points=\"0.0,1.0 1.0,0.5 2.0,1.5\" opacity=\"0.8\" /><polyline fill=\"none\" stroke=\"#66cc99\" stroke-width=\"0.0756\" points=\"1.0,0.5 0.0,2.0 -1.5,1.0\" opacity=\"0.8\" /></g></g></svg>"
      ],
      "text/plain": [
       "<shapely.geometry.multilinestring.MultiLineString at 0x7fa5ac932b00>"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "line1 = LineString([(0.0, 1.0), (1.0, 0.5), (2.0, 1.5)])\n",
    "line2 = LineString([(1.0, 0.5), (0.0, 2.0), ((-1.5, 1.0))])\n",
    "multi_line = MultiLineString([line1, line2])\n",
    "multi_line"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"100.0\" height=\"100.0\" viewBox=\"-1.7479999431039062 0.2520000568960938 3.995999940915595 1.9959998862078123\" preserveAspectRatio=\"xMinYMin meet\"><g transform=\"matrix(1,0,0,-1,0,2.5)\"><path fill-rule=\"evenodd\" fill=\"#66cc99\" stroke=\"#555555\" stroke-width=\"0.0799199988183119\" opacity=\"0.6\" d=\"M -0.027735009811261497,1.8613249509436929 L -1.4445299803774772,0.9167949705662156 L -1.4529526027295947,0.9117586128277628 L -1.4618283172361497,0.9075720678854278 L -1.471071645914485,0.9042756544556032 L -1.4805935704565107,0.9019011188016215 L -1.4903023895236738,0.9004713289998834 L -1.500104601880977,0.9000000547077824 L -1.5099058068649849,0.9004918345543763 L -1.5196116135138185,0.901941932430908 L -1.5291285496037155,0.9043363831021224 L -1.5383649618376436,0.9076521266991159 L -1.547231898516654,0.9118572307984795 L -1.5556439661933665,0.9169111979489928 L 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      ],
      "text/plain": [
       "<shapely.geometry.polygon.Polygon at 0x7fa5ac943be0>"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "multi_line.buffer(0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"100.0\" height=\"100.0\" viewBox=\"-1.70051724753217 0.2332767207612567 3.9162751535604716 2.0504455562046977\" preserveAspectRatio=\"xMinYMin meet\"><g transform=\"matrix(1,0,0,-1,0,2.5169989977272107)\"><path fill-rule=\"evenodd\" fill=\"#66cc99\" stroke=\"#555555\" stroke-width=\"0.07832550307120943\" opacity=\"0.6\" d=\"M -0.027735009811261497,1.8613249509436929 L -1.4445299803774772,0.9167949705662156 L -1.5554700196225228,1.0832050294337843 L 0.0277350098112618,2.138675049056307 L 1.015542483014356,0.6569638392516655 L 1.9292893218813452,1.5707106781186548 L 2.0707106781186546,1.4292893218813452 L 1.0197453049082135,0.3783239486709038 L -0.044721359549995794,0.9105572809000084 L 0.044721359549995794,1.0894427190999916 L 0.7079190373518112,0.7578438801990839 L -0.027735009811261497,1.8613249509436929 z\" /></g></svg>"
      ],
      "text/plain": [
       "<shapely.geometry.polygon.Polygon at 0x7fa5ac943f98>"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "multi_line.buffer(0.1, cap_style=CAP_STYLE.flat, join_style=JOIN_STYLE.mitre)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Convex hull"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"100.0\" height=\"100.0\" viewBox=\"0.92 0.92 2.16 2.16\" preserveAspectRatio=\"xMinYMin meet\"><g transform=\"matrix(1,0,0,-1,0,4.0)\"><g><circle cx=\"1.0\" cy=\"2.0\" r=\"0.0648\" stroke=\"#555555\" stroke-width=\"0.0216\" fill=\"#66cc99\" opacity=\"0.6\" /><circle cx=\"3.0\" cy=\"1.0\" r=\"0.0648\" stroke=\"#555555\" stroke-width=\"0.0216\" fill=\"#66cc99\" opacity=\"0.6\" /><circle cx=\"1.0\" cy=\"3.0\" r=\"0.0648\" stroke=\"#555555\" stroke-width=\"0.0216\" fill=\"#66cc99\" opacity=\"0.6\" /><circle cx=\"2.5\" cy=\"1.5\" r=\"0.0648\" stroke=\"#555555\" stroke-width=\"0.0216\" fill=\"#66cc99\" opacity=\"0.6\" /><circle cx=\"2.0\" cy=\"2.0\" r=\"0.0648\" stroke=\"#555555\" stroke-width=\"0.0216\" fill=\"#66cc99\" opacity=\"0.6\" /><circle cx=\"2.0\" cy=\"1.5\" r=\"0.0648\" stroke=\"#555555\" stroke-width=\"0.0216\" fill=\"#66cc99\" opacity=\"0.6\" /><circle cx=\"2.5\" cy=\"2.5\" r=\"0.0648\" stroke=\"#555555\" stroke-width=\"0.0216\" fill=\"#66cc99\" opacity=\"0.6\" /></g></g></svg>"
      ],
      "text/plain": [
       "<shapely.geometry.multipoint.MultiPoint at 0x7fa5ac953400>"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "points = MultiPoint([(1.0, 2.0), (3.0, 1.0), (1.0, 3.0), (2.5, 1.5), (2.0, 2.0), (2.0, 1.5), (2.5, 2.5)])\n",
    "points"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"100.0\" height=\"100.0\" viewBox=\"0.92 0.92 2.16 2.16\" preserveAspectRatio=\"xMinYMin meet\"><g transform=\"matrix(1,0,0,-1,0,4.0)\"><path fill-rule=\"evenodd\" fill=\"#66cc99\" stroke=\"#555555\" stroke-width=\"0.0432\" opacity=\"0.6\" d=\"M 3.0,1.0 L 1.0,2.0 L 1.0,3.0 L 2.5,2.5 L 3.0,1.0 z\" /></g></svg>"
      ],
      "text/plain": [
       "<shapely.geometry.polygon.Polygon at 0x7fa5ac943f60>"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "points.convex_hull"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Convex hull of a `Polygon`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"100.0\" height=\"100.0\" viewBox=\"-0.08 -0.08 2.16 1.6600000000000001\" preserveAspectRatio=\"xMinYMin meet\"><g transform=\"matrix(1,0,0,-1,0,1.5)\"><path fill-rule=\"evenodd\" fill=\"#66cc99\" stroke=\"#555555\" stroke-width=\"0.0432\" opacity=\"0.6\" d=\"M 0.0,0.0 L 0.5,0.5 L 0.0,1.0 L 1.0,0.7 L 2.0,1.5 L 2.0,0.0 L 0.0,0.0 z\" /></g></svg>"
      ],
      "text/plain": [
       "<shapely.geometry.polygon.Polygon at 0x7fa5ac953898>"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "polygon = Polygon([(0.0, 0.0), (0.5, 0.5), (0.0, 1.0), (1.0, 0.7), (2.0, 1.5), (2.0, 0.0)])\n",
    "polygon"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"100.0\" height=\"100.0\" viewBox=\"-0.08 -0.08 2.16 1.6600000000000001\" preserveAspectRatio=\"xMinYMin meet\"><g transform=\"matrix(1,0,0,-1,0,1.5)\"><path fill-rule=\"evenodd\" fill=\"#66cc99\" stroke=\"#555555\" stroke-width=\"0.0432\" opacity=\"0.6\" d=\"M 0.0,0.0 L 0.0,1.0 L 2.0,1.5 L 2.0,0.0 L 0.0,0.0 z\" /></g></svg>"
      ],
      "text/plain": [
       "<shapely.geometry.polygon.Polygon at 0x7fa5ac9430b8>"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "polygon.convex_hull"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Parallel offset"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "line = LineString([(0.0, 1.0), (1.0, 0.5), (2.0, 1.5), (3.0, -1.0)])\n",
    "parallel = line.parallel_offset(0.5, 'left', join_style=JOIN_STYLE.mitre)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fa5cc3f6588>]"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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CwiI+WbmfdxftpqKfN+OGtuDaVqH/2uu/kLfm7eTDZXv5dURHoupb7P6Hwxvgsz7Q6XHo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\nAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa5a46b1710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot([t[0] for t in line.coords], [t[1] for t in line.coords])\n",
    "plt.plot([t[0] for t in parallel.coords], [t[1] for t in parallel.coords])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Splitting a polygon"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The latter can be used to split a polygon along the line by computing the difference between the polygon and the buffer created out of the line.  The result is a `MultiPolygon`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "split = polygon1.difference(buffer1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"100.0\" height=\"100.0\" viewBox=\"0.92 1.02 2.16 2.06\" preserveAspectRatio=\"xMinYMin meet\"><g transform=\"matrix(1,0,0,-1,0,4.1)\"><path fill-rule=\"evenodd\" fill=\"#66cc99\" stroke=\"#555555\" stroke-width=\"0.0432\" opacity=\"0.6\" d=\"M 1.0,1.1 L 1.0,3.0 L 3.0,2.0 L 3.0,1.1 L 1.0,1.1 z\" /></g></svg>"
      ],
      "text/plain": [
       "<shapely.geometry.polygon.Polygon at 0x7fa5ac953f28>"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "split"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "ename": "TypeError",
     "evalue": "'Polygon' object does not support indexing",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-65-6797f170fdb0>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0msplit\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m: 'Polygon' object does not support indexing"
     ]
    }
   ],
   "source": [
    "split[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "ename": "TypeError",
     "evalue": "'Polygon' object does not support indexing",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-66-e914f78dd4b9>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0msplit\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m: 'Polygon' object does not support indexing"
     ]
    }
   ],
   "source": [
    "split[1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Interpolation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create a line out of two points, and use interpolation to tet a point on that line at a specified distance from the first."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [],
   "source": [
    "point1 = Point((1.0, 1.0))\n",
    "point2 = Point((3.0, 3.0))\n",
    "line12 = LineString([point1, point2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [],
   "source": [
    "point3 = line12.interpolate(1.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Verify that the distance between the first and the new point is 1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9999999999999999"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "point1.distance(point3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Check whether the new point is on the line defined by the first and second point."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "line12.contains(point3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "point4 = Point((3.1, 3.2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "line12.contains(point4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQgAAAD8CAYAAACLgjpEAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nTP8Pz3QSOxArYOAAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa5a15b1d30>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.axes().set_aspect('equal')\n",
    "plt.plot([1.0, 3.0], [1.0, 3.0], 'k-', linewidth=0.8)\n",
    "plt.plot([1.0, 3.0], [1.0, 3.0], 'bo')\n",
    "plt.plot([point3.coords[0][0]], [point3.coords[0][1]], 'ro');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Affine transformations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Import modules."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from math import cos, sin, pi\n",
    "import matplotlib.pyplot as plt\n",
    "from shapely import affinity\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define a function to plot a polygon."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def plot_polygons(*polygons):\n",
    "    plt.xlim(-10.0, 10.0)\n",
    "    plt.ylim(-10.0, 10.0)\n",
    "    plt.grid()\n",
    "    plt.axes().set_aspect('equal')\n",
    "    for polygon in polygons:\n",
    "        boundary = polygon.boundary.xy\n",
    "        plt.plot(boundary[0], boundary[1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create a rectangle to transform."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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h1JUiyZVJHp197tZifuoyMWHC0k/dnySfq6pTur/N4y5msbrX4TLgHOBk4Lzu9RqS07vX\naZLPNfkyM/92ZlvwT10mJkwanLqv5bcG2FlVD1XV08A1zLxeWkGq6lbg53M2L/inLhMTJgewCnh4\n1vp8p+5Pko8kubt7+znJv6Qe4mszWwE3Jbmr+7nHkBz0T1326+u3OYuyXKfuj9uB5glcDlzCzH+o\nlwCfBd6/fNVpAd5cVdNJjga2JLm/+3/5QXm+n7rst6LCpOdT91eMg51nki8CX++5nD5N3GuzEFU1\n3d0+muQ6Zj7WDSVMFvxTlyF8zDnYU/cnQvfC7XcuM188T6o7gNVJTkryEma+KN805pqaSHJYkiP2\nLwNnMdmv1VwL/qnLinpnciBJzgX+FTiKmVP3t1XVn1fV9iT7T93fx6xT9yfUp5OcwszHnF3ABeMt\nZ/Gqal+Si4AbgUOBK7ufWgzBMcB1SWDm39FXq+qG8Za0OEmuBtYCr+x+HvOPzPy05dokHwB+BLzz\neR/H0+kltTCEjzmSVgDDRFIThomkJgwTSU0YJpKaMEwkNWGYSGri/wGPTvorG7PbugAAAABJRU5E\nrkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7a892ff7b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rectangle = Polygon([(0.0, 0.0), (0.0, 2.0), (3.0, 2.0), (3.0, 0.0)])\n",
    "plot_polygons(rectangle)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Perform a rotation over 45 degrees."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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5r+ZMqpqdR+1hP1cCUyQdKml3YD6wJFdNZpZPznMm35E0lcphznrgdKgMDwr8MCLmRESP\npLOBu4ERwHURsSZjTWaWSc7hQU/uZ/nO4UGL+aXA0lptzax9NPvSsJmVhMPEzJJwmJhZEg4TM0vC\nYWJmSThMzCwJh4mZJeEwMbMkHCZmloTDxMyScJiYWRIOEzNLwmFiZkk4TMwsCYeJmSXhMDGzJBwm\nZpaEw8TMksj2tY2SbgU6itl9gT9GxNQa7dYDLwPbgZ6ImJ6rJjPLJ+d3wJ7UOy3pMuClAZrPiogX\nctViZvnlHtEPSQI+Bvxt7m2ZWfM04pzJXwPPR8ST/awPYLmkVcUQoGbWhrINDxoRtxfTC4CbB3ib\nD0REl6T9gWWSHo+I+2psy2MNt7Gy9q2s/RoKRUS+N5dGUhnu88iI2DCI9hcB3RHx3YHadXR0RGdn\nZ5oiW0iZx60ta9/K2i9Jq+q9GJL7MOdo4PH+gkTSGEnjeqeBY6k9jKiZtbjcYTKfPoc4kg6U1DuC\n30Tgl5J+A/wauDMi7spck5llkPVqTkR8ssayncODRsQ64D05azCzxvAdsGaWhMPEzJJwmJhZEg4T\nM0vCYWJmSThMzCwJh4mZJeEwMbMkHCZmloTDxMyScJiYWRIOEzNLwmFiZkk4TMwsCYeJmSXhMDGz\nJBwmZpaEw8TMknCYmFkSwwoTSf8gaY2kHZKm91n3ZUlrJXVKOq6f179F0jJJTxa/xw+nHjNrnuHu\nmTwCnAj8yaBZkg6n8s307wJmA/8uaUSN158P3BMRU4B7inkza0PDCpOIeCwiao2GNRe4JSJej4in\ngbXAjH7a3VBM3wB8ZDj1mFnz5BrqYjJwf9X8hmJZXxMjYmMx/RyVcXRqqh4eFHhdUhkH63or8EKz\ni8ikrH0ra7866n3BLsNkkOMJD1tEhKR+xyqNiEXAoqKmB+sdurAdlLVfUN6+lblf9b5ml2ESEUcP\noZYu4OCq+YOKZX09L2lSRGyUNAnYNIRtmVkLyHVpeAkwX9JoSYcCU6gM/1mr3SnF9ClAsj0dM2us\n4V4anidpA/B+4E5JdwNExBpgMfAocBdwVkRsL17zw6rLyN8GjpH0JJVBzr89yE0vGk7dLays/YLy\n9s39Kiii39MUZmaD5jtgzSwJh4mZJdE2YTLcW/fbhaSLJHVJWl38zGl2TcMhaXbxuayVVKo7nCWt\nl/Tb4nOq+1Jqq5B0naRN1fduDeVRl7YJE4Z/6347+X5ETC1+lja7mKEqPocrgeOBw4EFxedVJrOK\nz6md7zW5nsrfTrW6H3VpmzBJcOu+Nd4MYG1ErIuIbcAtVD4vayERcR/wYp/FdT/q0jZhMoDJwLNV\n8/3dut9OPivp4WL3s52fpC7jZ1MtgOWSVhWPe5TJoB916ZXr2ZwhadSt+802UD+Bq4CLqfxDvRi4\nDDi1cdVZHT4QEV2S9geWSXq8+L98qezqUZdeLRUmmW/dbxmD7aeka4E7MpeTU9t9NvWIiK7i9yZJ\nt1E5rCtLmNT9qEsZDnMGe+t+Wyg+uF7zqJx4blcrgSmSDpW0O5UT5UuaXFMSksZIGtc7DRxLe39W\nfdX9qEtL7ZkMRNI84HJgApVb91dHxHERsUZS7637PVTdut+mviNpKpXDnPXA6c0tZ+giokfS2cDd\nwAjguuJRizKYCNwmCSp/RzdFxF3NLWloJN0MzATeWjwe83Uqj7YslvRp4BngY7t8H99Ob2YplOEw\nx8xagMPEzJJwmJhZEg4TM0vCYWJmSThMzCwJh4mZJfH/TLiiXts1MloAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7a8926a400>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_polygons(rectangle, affinity.rotate(rectangle, 45))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Or over $\\pi/4$ radians."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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5r+ZMqpqdR+1hP1cCUyQdKml3YD6wJFdNZpZPznMm35E0lcphznrgdKgMDwr8MCLmRESP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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7a8939aa20>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_polygons(rectangle, affinity.rotate(rectangle, pi/4, use_radians=True))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By default, the rotation's center is that of the shape to be rotated, however, it can be set explicitly."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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IlfvWVnVsVR2XpHXNXgzJfZhzPPDccEEiaW9JUwangRNp3EbUzCa43GGyjCGHOJIOljTY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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7a892750f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_polygons(rectangle, affinity.rotate(rectangle, 45, origin=(0.0, 0.0)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Scale the rectangle."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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bTZ8LNNvTkTRcCz00fFaSrcCbgRuT3AxQVRuB64AHgJuAC6rq+e45X5lyGPkzwGlJHmHQ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\nZA/mYk9IAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7a89157a20>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_polygons(rectangle, affinity.scale(rectangle, 2.5, 2.5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Scale only along the $y$-axis."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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kLaz5nho+M8k24K3A9UluBKiqzcC1wH3ADcD5VbW7e8/XJ51G/jxwSpKHGBQ5//w+bnrd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\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7a890bf8d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_polygons(rectangle, affinity.scale(rectangle, yfact=2.5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Skewing the rectangle in the $x$-direction."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7a8902c208>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_polygons(rectangle, affinity.skew(rectangle, xs=30.0))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Skewing in both $x$ and $y$ direction."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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Rls/eQzYejpmVVK36M/k0cH0nrwVwq6Q1wGX5yH3/wMODNraytq2s7eoJZSNN9HBj6VZg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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7a89057cc0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_polygons(rectangle, affinity.skew(rectangle, xs=30.0, ys=30.0))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Setting the origin explicitly for the skew."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7a891fb4a8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_polygons(rectangle, affinity.skew(rectangle, xs=20.0, origin=Point((0.0, 0.0))))"
   ]
  }
 ],
 "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": "3.5.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}