reeks4.ipynb

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    "# Map Projections and Coordinate Systems\n",
    "Course at Ghent University, Department of Geography  \n",
    "Professor Frank Canters\n",
    "\n",
    "## Session 4: Cylindrical Projections and Projection Optimisation\n",
    "Performed in Python using Jupyter Notebook.\n",
    "\n",
    "Written by Manuel Claeys Bouuaert, <manuel.claeysbouuaert@ugent.be>, 2015"
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   "source": [
    "Choosing a good projection for a specific area means choosing:\n",
    "\n",
    "- projection type\n",
    "- projection parameters (standard parallel, Miller coefficients, ...)\n",
    "- coordinates of pseudopole  \n",
    "\n",
    "In this session, we will work with small scale projections (world maps) and only change projection parameters."
   ]
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    "A good choice of the projection parameters is one which minimises the average distortion over the concerned area:  \n",
    "\n",
    "$$ E = \\frac{1}{S} \\int_S e \\,\\mathrm{d}S $$\n",
    "\n",
    "- $E$ is the average distortion\n",
    "- $S$ is the concerned area on the sphere or ellipsoide\n",
    "- $e$ is the local distortion\n",
    "- $\\mathrm{d}S$ is the area of a local infinitessimal surface element"
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    "In practice, this integral is approximated by a discrete sum, by dividing the sphere into small cells:\n",
    "\n",
    "$$ E = \\frac{1}{S} \\sum_{i} e_i \\, S_i $$\n",
    "\n",
    "$$ S = \\sum_{i} S_i $$\n",
    "\n",
    "- $E$ is the average distortion\n",
    "- $S$ is the concerned area on the sphere or ellipsoide\n",
    "- $e_i$ is the local distortion at the center of a cell\n",
    "- $S_i$ is the area of a local cell"
   ]
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   "source": [
    "Most (non-poly/pseudo) cylindrical projections are invariant under mirroring of $\\phi$ or translation in $\\lambda$, and hence their distortions are too. In those cases, this symmetry means we must only perform a summation over one half meridian: $\\phi = 0 \\rightarrow \\frac{\\pi}{2}$. "
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   "source": [
    "When deviding the globe ($\\phi = -\\frac{\\pi}{2} \\rightarrow \\frac{\\pi}{2}$ and $\\lambda = -\\pi \\rightarrow \\pi$) into cells of $\\Delta\\lambda$ and $\\Delta\\phi$, the area of a cell scales with $\\cos\\phi$.  \n",
    "The area of the cell at $(\\phi_i,\\lambda_i)$ is $S_i = P_i \\cos{\\phi_i} \\, \\Delta\\phi \\, \\Delta\\lambda$  \n",
    "Since $\\Delta\\phi$ and $\\Delta\\lambda$ are independent of $i$ and also appear in $S$, they cancel out, leading to $S_i = P_i \\cos{\\phi_i}$.\n",
    "\n",
    "The weight factor $P_i$ can be used to select a specific area, by putting $P_i = 1$ there and $P_i = 0$ elsewhere."
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    "Which local distortions are used?  \n",
    "\n",
    "- Local angular distortion: $e_{an,i} = 2\\Omega_i = 2\\arcsin(\\frac{a_i-b_i}{a_i+b_i})$\n",
    "- Local area distortion: $e_{ar,i} = \\sigma_i^p-1 = (a_ib_i)^p-1$\n",
    "- Local mean linear scale distortion: $e_{ab,i} = \\frac{a_i^q+b_i^r}{2}-1$\n",
    "\n",
    "To ensure that enlarging and shrinking with the same factor has the same weight, we use\n",
    "\n",
    "- $p$ = 1 if $\\sigma_i$ ≥ 1 and $p$ = -1 if $\\sigma_i$ < 1\n",
    "- $q$ = 1 if $a_i$ ≥ 1 and $q$ = -1 if $a_i$ < 1\n",
    "- $r$ = 1 if $b_i$ ≥ 1 and $r$ = -1 if $b_i$ < 1.\n",
    "\n",
    "These are all 'linear distortions' and are used in (1). Other authors have used similar measures, but square the distortions in order to obtain positive values ($e_{an,i}=(\\sigma-1)^2$, $e_{ab,i}=\\frac{1}{2}((a-1)^2+(b-1)^2$). This gives more weight to large distortions, which is yields unfavorable results for small scale projections. Some authors have proposed to use the logarithmic function in distortions. This also allows to attribute enlarging and shrinking with the same factor the same weight, and additionally weights extreme distortions relatively less.  \n",
    "(1) *Canters, F., 2002. Small-scale Map Projection Design. CRC Press.*"
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    "- Equivalent projections ($e_{ar,i} = 0$) $\\rightarrow$ minimise $e_{an,i}$\n",
    "- Conformal projections ($e_{an,i} = 0$) $\\rightarrow$ minimise $e_{ar,i}$\n",
    "- other projections $\\rightarrow$ minimise $e_{ar,i}$, $e_{an,i}$ and/or $e_{ab,i}$"
   ]
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   "source": [
    "### Exercises in Python"
   ]
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   "source": [
    "Load modules"
   ]
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   "source": [
    "%matplotlib inline \n",
    "# show plots inline\n",
    "import os # to access UNIX commandline\n",
    "import urllib.request # to download files\n",
    "import zipfile # to unzip files\n",
    "import numpy as np # to use arrays etc\n",
    "from scipy import misc # to read images etc\n",
    "import matplotlib.pyplot as plt # to plot\n",
    "import matplotlib.cm as cm # to have colormaps\n",
    "from ipywidgets import * # to have interactive widgets\n",
    "from IPython.display import display, HTML # to have interactive widgets"
   ]
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   "source": [
    "Define latitudes $\\phi$ of cells. (From here on, we drop the index $i$)  \n",
    "We are asked to use cells with an interval of 1° (in stead of 2.5°)."
   ]
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     "text": [
      "[  0.5   1.5   2.5   3.5   4.5   5.5   6.5   7.5   8.5   9.5  10.5  11.5\n",
      "  12.5  13.5  14.5  15.5  16.5  17.5  18.5  19.5  20.5  21.5  22.5  23.5\n",
      "  24.5  25.5  26.5  27.5  28.5  29.5  30.5  31.5  32.5  33.5  34.5  35.5\n",
      "  36.5  37.5  38.5  39.5  40.5  41.5  42.5  43.5  44.5  45.5  46.5  47.5\n",
      "  48.5  49.5  50.5  51.5  52.5  53.5  54.5  55.5  56.5  57.5  58.5  59.5\n",
      "  60.5  61.5  62.5  63.5  64.5  65.5  66.5  67.5  68.5  69.5  70.5  71.5\n",
      "  72.5  73.5  74.5  75.5  76.5  77.5  78.5  79.5  80.5  81.5  82.5  83.5\n",
      "  84.5  85.5  86.5  87.5  88.5  89.5]\n"
     ]
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   ],
   "source": [
    "lat_deg = np.arange(0,90,1)+1/2\n",
    "lat=lat_deg/180*np.pi # use radians by default\n",
    "print(lat_deg) # only use degrees when printing or plotting"
   ]
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   "source": [
    "Define weight factor $P$.  \n",
    "We will consider the entire sphere by default."
   ]
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   "source": [
    "P_ones = np.ones(len(lat))\n",
    "P = P_ones"
   ]
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   "source": [
    "The values `lat` and `P` will be used as global parameters"
   ]
  },
  {
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   "source": [
    "Define functions for different local distortions $e_{..}$ times the local surface area $S$.  \n",
    "They take the local values of $h$ and $k$ as input: the scale factor along the meridians and parallels.  "
   ]
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   "source": [
    "def eanS(h,k):\n",
    "    global lat, P\n",
    "    a = np.amax(np.vstack((h,k)),axis=0)\n",
    "    b = np.amin(np.vstack((h,k)),axis=0)\n",
    "    twoOmega = 2*np.arcsin((a-b)/(a+b))\n",
    "    eanS = twoOmega*P*np.cos(lat)\n",
    "    return eanS\n",
    "def earS(h,k):\n",
    "    global lat, P\n",
    "    a = np.amax(np.vstack((h,k)),axis=0)\n",
    "    b = np.amin(np.vstack((h,k)),axis=0)\n",
    "    sigma = a*b\n",
    "    p = np.sign(sigma-1)\n",
    "    earS = (np.power(sigma,p)-1)*P*np.cos(lat)\n",
    "    return earS\n",
    "def eabS(h,k):\n",
    "    global lat, P\n",
    "    a = np.amax(np.vstack((h,k)),axis=0)\n",
    "    b = np.amin(np.vstack((h,k)),axis=0)\n",
    "    q = np.sign(a-1)\n",
    "    r = np.sign(b-1)\n",
    "    eabS = ((np.power(a,q)+np.power(b,r))/2-1)*P*np.cos(lat)\n",
    "    return eabS"
   ]
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   "source": [
    "The average distortion $E$ is the average of the local distortion $e$"
   ]
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   "source": [
    "def E(eS):\n",
    "    global lat, P\n",
    "    S = np.sum(P*np.cos(lat))\n",
    "    E = np.sum(eS)/S\n",
    "    return E"
   ]
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   "source": [
    "### Ex. 1: Equivalent Cylindrical Projection with varying standard parallel  \n",
    "Given the secant Equivalent Cylindrical Projection, which value for standard parallel yields the lowest angular distortion over the entire sphere?"
   ]
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   "source": [
    "The Equivalent Cylindrical Projection:\n",
    "\n",
    "$ x = R \\lambda \\cos \\phi_0 $  \n",
    "$ y = \\frac{R}{\\cos \\phi_0} \\sin \\phi $  \n",
    "\n",
    "With (see course notes slide 4.23)\n",
    "\n",
    "$ h = \\frac{\\cos \\phi}{\\cos \\phi_0} $  \n",
    "$ k = \\frac{\\cos \\phi_0}{\\cos \\phi} $"
   ]
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   "source": [
    "Define $h$ and $k$ for this projection, for varying $\\phi_0$"
   ]
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   "source": [
    "def h_ecp_lat0(lat0):\n",
    "    global lat\n",
    "    h = np.cos(lat)/np.cos(lat0)\n",
    "    return h\n",
    "def k_ecp_lat0(lat0):\n",
    "    global lat\n",
    "    k = np.cos(lat0)/np.cos(lat)\n",
    "    return k"
   ]
  },
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   "source": [
    "Plot the local angular distortion $\\times$ surface: $e_{an} \\, S$."
   ]
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PSaOBiRExJts+C4iIuLDdcfkHa2bWgLozhUe9JIp1gcXAAcCzwD3AURGxMNfA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\nAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x117341358>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def plt_eanS_ecp_lat0(lat0):\n",
    "    global lat, P\n",
    "    h = h_ecp_lat0(lat0)\n",
    "    k = k_ecp_lat0(lat0)\n",
    "    f, ax = plt.subplots(1) # make figure and axis\n",
    "    ax.plot(eanS(h,k)*180/np.pi,lat*180/np.pi)\n",
    "    ax.set_xlabel('Local angular distortion $e_{an}$ (deg) * surface $S$')\n",
    "    ax.set_ylabel('Latitude $\\phi$ (deg)')\n",
    "    ax.set_xlim(xmin=0,xmax=180)\n",
    "    plt.show()\n",
    "    \n",
    "v = interactive(plt_eanS_ecp_lat0, lat0 = (0,90/180*np.pi,5/180*np.pi));\n",
    "display(v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Notice:\n",
    "\n",
    "- Zero local angular distortion at $\\phi = \\frac{\\pi}{2}$ because $\\cos(\\phi) = 0$  \n",
    "- Zero local angular distortion at $\\phi = \\phi_0$ because no distortion at standard parallel\n",
    "\n",
    "For which $\\phi_0$ is the surface between the curve and the $y$-axis minimal?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Compute and plot average angular distortion $E_{an}$ for $\\phi_0$ = 0°, 5°, 10°, ... 60°."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "lat0_range = np.arange(0,95,5)/180*np.pi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "Ean_ecp_lat0 = np.zeros(0) # create empty array\n",
    "\n",
    "for lat0 in lat0_range:\n",
    "    h = h_ecp_lat0(lat0)\n",
    "    k = k_ecp_lat0(lat0)\n",
    "    Ean_ecp_lat0 = np.append(Ean_ecp_lat0,E(eanS(h,k)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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N+M+eCKxWOVGYWbUpZh2FaCkMSPa45AUcvZUThZlVm2J6FDcCT0u6O3t+AqnU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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1170ba1d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f, ax = plt.subplots(1) # make figure and axis\n",
    "ax.plot(Ean_ecp_lat0*180/np.pi,lat0_range*180/np.pi,'b.-')\n",
    "ax.set_xlabel('Average angular distortion $E_{an}$ (deg)')\n",
    "ax.set_ylabel('Latitude of standard parallel $\\phi_0$ (deg)')\n",
    "ax.set_xlim(xmin=0)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "The average angular distortion is the smalles for $\\phi_0$ = 30°."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Compare this to the various known secant Equivalent Cylindrical Projections:\n",
    "\n",
    "- Lambert $\\phi_0$ = 0°\n",
    "- Behrmann $\\phi_0$ = 30°\n",
    "- Gall-Peeters $\\phi_0$ = 45°\n",
    "\n",
    "Behrmann correctly computed position for the standard parallels in 1910, by minimising the angular distortion!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "The Behrmann Projection  \n",
    "![Behrmann Projection](https://upload.wikimedia.org/wikipedia/commons/f/f5/Behrmann_projection_SW.jpg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Ex. 1-bis: Equivalent Cylindrical Projection with varying Standard Parallel, weighted with landmass\n",
    "Redo these calculations while ony averaging over land areas."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Download land polygons Shapefile from Natural Earth database"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "if not os.path.exists('downloads'):\n",
    "    os.makedirs('downloads')\n",
    "if not os.path.isfile('downloads/ne_110m_land.shp'): # Do this only if it doesn't already exist\n",
    "    urllib.request.urlretrieve('http://www.naturalearthdata.com/http//www.naturalearthdata.com/download/110m/physical/ne_110m_land.zip','downloads/ne_110m_land.zip')\n",
    "    zf = zipfile.ZipFile('downloads/ne_110m_land.zip')\n",
    "    zf.extractall('downloads')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Convert Shapefile to raster image."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "os.system('gdal_rasterize -a scalerank -ot Byte -te -180 -90 180 90 -tr 1.0 1.0 -l ne_110m_land downloads/ne_110m_land.shp land.tif')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "This calls the `gdal_rasterize` function through a terminal command (UNIX). It reads the `ne_110m_land.shp` file and creates a rasterized version: the `land.tif` file. The following options are specified:  \n",
    "* `-a scalerank` : the name of the attribute used in the input file\n",
    "* `-ot Byte` : the output is binairy (avoid using the default Float64)  \n",
    "* `-te -180 -90 180 90` : the input is considerd in the range x=-180...180 and y=-90...90 (default projection EPSG::4326)  \n",
    "* `-tr 1.0 1.0` : the target resolution is one pixel per 1.0-by-1.0 input block.  \n",
    "* `-l ne_110m_land` : the name of the layer to be considered\n",
    "\n",
    "Alternatively, one could use the osgeo.gdal package:  \n",
    "`from osgeo import gdal`  \n",
    "`raster = gdal.RasterizeLayer(...)`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Load the image into a variable. It is stored as a `numpy.ndarray`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "numpy.ndarray"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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DTE1NpaZr1aoVrly5Am9vbxw+fBjVqlUDAPzwww8YNWoUAFF7ad++PUxNTZGT\nk6MWnbt16ybmctnKygpmZmZqyVtfsLe3x7t375Ceng4DAwN07doVeXl52LNnDx/n119/FVt/rggz\nMzNYWVlpQl3NIM/iKxIANYv/OkA0rt4FwJsycV7LSEsBAQG86ML6z4qi6Jg4XZNGjRoJXWWCoKin\nXp56SUxMpLt378otr0aNGrRv3z4+XN3tpX379hJj9FXtpKWoqCgqLCyku3fvUmJiIl8PoaGh5OTk\nRABo9uzZ9PbtW7nvMycnh+7evUtr1qwhPz8/lduDOomIiBCzlYAGh19I3EgHAJgF4A7Eh1/uyIiv\n+drQMqNHj+Z/fuuDCD3hIxRHjhwhLy8v8vLyIgcHB2rUqBF/7eXlRdnZ2VRQUED//POPQnn37h0R\nEQ0cOFDmaTlHjhyh2rVrS7huHT16NHl5eRHHcWRgYECenp4EiM6g9fLyKpd7Z29vbzH9Bg0aJHg7\n07ZERkaSgYEBDRo0SKy+pfnRL0vJe9+5cycfV2ijXhZoyqgDsABgVfzZEsB5AH0gmiidS1VsorSE\nMWPG6M1hwlXVqJfG19dXbGz87Nmz9Pfff9ORI0eUNiDKsHfvXrFeemkGDx5MgwcPptTUVOrbty8/\nfsuOwSu/DB48WKKeT548SY0bNyZA0qjLe+/6ZtQr4vvFCcDh4pl4IwC7iSiM47jLAPZxHDcJwGMA\nwytQht6xa9cuGBoaYseOHUKrIpchQ4bo5RyAOrl+/TqSkpJw7tw55ObmAgC++uor/qg7RXTu3Fnp\nOYnhw2V/DY4ePcp/Dg0NVSo/hnxK12kJjRs3xhdffIHbt2+jYcOGAIAzZ84gNTUVX331FXr16oX9\n+/fD3NwcQ4YM4dPpi0tpHnkWX5MiKrryMm3aNBo3bpzgPZayMmTIEJo2bZrCMcXKzpUrV6hXr14V\nqsvJkydX6ABheSQmJtLo0aMFby/6Khs3bpSo03PnztHhw4f567///pvc3Nz4NFlZWTRt2jT6/vvv\nNfJO1QWgpTF1VaVYsUpNenq6Tk1U+fj40IMHD4SuFsG5du0a9e3bVy116ufnR0uWLOFFVnlLliyh\n//3vf0rrWLKUjkn5RJmlusOGDePf4ffff68Ta9CVAWBGXVBu3LhBAKhFixb06aefCtbIP/vsM0pI\nSBC6OgQnLi6O+vTpo7F6LtvLK11e165dafbs2bw8e/aMj1c2HTPqFRMjIyP65Zdf5LaFYcOG0dSp\nU1U+UFoUtCmxAAAgAElEQVRoAM2NqTOUoFatWpg1axYePnyIxYsXAwAOHjyodT369u2Lxo0ba71c\nXePevXsICwvTWP7BwcH8QSlNmzZF/fr1+fKio6MRHR3Nx33w4AF/fN6ff/4pdsDKo0ePNKZjVcDA\nwIAfN5fF9OnT4e7uDgcHBy1ppR2YUdcwdnZ28Pf3x+7duxEZGYklS5YgPz8fx44dK3ees2fPhpeX\nF389ZswYAMDu3bsl4h45ckRqOEMzFBYWYuvWrQAABwcHuYbl0KFDYteHDx/GDz/8gNmzZ6tcbp06\ndbB8+XLExsbyJ1JVZQoLCxEZGclv9pKGPh69qAzMqGsBR0dHjBkzBjt37kR0dDSaNGlS7ry+++47\nzJkzR2zVhb29PQCge/fuEvGTkpLKXVZl4/r161i4cKHWyktLS0NaWprS8T98+IB9+/ahVatWGDp0\nqEq62tjY4LPPPkPXrl1RVFSElStXlkflSkNhYSFOnDghtBrCIG9sRpMiKrrqEB4eTnXq1CEnJyda\nsmQJDR06VOVxwm+++YbS0tJUKjcwMJAAUJ06dejMmTMaejrd5sWLF+Tp6UmNGjUSfKxXkbi5uVFC\nQgK9efOGfvzxR6XTmZmZ0eTJk4mIKC0tja5cuaKTq6+0JWZmZhQXFydwy9MMUDCmzoy6lsjMzKSk\npCTy9/enoKAgSk1NpaSkJGrfvr3SDbXsjkRlePv2LU2ePJkAkQOqkSNHytzOXhnJysqiOnXqCG5k\nlBUTExMaPXo0Eam++cjMzIymTZvGP/urV68oKSmJl5KNN1VBKvMqL0VGnQ2/aAlLS0s0aNAACxcu\nREBAAEJDQzF27FiNO1yqXr06Vq1ahezsbADApk2bYGFhodEydQkXFxe5zrl0jby8POzbtw9GRkbY\ntGkTnj59qvQwTE5ODl6+fIkdO3bgwYMHWLBgAT80BwCXLl1CYWEhXFxc1OZETBd5+vQpXFxchFZD\nMCrqpZGhIiVGvGQHY1RUFDw8PBSm8/f3h7+/f7nKtLCwwM6dO+Hj44Nq1apVmZ2kRKRXBr2EgoIC\nZGZmwsTEBObm5uA4Tum0hw4dQkxMDAICAkBE8PPzA8dx4DgOjx49QvXq1ZGVlSWzlxcYGKjBJ9MO\ntra2+PDhA2rXri20KsIgrxuvSUEVG35RRLNmzeT+nCyPP+ecnBzKysqirKwsevv2Lb19+5by8/M1\noL1uoi4/5UJJaZ8j6tzEVnp9vDR+/vlnsrGxUbtzutKnTalTLCwsyMbGRuwousp8TgAUDL+wnrqO\ncPv2bTRt2lTm/RIf0apw+PBhrFmzBqtXr4anpyc8PT3x119/4eHDhygoKKioyjrNkydPSjoPesv7\n9+/x5s0btefbsWNHufcXLlyI9PR0DBw4sELluLq6ol69eqhXrx6aNWuGW7dulTsvY2NjODs7AxD5\niS/Jt169ejh48CDS09Ph7u7Ox69Xr16FdNdr5Fl8TQpYT12Cq1evyu2RyJsovXLlCl25coVu374t\ntww/Pz/y9PSk8PBw/gT6yoisY9/0TUp+oS1cuJBsbGzUkqey3jlnzZpFnp6e5OnpWa5ee9nVJy9e\nvKCmTZuqnI+xsTENHz6cLl68SJ6enrRo0SK1txd9Agp66pXOqN++fZtevnypkbw1Tb9+/eQ27i+/\n/FKqA6mIiAg+TukDiOXx+eefU/fu3YmIKDs7m86cOUNnzpypVMvA9H34pbRRJxINi5Q+BLk88vHH\nH1NOTo7Kdeni4qJ0GS1atKDu3btTUlKSRD6JiYm833hlxMDAgD+i782bN3TmzBm6c+dO+RtFJaDK\nGfV169bRP//8o5G8tcGQIUPkNvJp06bxp9uEhYXRgQMHyMTEhACRE6Nvv/1WpfLy8/Np48aNfP5e\nXl504MABiouLo7i4OL1eGlYZjPrQoUPFlqBW5CzcXr160atXr8pVl8oa9TZt2ij8/p09e1ZpnUuP\njZ89e5ZcXFzoP//5T7meobKgyKhXuiWN06dPF1qFCrF371589dVXAIDnz5/j5MmTYvfXr18PAGjT\npg0WLFiAp0+f8vfs7OywYsUKlcorLCzExYsX4evry4f9/fff/Hi0k5MT6tevX65nYVScI0eOwNDQ\nEEuWLEGTJk0wYMAAxMfHK+3zHQB69uyJevXqSSxxVDft2rXD8uXL0aFDB7XlWVhYiJCQEIwaNQo1\na9bE/PnzMW3aNIXptmzZAktLS7luAiot8iy+JgUa6qlXJu7fv0+zZs2i3r17K9WrcXV1FVplnWH9\n+vViJ8jru5Q+qee3334jKysrhWm6du1Ks2bNUsuQmqKeert27ejs2bNK5aVKTx0A2djY0J9//qmS\nvpX5+wAFPXWOintk2objOBKqbH3j2rVrEj12AFi7di1evnzJX1erVg1ff/01f/3RRx9h2LBhWtFR\n13B1dcWzZ8+EVkNt9O/fH8HBwfwKD0XP16VLFwQFBSlc6aIsa9aswaxZsyRWTTVv3hyjR49Gx44d\n0aNHD6XyevTokZiTuVOnTiEmJkZuGkdHR/z6669ivyjlwXEcXF1dkZKSolR8fYLjOBCR7M0L8iy+\nJgWsp15hQkJC5K6IaNasmdhJL1UJVSb29EX69u1L9+7dk/l8jRo1onXr1tG6detUOpBDWaTNUfj4\n+FQ43zlz5ij1/N7e3krniSrcU690Y+pViZEjR8LU1BTZ2dm8+93SODs7V+31upWMU6dOYdq0afjj\njz/Ewu3s7LB27Vo4OTmhZ8+eGiv/zz//xMiRIzWWvzrZvXs3cnJyMHr0aDRv3hw//vij0CppDWbU\n9ZyS4ZWSwxYePnzIT7Tev38fV69eRevWrQXTj6Fezpw5g9GjR2PVqlWwsrICIHI9Ic3tsroZMWKE\nRo36/Pnz0alTJ0yfPr3Ch4SMHj0aOTk5qFmzJlJSUrB48eKqY9jldeM1KWDDLxohMzOTfv/9dwJA\ngwcPpsePHytM8/LlSxozZowWtNMeV65cIVNTU8GHTDQlJctaS3P58mXy8vKiX3/9VWP1WnpPBNQ0\n/PLo0SMaNGgQHTx4kIhER0A6ODhUaPiF6N92nZ6eThs2bNBovWgTKBh+YUa9ElKyusDGxoaWLVsm\nM16rVq2oSZMm1LBhQ/Lw8KDY2Fhq0qSJmOzbt4+IiL744gtKSEigrl27UnZ2trYepULcvXtXcOOr\nKWnYsCFlZ2eLvavatWsTIO4zRt18+PBB7UZ92bJltGbNGsrIyODD7t+/z/9TdnJyojt37ijVQSES\ntWsiory8PH6fRUZGhkKfN/oCM+pVkNzcXHrx4gX99ttvZGFhQb///jsREU2cOJHOnz/Pxys98eXh\n4SF1qVm1atXI0dGRjI2Nyc7OjgwMDJTeZq4LPHz4UHADrClxdHSUGq4to/7xxx/Trl276LvvvqtQ\nnnPmzBFbslnCixcvyNzcXKWDoVu1akWPHj2SCD9+/HiF9dQVFBl15tCrEmJiYgInJydYW1sjKysL\nM2bMgImJCfr3748OHTqgTZs2MDExEfOpffPmTfTp00cir/fv3yM1NRX5+fl48+YNioqKYGNjo83H\nqRB169attEf6lT6ouoShQ4fyG9Q0gbm5OZKSktCiRQtERkbCzMwMGRkZFc536tSpOHLkiFiYk5MT\n0tPTVToY+uLFi6hTpw5/HR8fjz59+iA3N1cteuoDbKK0EjN58mRMnjwZgGinrbGxMQwMDGBkZCSx\n3piIlPbcmJ+fDwsLC7x+/Rrm5uZq11udcBwHU1NTmJubo6ioiPdjXxkxMDCAubk5jIw097XmOA4N\nGjRAfHw8H7ZlyxbY2dkhKCioXHkGBwcjMzMTeXl5yMrKgpmZGQwMRP1NExMTpfLIyclBUVERzMzM\neP/zT58+ha+vL4KCgtC7d2++s7N48eJy6ak3yOvGa1LAhl8E48WLF+Ts7Fzhn//Ozs6UkpIiJunp\n6UI/nkzOnz9PdnZ2Sj2bi4uLmCibTkhRdSJRHZw4cYJcXFxo8eLFFc5r2LBhBIDCw8OpoKBApbQl\nPnEuX75MRUVFYvfOnz+v0SEpbQMFwy/MqFdBhg4dqrEt9JMmTVJpDFTbhIWFkbu7Ozk5OZGTkxO5\nu7tLFWnpdHlDk6mpKU2cOFGAGi0/r1+/pvj4eL69zJgxgywsLAiQvrpHGikpKRQfH09t27bl60Kf\n5nzKAzPqDKlo0kBNmzaNnj9/LvQjymXXrl20e/duldIcPXqUOnXqxIu9vb3gxhwQHVY9btw4DdWU\n5jh69CjVrVtXrL2U9LiVMeoPHjyggQMH8vXg6elJnTp1KpdrYX2CGXWGGOfPn6djx45pfDhB2mqG\nykbJcIGQYmhoSBMmTBC6KsrNgQMHqGHDhvR///d/9OzZM5o3bx5Vq1aNtm3bJnMI5tGjR3Ts2DHy\n8fERq4s1a9ZU+l46ETPqjDJUxB+3KjJp0iSl1xXrI+fPnxf7yS+U+Pr6Cl0VFWbPnj3k5uZGX3/9\nNW3fvp2mTZtG5ubmEgb6+fPntH37dpo4caLM+lCmh3/58mXeh44+osios9UvVYgTJ06I+V/XJH/8\n8QcGDBggtrysMvHHH3/gypUrgpU/adIkWFpaYs2aNYLpoC5GjBiB06dPY+3atQCAuXPnYurUqTA0\nNBSLd+fOHYwfP16pPEvXS61ateDj48Nf7927F40bN0ajRo3UoL3uwdapVyG2bt2q1TXb+/btq7AP\nD4YkU6dOxcqVKyuFQS9h0KBB/MHrQUFB4DiOX5pYHl6/fs1L2fXpvXr1gqenJ3+9fv36yrWGXV43\nXpMCNvyiVUJCQqhhw4ZaHx747LPPaMaMGbxUFnx9fQUbcqlM58iW5vjx42JttOzwy5MnT2jChAky\n68Xf35/ev39Ps2fPphkzZlBAQIDCMtesWUP29vbk6+tLubm5Gnoy9QIFwy/MqFcRhDRCpSUnJ0fl\nc1R1kXPnzgkypj5v3jx68+aN0I+vMUrP+XzxxRc0efJksfsHDhxQ+M+uxP2FMv7UT548Sdu2baPq\n1atTVlYW5eTk0Pjx43lZu3atRp6zIjCjziAiomvXrlGHDh0EN+oDBgwgMzMzGjx4MC8lvmn0jQsX\nLpC7u7vW6m7u3LmUlpYm9GNrlJiYGKpTpw7/zKUPniYievr0KR05coSXXr16EQBavHgxvXv3johE\nRn3v3r0SRv3UqVNix+ItXbqU4uPjiUi0xHfAgAE0YMAAsTqvWbMm305LvEgKDTPqDCIiCgoK0pl1\n1WXF2dmZ1q9fL3QVlQttrSaaM2dOpTfoJVy+fJlq1Kgh1aiXJSEhgaKioujt27d8WExMDBUUFNCx\nY8fElns+e/aMkpKS+Otbt27x6f755x+KioqiU6dOyXwH9evXpy5dulBUVJSan1g1mFFnEJHuDL/I\nEgcHB/Lw8CAPDw86ffq00NWlNImJieTm5qbx+qkK6/5Lc/v2bTI1NVVo1OWRnJxMHh4eSsdv164d\nFRYW0o0bN2j//v0y30XdunXp0qVLRETUs2dPys7OJg8PD+rVq1e5dVUFRUZd4ZJGjuO2AhgI4CUR\ntSwOswWwF0BdAI8ADCeid8X3vgcwCUABgBlEFKaoDAYjLS0NaWlpAKBXKxEaNmwIMzMzodWodDRr\n1gyJiYnIzs5G7dq1AQAff/wx9u7dq3Qezs7OCAtT3vzEx8fDwMAALVu2lBvv8ePH6N+/P0xNTXH0\n6FGYmJjgxIkTYkswT548iaioqHI7OasQ8iy+6J8COgNoDSCuVFgQAP/iz3MBBBZ/bg7gGkTeH+sB\nSATAychXK//VGCKys7Np0KBBgvfIlRFdGbtUlg8fPlDNmjVZT10DFBUV8YedGBoakqWlJS8rVqxQ\na1nv37/nP9+4cUPhO4mNjaXCwkKpeeXn5/OHycyfP19sLL+iQEFPXdmhkrpljPpdAE7Fn50B3C3+\nPA/A3FLx/gbgJSNPtT0kQ3m0NQZcXjEwMKBDhw4JXU0qo2lnXxs2bJBpQKoyX375Jd9eWrduTYaG\nhpSRkUHVqlWj5ORk8vT0pMLCQiooKKDNmzfTvHnzVMr/4MGDZGhoSIaGhmKniHXv3p0A0KNHjyS8\nQsqioKCA8vPzlY4vC00Z9Tdl7r8p/rsWwOhS4VsA+MjIs0IPxigfgwcPJkNDQ8GNtyzR1x6pNjw4\n6tsvGKGoXbu22HVMTIxG/ON07dqVatSoQVlZWUp5Jh0/fjzVqFGDbt26VaFyFRl1de0oJTXlw9Aw\nR48exSeffCKxBVsXqFGjhl6dqlSaBg0aVGgHJEN9PHnyROy6c+fO2LZtm8z4jx49UnhAzJs3b/Dm\nzRuxsKioKKSlpcHc3By9evVSqNeff/6JtLQ0NG/eXGHcilBe3y8vOY5zIqKXHMc5Ayg5V+spgNql\n4tUqDpPKggUL+M/e3t7w9vYupzoMVTh+/DhcXV3x7NkzoVXhqVmzJoKCgvD5558LrUq5WLNmDTp2\n7KjRk5Xu37+PjIwMWFtbS72fkZGBly9fVlqfJprgzp07GDRoEFatWgVHR0cAotOWWrduLRZv8+bN\n4DgO/v7+UvO5ceOGxnSMjIxEZGSk8gnkdePp36GSegDiS10HoXjsHNInSk0A1AebKNVZRo8eTQYG\nBoIPtwCinX9btmwRukoqhLYO0IiMjJSpw9mzZ8nHx0eLT63fXL16lTw8PMTq18DAgEaPHi0RNzAw\nkHx9fSk5OVkATcVBRYdfOI77L4D/AWjMcdwTjuMmAggE0JvjuAQAPYuvQUS3AewDcBvASQDTi5Vg\n6Bi7d+9W+vxHTfPJJ5/A19dXaDXKzd9//43s7GxBdUhPT8fZs2cF1UHXSUpKwp49e3Dnzh1cvHgR\nU6ZMETtrFQBMTU2xe/duqem3bt2KRYsWITk5WRvqlhuFwy9ENFrGLamDSES0FMDSiijF0A5+fn74\n/fffhVZD7/H398fbt281Xk6fPn1Qs2ZNifDMzEwsX74cixcvFnMxyxDx9OlTHD9+HOfOncPu3bsx\nbNgwpKam4tKlSyrntXnzZvTr149fO6+LMH/qVZi1a9fCysoK79+/x7p164RWh6GAKVOmoHHjxvz1\nsmXLUFBQgHfv3gmzyUVPuHfvHqZNm8ZfHz58WGbcgoICLF26FI6Ojvyvx+vXryMiIoKPc+jQIXz0\n0UdqOysgIyMD+/fvV9uvVWbUqzhLly7F+/fvUadOHSQlJWHTpk1Cq8QoQ//+/dGtWze0aNFCLPzn\nn39GTk4Of92gQQOMHTtW2+rpLOnp6fj1118lVsPIIz8/Hz/88AMcHByQkJAADw8P5OTk4NSpU3yc\n3bt3Y/LkyWoz6hzHwcBAuYWIS5YsURiHGXUGqlWrBn9/fzx79gwcx2Hjxo1aK9vT05MZIhl07twZ\nY8eOhZeXF1q1aqUwfp06dTB06FAtaKYfZGZmYtmyZeVKm5aWhmXLlqFevXoaH2qpVq0aJk6cqFTc\nevXqKY4kbxZVkwK2+kUnkeevWhNSGVZrnDp1imxtbdVaL15eXrzTKFmU+A0vEW9vby09se6Tk5Mj\n4UZXndKtWzfBzuCFgtUvrKfOYFSQPn36wNzcXK2Tpa6urvjoo4/Ull9VYeTIkXj16hWKiorExsHV\nTVRUFIYPHw4rKyucOHECpqamKucxYMAAnDhxQu26MaPOEIy2bdtWmgm+kydPwsvLS6Obj0rTv39/\nsbLc3NywYcMGrZSta8yZM4c34HFxccjPz9dKuRcuXAAAFBUVlSv9L7/8gs6dO+PcuXPqVIsNvzDE\n0dbwS5MmTSglJUXox1UrZYdDKiKWlpb0/fffyyyr7GYnVfyGVybmzZtHlpaWWh0yLCv169cvt/6l\nD+1QFrDhF4Yu8fr1awCAoaGh3vp50Qbdu3fH/PnzZd6/efMmXFxcxFa/VEXS0tLw4cMHQXV4+PAh\natWqhZSUFJXTNmjQQO36qMuhF6MSEBERoTHfK0+ePEFhYSHs7OxgZ2dXKQ26Oo2LiYkJLC0tZd63\ntbXlP7u6uuL69etqK1tfmDt3LrZu3Sq0GgBEG5zUtcSxojCjzhCD1OzVwdLSEjdv3kTt2rWVXour\nr6jj+Tw8PEBEOHjwoMK42dnZcHZ2RkpKSqWvW2ksWLBAp5bDJicnKzw1SRtUvZbA0ConTpyAu7u7\n0GpUWp4/fy60CoKxatUqubtDqyrMqDMYasTDw6PcaY2NjdG0aVM1alO5+f777zF79mydGsrLyclB\nYmKioDowo87QGC1atED16tWFVkOrXLx4ER07dlQ5nYGBAYYOHYp9+/ZpQKvKS0BAAHr06CG0GjzP\nnj3Df/7zH0F1YEadAQB49+4dYmJi1JrnwoULldreXpkwMzMrl2E2NTVlBr0S0KBBA6xYsUJQHZhR\nZwAAHj9+jICAALXl16lTJ9StW1dt+ekTFhYWKvtgGTdunIa0qfz07NkTzs7OQquhM3DqXu2gdMEc\nR0KVzZDk2bNnmDlzJvbv31/hvLp06YJly5bBy8tLDZrpJ3FxcUr/Svnmm2+wevVqDWtUuenWrRui\no6PlxvHw8EDv3r1l3l+zZo3Cs0rlYWVlhYCAAMyePbvceSgDx3EgIpkH4rLNRwwAojHd8vivKEuX\nLl0QFBRUpQ26svzf//0fnJycMHfuXKFV0Wv++9//4tGjRzLv169fH5MmTYKnpycGDBggM56DgwO+\n//57WFtbY+zYsSqdMWBsbIzFixdjxowZqqiuEVhPnQFAtZ5lCdWrV8fUqVMRGBgIAPj444+xbNky\nfPzxx5pQUa949eoVFi1ahLVr1+Ljjz/GqFGjJOKMGjUK9vb2AmhXuTh58iRmzJjBrzpZsWIFjI2N\n+fuurq4YNmyYUnmZmJggODgYFy5cwJ49e5TWwdzcHFlZWaopXk5YT52hkDdv3uD7779XOV1ubi7e\nvn2LXbt2ARA5lWI9dBE1atTADz/8AC8vL1YvGqZ///4wNjZGamoqAGDEiBEwMiqfaduxYwcGDx6M\n7OxslYz6H3/8Ua7y5BESEgJnZ2d0795dpXSsp85ASkpKuQ8C8PDwQFxcnJo1YjCEJSMjA+fOncP5\n8+fx66+/KoyvCVt29+5dbNmyBcOGDUOnTp34cNZTZ8glJyeHnZbDYJTB2toa/fv3R3Z2tmA6NG3a\nFFOnTsWyZctgYWEBT09PpdKxnnoVp6ioCFFRUSpv4DAzM8PVq1dhamqqEU9zDIYukJGRgadPnwIA\nfvvtN9SrVw/R0dEIDw/HkSNH+IPAmzVrpjEdfHx8cOXKFVhYWCAyMhLOzs5ye+rMqDNUGn65evUq\natasCY7j4OTkpGHNGAzd4f379zA0NERubi5yc3Nhb28vNiGrCWbNmoV169YhNzcXZ8+eRdeuXWFk\nZMSMOkM2WVlZsLOzU/rEnpSUFLi6umpYKwaDAQD5+fn4/PPPcfToUcTFxcHDw0PhmDrbUcpQaNBD\nQkKQlZWFwsJCZtAZVR4fHx9wHAeO4/ihGXVSUFCAxYsXg+M4bN++HWZmZoiOjlbaWRwz6lUcZYZR\npk+fjoYNGyIhIUFLWjEYuoudnR1MTEzg4OCgdj/22dnZWLlyJX766SdUq1YNFhYW2LNnD7p06aJ0\nHmz4pYpTVFSEyMhI9OzZU2acgwcPwsfHR4taMRi6zfjx4zF//nx+orSi5OTk4MGDBzh9+jQWLFiA\nmjVrYtasWfD19ZWIy5Y0MuQSERGBXr16Sb1Xt25duLq6sl2PDEYZtm/frra8cnNzsXfvXkyYMAEA\n4Ofnh82bN5c7P2bUqziyDDogGnbx9/fXojYMRtXj/fv3/Fmrzs7OFTpoBWBj6gw5XLt2TfBTXBiM\nyk6NGjWwa9cujBkzBvPmzcM333xTofxYT70Ks379ern3nzx5gtTUVLi5uWlJIwajalKnTh3eh1JF\nYROlVRhzc3Pk5ORIvdexY0cEBQWpNOvOqDz89ddfcHNzY2em6iBsnTpDJkFBQVLDvby8mEGv4piY\nmMDQ0FBoNRjlgPXUqzgcJ/kPX5nZ94kTJ8LW1lbw8xgZjKoG66kz5BISEiJ23aFDB0yfPl1husGD\nB6NPnz6aUovBYJQT1lOv4mRlZcHS0pK/rlGjBtzd3TFq1ChMmTJFQM0YDIY0FPXUmVGv4hQVFSE+\nPh737t3D8OHD+XAHBwfUrFkTAQEBbDcpg6FDMKPOUIq8vDw8f/4cgGhI5vHjx5g3bx7s7OxQrVo1\ngbVjMBglVNiocxy3FcBAAC+JqGVxWACAyQBSi6P9QEShxfe+BzAJQAGAGUQUJiNfZtR1lLy8PBQV\nFcHMzExoVRgMRhnUYdQ7A8gEsKOMUX9PRCvKxG0G4L8A2gGoBSAcQCNp1psZdQaDwVCdCq9+IaJz\nAN5Ky1tK2BAAe4iogIgeAbgPoL2SujIYDAajglRkSeP/cRx3neO4LRzH2RSHuQJILhXnaXEYg8Fg\nMLRAeX2//AfAIiIijuMWA1gOwE/VTBYsWMB/9vb2hre3dznVYTAYjMpJZGQkIiMjlY6v1OoXjuPq\nAjheMqYu6x7HcfMAEBEFFd8LBRBARBekpGNj6gwGg6Ei6tpRyqHUGDrHcc6l7vkAuFn8+RiAkRzH\nmXAcVx+AG4CLqqnM0AcKCgqwb98+nDp1SmhVGAxGKRQadY7j/gvgfwAacxz3hOO4iQCCOY6L4zju\nOoBuAL4FACK6DWAfgNsATgKYzrrjlYtdu3ahoKAARUVFOHPmDC5eZP+zGQxdgm0+YijFf//7X/Tu\n3RubNm2Cv78/jI2NhVaJwVA7K1aswHfffSe0GnJhO0oZ5ebkyZOIjo4GAFhZWeHLL7+Eo6OjwFox\nFHHx4kUcOnRIInzmzJlwdnaWkkJ32LhxIx4+fMhfBwYGSsQp+3xffPEFWrRooXJZO3fuxK1bt8TC\n1qxZg6ysLJXz0ibs4GlGuTh16hR++OEH3LhxA1OmTME333zDDLoOk5KSgl9++QUAcPfuXf6fcWmS\nksHsoH8AAAquSURBVJJgZ2cHQGQ81cmpU6ek/iNRlePHj/PuKgDg7VvJLTJln+/q1auoX7++ymWF\nhYXh0aNHYmFGRkZKObKbNWsWGjdurHKZ2oD11BlSCQoKwrx58wAABw8eZE69dIxp06bhzZs3/PW7\nd+9UmrQu7bxNGlOmTEGPHj3kxrl16xYWLVoEQGRo4+LilC5f3+nevTscHBz4671792qtbDb8wlCK\nBw8eYOrUqejcuTN+/vlnMaPeqlUrODo6YuPGjeXqETGUIy0tDWPGjJEI9/Pzw/Dhw7Fo0SKcO3cO\nABAdHY3c3FyN6dK8eXNs3LgRnTt3lqnn27dvcfnyZY3poE/07t1b5r2S96cumFFnKEVcXBxatWoF\nW1tbzJo1C0ZGRrxRL8Hd3R1nz56tssMwR44cwZIlS/jr4OBgdO/eHaNGjUJiYqLctOPGjcPXX3+N\nxYsX4+jRo1Lj5Ofn48aNGxLhtWrVgrOzMxITE5Genl6xh1CCr7/+GmPHjoWbmxtsbW0BAAMGDEBq\naqpcPRnSKXl/gGjYq02bNhXKjxl1hlwaNWoEQOSZ8cmTJwAAa2trfPXVV0hLS8OWLVvE4terVw93\n7tzRSw+OsbGxGDt2LIYPH44lS5YgODhY4bF9pcnIyOANGwDUrFkTlpaWePLkCfLy8uSmtbW1hb29\nPVJTU5GRkVHuZ9AGQUFB8Pf3568HDhyI0NBQFBYWCqhV5cDV1RXm5uZy45S0T1kwo86Qy6tXr6SG\nm5ubY+HChVi2bJnEvaysLIUNU5d49uwZWrZsifz8fGRkZMDU1BRWVlbIyspCdna20OrpHBYWFmLv\nNz09nRl0LaLojGBFRh1EJIiIitZv5s6dSwYGBnTo0CEiImrVqhUZGBhQVlZWhfJNTk4mQ0NDMjAw\n4GXevHn8/U8//VTsnoGBAR+/5O9nn30mlufkyZP5e3FxcQrLMzAwII7jCICEVPT5tIWlpSX/XNKe\ngwkTXZJu3bpRQUEBFRYWym3XAIjk2NYq2VMvKChQqofm6OiInJwcuXHWrl2LsLAwHD9+HIBoFYCL\niwusra2lxp85cyaICG5ubhg/frzcvKOjozFw4ECFemoaMzMzGBkZITY2Fu7u7kKrI5esrCy+V6nM\n+2MwhMLKygoWFhZ4+fKlSunY8IsUIiIiFC7XqqqYm5ujRo0aYmEbN27EJ598IpBG/1JQUIBnz57J\njTN8+HBcuCDhP47B0DnKO4xZZTYfpaamim1akEdSUpKGtdEfWrYUd7zZq1cvLF++XCBtJLl9+zby\n8/MBAC9fvkTfvn0F1ojB0G30zqgXFhbya3WrV6+OVq1aAQDCw8Mxd+5cpKSkCKmeXtC2bVtYWloC\nAKKiojRa1vXr1/Hu3Tv+2sDAAF26dJEZPyEhAS9evOCvP/vsM5mTuQyGPhMTEwNTU1OYmpqiQ4cO\nastX74ZfcnNzMWLECGRkZCAtLQ2//PIL6tatC09PT2zfvh2HDx+Wmq6wsBB//fVXRdWuFKxatQpf\nfvmlRlewXLt2DY8fP8aPP/4o5l/DxMQEx44dk9njXrduHU6fPi0RLmttN4OhT3Tu3Bn29vZiYfb2\n9ti6davSeVTaMfXk5GT8/PPPAIBu3bphwoQJcuPn5uZi6tSpePv2LTMQAJYvX85vLLGwsMCIESOk\nxjt//jzu3bsnET5x4kSJsISEBPzvf/8DAGzbtg0xMTFi98eOHQtDQ0PY2tpixYoVEunlwXGyV3Ax\nGPpA9+7dsX79ejRp0qRC+VTaMfXatWtj27ZtUu8dP35c6g6/li1bKpxoqyrMmjWL/2xjYyOzXvbu\n3QsHBwe4ubkpzPPNmze8/4+2bduibdu2YveDgoJgYmLCXycnJ+PAgQPlUZ/B0Dvq1q3LD3tqEr3t\nqctj3759uHv3LtatWye2A5BRPj7//HM0bdpULMzY2Bg//fQTf3379m3s2bNHpXwfPXqEnTt3qkVH\nBkMfGDduHOrWrQsA+Omnn8p1LkGl7anLY/jw4Vi9ejXbBacm9u/fLxFmZGQk9g8zKSkJf//9tzbV\nYjD0jh07dvCf09LSYGT0rwlu1KgRvvnmmwqXIWhP/YsvvuCv27VrBw8PD/zxxx9S469evZr3Ba0M\nYWFhUhf1JycnY/78+aorzGAwGBrEyclJrrdHQGQH7e3tdXeitPS1s7MzHBwcEB8fLzV+z549K+RE\nytTUFBs3bsSnn34q9QABBoPB0HV69uyJM2fO6IdR1zQGBgZo06YN8//MYDD0HmbUGQwGoxIhz6gb\naFMRBoPBYGgWZtQZDAajEsGMOoPBYFQiBBtTZzAYDIb6YT11BoPBqEQwo85gMBiVCEGMOsdx/TiO\nu8tx3D2O4+YKoYMqcBz3iOO4GxzHXeM47mJxmC3HcWEcxyVwHHeK4zgbHdBzK8dxLzmOiysVJlNP\njuO+5zjuPsdxdziO6yOM1rwu0nQP4DguheO4q8XSr9Q9ndCd47haHMed5TjuFsdx8RzHfVMcrtP1\nLkXvr4vD9aHOTTmOu1D8fYznOC6gOFzX61yW3uqtc3kHmGpCIPpHkgigLgBjANcBNNW2Hirq/ACA\nbZmwIAD+xZ/nAgjUAT07A2gNIE6RngCaA7gGkf+fesXvhNMx3QMAfCclbjNd0R2AM4DWxZ+tACQA\naKrr9S5Hb52v82J9LIr/GgKIBdBe1+tcjt5qrXMheurtAdwnosdElA9gD4AhAuihChwkf9UMAbC9\n+PN2AEO1qpEUiOgcgLdlgmXpORjAHiIqIKJHAO5D9G4EQYbugKjuyzIEOqI7Eb0gouvFnzMB3AFQ\nCzpe7zL0di2+rdN1DgBElFX80RQio0fQ8ToHZOoNqLHOhTDqrgCSS12n4N/GpKsQgNMcx13iOM6v\nOMyJiF4Coi8IAEfBtJOPoww9y76Hp9DN9/B/HMdd5zhuS6mf0zqpO8dx9SD6tREL2e1D53QvpXfJ\nid06X+ccxxlwHHcNwAsAp4noEvSgzmXoDaixztlEqXJ0IqI2APoD+IrjuC749z9sCfqyNlRf9ASA\n/wBoQEStIfoS6M6J2GXgOM4KwAEAM4p7vnrRPqTorRd1TkRFROQJ0a+i9hzHuUMP6lyK3s2h5joX\nwqg/BVCn1HWt4jCdhYieF/9NA3AEop9ALzmOcwIAjuOcAejqaRyy9HwKoHapeDr3HogojYoHFwFs\nxr8/PXVKd47jjCAyjDuJqOSsRJ2vd2l660udl0BEGQAiAfSDHtR5CaX1VnedC2HULwFw4ziuLsdx\nJgBGAjgmgB5KwXGcRXFvBhzHWQLoAyAeIp0nFEcbD0BXDj7lID4+J0vPYwBGchxnwnFcfQBuAC5q\nS0kZiOle/MUswQfAzeLPuqb7HwBuE9HqUmH6UO8SeutDnXMcV6NkiILjOHMAvSGaE9DpOpeh9121\n17lAM8D9IJptvw9gnhA6qKBrfYhW6FyDyJjPKw63AxBe/BxhAKrrgK7/BfAMQC6AJwAmArCVpSeA\n7yGaUb8DoI8O6r4DQFxx/R+BaMxUp3QH0AlAYak2crW4fctsH7qguxy99aHOPYr1/f927tAGABAG\ngODgLMlImFocgnzuZFVT8YaEPbuumf9+89veT2/umwCAEA+lACGiDhAi6gAhog4QIuoAIaIOECLq\nACGiDhByABnTEpIJrLUxAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11897acf8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "land_longlat = misc.imread('land.tif')\n",
    "plt.imshow(land_longlat,cmap=cm.Greys)\n",
    "type(land_longlat)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Count number of longitudes covered with land, by lattitude."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "land_lat=land_longlat.sum(axis=1) # Sum over second dimension = longitudes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Plot the result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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3vOMdNi21MeagePaJISKfAtKqeuO4O5sxpVIp4vE41157LdFo1OtwjDEFoqen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\nDwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a86da58>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f, ax = plt.subplots(1) # make figure and axis\n",
    "# ax.plot(land_lat,np.arange(0,180),'k')\n",
    "ax.fill_betweenx(np.arange(0,180), np.zeros(len(land_lat)), land_lat, color = 'k')\n",
    "ax.set_xlabel('Number of longitudes covered with land')\n",
    "ax.set_ylabel('Polar distance')\n",
    "ax.set_xlim(xmax=360)\n",
    "plt.gca().invert_yaxis()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Convert this into values for $P$: a ratio of land, for latitudes from 0 to 90  \n",
    "Plot $P$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "P_land = (land_lat[-91:-181:-1] + land_lat[90:180:1])/(2*360)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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n8s1vfpP6+vrQkUQkgGw2Szab3fF84cKFZVsgVgHN7t5iZoOB5e5+yG7WVYHo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\nZ8A54ZgAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10dc74278>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f, ax = plt.subplots(1) # make figure and axis\n",
    "# ax.plot(P_land,np.arange(0,90)+0.5,'k')\n",
    "ax.fill_betweenx(np.arange(0,90), np.zeros(len(P_land)), P_land, color = 'k')\n",
    "ax.set_xlabel('Fraction of land P_land')\n",
    "ax.set_ylabel('Latitude $\\phi$')\n",
    "ax.set_xlim(xmin=0,xmax=1)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "We use this $P$, and redo the calculations of the average angular distortion $E_{an}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "P = P_land"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "Ean_ecp_lat0_land = np.zeros(0)\n",
    "\n",
    "for lat0 in lat0_range:\n",
    "    h = h_ecp_lat0(lat0)\n",
    "    k = k_ecp_lat0(lat0)\n",
    "    Ean_ecp_lat0_land = np.append(Ean_ecp_lat0_land,E(eanS(h,k)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Tk+NmW6vCzp3u9RG++gp+/NHt8PXXEXYwxpjy4ae/og9wjYisBXYDgo9igMaf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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10dc14d30>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f, ax = plt.subplots(1) # make figure and axis\n",
    "ax.plot(Ean_ecp_lat0*180/np.pi,lat0_range*180/np.pi,'b.-')\n",
    "ax.plot(Ean_ecp_lat0_land*180/np.pi,lat0_range*180/np.pi,'r.-')\n",
    "ax.set_xlabel('Average angular distortion $E_{an}$ (deg)')\n",
    "ax.set_ylabel('Latitude of standard parallel $\\phi_0$ (deg)')\n",
    "ax.set_xlim(xmin=0)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "The optimal value for $\\phi_0$ is still around 30°.  \n",
    "\n",
    "There is quite some landmass is at high latitudes. For normal, low values of $\\phi_0$, the angular distortion is large at high latitudes. Therefore, the average angular distortion $E_{an}$ is higher for these $\\phi_0$ then when landmass is ignored, and the optimal value for $\\phi_0$ shifts (slightly) towards higher latitudes (but stays near 30°)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "We reset $P$ to ones, and ignore landmass in the following exercises"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "P = P_ones"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "We have chosen $\\phi_0$ to minimise the average angular distortion $E_{an}$ in the Equivalent Cylindrical Projection ($E_{ar}=0$).  \n",
    "\n",
    "Could we choose $\\phi_0$ to minimise the average area distortion $E_{an}$ in the Conformal Cylindrical Projection (Mercator) ($E_{an}=0$)? Would this be a usefull way to obtain a good worldmap?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "The problem with Mercator is the ever increasing distance between the parallels. \n",
    "\n",
    "- This makes it a bad projection for a worldmap in any circumstance.\n",
    "- The local area distortion $e_{ar}$ is so large at the poles, that this will dominate the average area distoriton $E_{ar}$ and lead to a bad choice for $\\phi_0$.\n",
    "\n",
    "Possible solutions?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- Stop integration at a $\\phi_{max}$, example $\\phi_{max}=80°$\n",
    "- Only integrate over landmass (excluding Antactrica)\n",
    "- Use logarithmic distortion measures (will make extreme distortions count less, but still a lot)\n",
    "- Modify projection to bring poles at finite distance (see Ex. 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Ex. 2: Conformal Cylindrical Projection with varying Miller Coefficients - Miller Projection\n",
    "Given the (non-secant) Conformal Cylindrical Projection (Mercator Projection), addapted with Miller coefficient $C$, which value for this coefficient yields the lowest angular and areal distortion over the entire sphere?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "The American cartographer O.M. Miller added a parameter $C$ to the equations of the Mercator Projection:\n",
    "\n",
    "$ x = R \\lambda $  \n",
    "$ y = C R \\ln \\tan ( \\frac{\\pi}{4} + \\frac{\\phi}{2C} )$  \n",
    "\n",
    "$C$ allow to reduce the wide spacing of the parallels towards the poles, and brings the poles to finite distance.\n",
    "\n",
    "With $C = 1$ for normal Mercator. For high $C$ the projection starts to behave like the Equidistant Cylindrical Projection."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Because of the added $C$, we must first again derive $h$ and $k$ analyticaly.\n",
    "\n",
    "Determine\n",
    "\n",
    "$h = \\frac{\\mathrm{d}y}{R\\mathrm{d}\\phi}$  \n",
    "$k = \\frac{\\mathrm{d}x}{R\\cos\\phi\\mathrm{d}\\lambda}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Solution\n",
    "\n",
    "$h = \\frac{1}{\\cos\\frac{\\phi}{C}}$  \n",
    "$k = \\frac{1}{\\cos\\phi}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Define $h$ and $k$ for this projection, for varying $C$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "def h_ccp_C(C):\n",
    "    global lat\n",
    "    h = 1/np.cos(lat/C)\n",
    "    return h\n",
    "def k_ccp_C(C):\n",
    "    global lat\n",
    "    k = 1/np.cos(lat)\n",
    "    return k"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Plot the local angular and area distortion $\\times$ surface: $e_{an} \\, S$ and $e_{ar} \\, S$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
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lCLjySvjwh7OOxOwNjZxj+xZwKHBKen8xcFnbIzIrvsLk0rRpsHgxHHbY8M8165ZGzrEd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\nSUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10dd3f5f8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def plt_eanS_earS_ccp_C(C):\n",
    "    global lat, P\n",
    "    h = h_ccp_C(C)\n",
    "    k = k_ccp_C(C)\n",
    "    f, (ax1, ax2) = plt.subplots(1, 2) # make figure and axis\n",
    "    f.tight_layout()\n",
    "    ax1.plot(eanS(h,k)*180/np.pi,lat*180/np.pi)\n",
    "    ax1.set_xlabel('Local angular distortion $e_{an}$ (deg) * surface $S$')\n",
    "    ax1.set_ylabel('Latitude $\\phi$ (deg)')\n",
    "    ax1.set_xlim(xmin=0,xmax=180)\n",
    "    ax2.plot(earS(h,k),lat*180/np.pi)\n",
    "    ax2.set_xlabel('Local area distortion $e_{ar}$ * surface $S$')\n",
    "    ax2.set_ylabel('Latitude $\\phi$ (deg)')\n",
    "    ax2.set_xlim(xmax=10)\n",
    "    plt.show()\n",
    "    \n",
    "v = interactive(plt_eanS_earS_ccp_C, C = (1,1.8,0.1));\n",
    "display(v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Notice how the local angular distortion $e_{an}$ grows and local area distortion $e_{ar}$ shrinks as $C$ grows.\n",
    "\n",
    "Also (see earlier remarks on Mercator Projection):  \n",
    "For $C=1$, at $\\phi = \\frac{\\pi}{2}$, $S = P \\cos \\phi = 0$, but this is not able to bring the local area distortion $e_{ar}$ to finite values!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Compute and plot the average angular distortion $E_{an}$ and average area distortion $E_{ar}$ for $C$ = 1, 1.1, ..., 1.7."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "C_range = np.arange(1,1.8,0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "Ean_ccp_C = np.zeros(0)\n",
    "\n",
    "for C in C_range:\n",
    "    h = h_ccp_C(C)\n",
    "    k = k_ccp_C(C)\n",
    "    Ean_ccp_C = np.append(Ean_ccp_C,E(eanS(h,k)))\n",
    "    \n",
    "Ear_ccp_C = np.zeros(0)\n",
    "\n",
    "for C in C_range:\n",
    "    h = h_ccp_C(C)\n",
    "    k = k_ccp_C(C)\n",
    "    Ear_ccp_C = np.append(Ear_ccp_C,E(earS(h,k)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
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JUKWzH9dqfYW7LwFuMLOjgFtiiOX3wLkt/t9mMg4bNuzTy/X19dTX18cQgmRp8uTwCx7g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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10e39c9b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f, (ax1, ax2) = plt.subplots(1, 2) # make figure and axis\n",
    "f.tight_layout()\n",
    "ax1.plot(Ean_ccp_C*180/np.pi,C_range,'b.-')\n",
    "ax1.set_xlabel('Average angular distortion $E_{an}$ (deg)')\n",
    "ax1.set_ylabel('$C$')\n",
    "ax1.set_xlim(xmin=0)\n",
    "ax2.plot(Ear_ccp_C,C_range,'b.-')\n",
    "ax2.set_xlabel('Average area distortion $E_{ar}$')\n",
    "ax2.set_ylabel('$C$')\n",
    "ax.set_xlim(xmin=0)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "The same trend is observed: the average angular distortion $E_{an}$ grows and the avergae area distortion $E_{ar}$ shrinks as $C$ grows. The projection is conformal for $C=1$, and loses its conformaty for $C>1$ in return for lower area distortion.\n",
    "\n",
    "Notice how $C=1.1$ leads to a dramatic reduction of $E_{ar}$ and only a relatively small increase in $E_{an}$.\n",
    "\n",
    "It is difficult to find an optimal value for $C$ by comparing both measures, since they have different units (degrees/radians vs. dimensionless). This can, for example, be done using a uniform measure like the average mean linear scale factor (Ex.4.).\n",
    "\n",
    "Miller proposed two values for $C$ (1942): \n",
    "\n",
    "- $C = 1.25$, known as 'the' Miller Cylindrical Projection (EPSG:54003)\n",
    "- $C = 1.5$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "The Miller Projection  \n",
    "![Miller Projection](https://upload.wikimedia.org/wikipedia/commons/5/5f/Miller_projection_SW.jpg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Ex. 3: Equivalent Cylindrical Projection with varying Miller Coefficients\n",
    "Given the (non-secant) Equivalent Cylindrical Projection, addapted with Miller coefficient $C$, which value for this coefficient yields the lowest angular and areal distortion over the entire sphere?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "We add a coefficient $C$ to the Equivalent Cylindrical Projection, in the same fashion as with Miller:\n",
    "\n",
    "$ x = R \\lambda $  \n",
    "$ y = C R \\sin \\frac{\\phi}{C}$  \n",
    "\n",
    "This time, $C$ allow to enlarge the narrow spacing of the parallels towards the poles.\n",
    "\n",
    "With $C = 1$ for normal equivalent. For high $C$ the projection also starts to behave like the Equidistant Cylindrical Projection."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Because of the added $C$, we must first again derive $h$ and $k$ analyticaly.\n",
    "\n",
    "Determine\n",
    "\n",
    "$h = \\frac{\\mathrm{d}y}{R\\mathrm{d}\\phi}$  \n",
    "$k = \\frac{\\mathrm{d}x}{R\\cos\\phi\\mathrm{d}\\lambda}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Solution\n",
    "\n",
    "$h = \\cos\\frac{\\phi}{C}$  \n",
    "$k = \\frac{1}{\\cos\\phi}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Define $h$ and $k$ for this projection, for varying $C$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "def h_ecp_C(C):\n",
    "    global lat\n",
    "    h = np.cos(lat/C)\n",
    "    return h\n",
    "def k_ecp_C(C):\n",
    "    global lat\n",
    "    k = 1/np.cos(lat)\n",
    "    return k"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Plot the local angular and area distortion $\\times$ surface: $e_{an} \\, S$ and $e_{ar} \\, S$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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1S5bAgQe6qJmZ2R/0dGG791447LCiozAzszLp+cJ26KFFR2FmZmXS04Xtvvvg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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10e0c2be0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def plt_eanS_earS_ecp_C(C):\n",
    "    global lat, P\n",
    "    h = h_ecp_C(C)\n",
    "    k = k_ecp_C(C)\n",
    "    f, (ax1, ax2) = plt.subplots(1, 2) # make figure and axis\n",
    "    f.tight_layout()\n",
    "    ax1.plot(eanS(h,k)*180/np.pi,lat*180/np.pi)\n",
    "    ax1.set_xlabel('Local angular distortion $e_{an}$ (deg) * surface $S$')\n",
    "    ax1.set_ylabel('Latitude $\\phi$ (deg)')\n",
    "    ax1.set_xlim(xmin=0,xmax=180)\n",
    "    ax2.plot(earS(h,k),lat*180/np.pi)\n",
    "    ax2.set_xlabel('Local area distortion $e_{ar}$ * surface $S$')\n",
    "    ax2.set_ylabel('Latitude $\\phi$ (deg)')\n",
    "    ax2.set_xlim(xmax=10)\n",
    "    plt.show()\n",
    "\n",
    "v = interactive(plt_eanS_earS_ecp_C, C = (1,1.8,0.1));\n",
    "display(v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Notice how the local angular distortion $e_{an}$ shrinks and local area distortion $e_{ar}$ grows as $C$ grows."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Compute and plot the average angular distortion $E_{an}$ and average area distortion $E_{ar}$ for $C$ = 1, 1.1, ..., 1.7."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "C_range = np.arange(1,1.8,0.1)\n",
    "\n",
    "Ean_ecp_C = np.zeros(0)\n",
    "\n",
    "for C in C_range:\n",
    "    h = h_ecp_C(C)\n",
    "    k = k_ecp_C(C)\n",
    "    Ean_ecp_C = np.append(Ean_ecp_C,E(eanS(h,k)))\n",
    "    \n",
    "Ear_ecp_C = np.zeros(0)\n",
    "\n",
    "for C in C_range:\n",
    "    h = h_ecp_C(C)\n",
    "    k = k_ecp_C(C)\n",
    "    Ear_ecp_C = np.append(Ear_ecp_C,E(earS(h,k)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10dc85c88>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f, (ax1, ax2) = plt.subplots(1, 2) # make figure and axis\n",
    "f.tight_layout()\n",
    "ax1.plot(Ean_ecp_C*180/np.pi,C_range,'b.-')\n",
    "ax1.set_xlabel('Average angular distortion $E_{an}$ (deg)')\n",
    "ax1.set_ylabel('$C$')\n",
    "ax1.set_xlim(xmin=0)\n",
    "ax2.plot(Ear_ecp_C,C_range,'b.-')\n",
    "ax2.set_xlabel('Average areal distortion $E_{ar}$')\n",
    "ax2.set_ylabel('$C$')\n",
    "ax.set_xlim(xmin=0)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "The same trend is observed: the average angular distortion $E_{an}$ shrinks and the avergae area distortion $E_{ar}$ grows as $C$ grows. The projection is equivalent for $C=1$, and loses its equivalence for $C>1$ in return for lower angular distortion."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Ex. 4: Conformal Cylindrical Projection and Equivalent Cylindrical Projection with varying Miller Coefficients, measured using Average Linear Scalefactor\n",
    "For all projections in Ex. 2 and Ex. 3, compute and plot the average mean linear distortion vs. $C$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Try it yourself!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Takeway from this course\n",
    "There will be no coding or pc-session in the exam. \n",
    "\n",
    "However, you are expected to understand the main insights about projection optimisation from this course. Make sure you: \n",
    "\n",
    "- know which parameters one can tune\n",
    "- how average distortion is computed from local distortion\n",
    "- which distortion measures one can use\n",
    "- what are the options to optimise equivalent, conformal and other projections\n",
    "- how you expect the local distortion * surface to vary by latitude for the main projection types and how and why this changes with the tuned parameter value\n",
    "- how the average distortion depends on the parameter\n",
    "- ..."
   ]
  }
 ],
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