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<!DOCTYPE html>
<html>
<head><meta charset="utf-8" />
<title>reeks4</title>
<script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
<style type="text/css">
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*
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<h1 id="Map-Projections-and-Coordinate-Systems">Map Projections and Coordinate Systems<a class="anchor-link" href="#Map-Projections-and-Coordinate-Systems">¶</a></h1><p>Course at Ghent University, Department of Geography<br>
Professor Frank Canters</p>
<h2 id="Session-4:-Cylindrical-Projections-and-Projection-Optimisation">Session 4: Cylindrical Projections and Projection Optimisation<a class="anchor-link" href="#Session-4:-Cylindrical-Projections-and-Projection-Optimisation">¶</a></h2><p>Performed in Python using Jupyter Notebook.</p>
<p>Written by Manuel Claeys Bouuaert, <[email protected]>, 2015</p>
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<p>Choosing a good projection for a specific area means choosing:</p>
<ul>
<li>projection type</li>
<li>projection parameters (standard parallel, Miller coefficients, ...)</li>
<li>coordinates of pseudopole </li>
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<p>In this session, we will work with small scale projections (world maps) and only change projection parameters.</p>
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<p>A good choice of the projection parameters is one which minimises the average distortion over the concerned area:</p>
$$ E = \frac{1}{S} \int_S e \,\mathrm{d}S $$<ul>
<li>$E$ is the average distortion</li>
<li>$S$ is the concerned area on the sphere or ellipsoide</li>
<li>$e$ is the local distortion</li>
<li>$\mathrm{d}S$ is the area of a local infinitessimal surface element</li>
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<p>In practice, this integral is approximated by a discrete sum, by dividing the sphere into small cells:</p>
$$ E = \frac{1}{S} \sum_{i} e_i \, S_i $$$$ S = \sum_{i} S_i $$<ul>
<li>$E$ is the average distortion</li>
<li>$S$ is the concerned area on the sphere or ellipsoide</li>
<li>$e_i$ is the local distortion at the center of a cell</li>
<li>$S_i$ is the area of a local cell</li>
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<p>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: $\phi = 0 \rightarrow \frac{\pi}{2}$.</p>
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<p>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 $\cos\phi$.<br>
The area of the cell at $(\phi_i,\lambda_i)$ is $S_i = P_i \cos{\phi_i} \, \Delta\phi \, \Delta\lambda$<br>
Since $\Delta\phi$ and $\Delta\lambda$ are independent of $i$ and also appear in $S$, they cancel out, leading to $S_i = P_i \cos{\phi_i}$.</p>
<p>The weight factor $P_i$ can be used to select a specific area, by putting $P_i = 1$ there and $P_i = 0$ elsewhere.</p>
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<p>Which local distortions are used?</p>
<ul>
<li>Local angular distortion: $e_{an,i} = 2\Omega_i = 2\arcsin(\frac{a_i-b_i}{a_i+b_i})$</li>
<li>Local area distortion: $e_{ar,i} = \sigma_i^p-1 = (a_ib_i)^p-1$</li>
<li>Local mean linear scale distortion: $e_{ab,i} = \frac{a_i^q+b_i^r}{2}-1$</li>
</ul>
<p>To ensure that enlarging and shrinking with the same factor has the same weight, we use</p>
<ul>
<li>$p$ = 1 if $\sigma_i$ ≥ 1 and $p$ = -1 if $\sigma_i$ < 1</li>
<li>$q$ = 1 if $a_i$ ≥ 1 and $q$ = -1 if $a_i$ < 1</li>
<li>$r$ = 1 if $b_i$ ≥ 1 and $r$ = -1 if $b_i$ < 1.</li>
</ul>
<p>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.<br>
(1) <em>Canters, F., 2002. Small-scale Map Projection Design. CRC Press.</em></p>
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<ul>
<li>Equivalent projections ($e_{ar,i} = 0$) $\rightarrow$ minimise $e_{an,i}$</li>
<li>Conformal projections ($e_{an,i} = 0$) $\rightarrow$ minimise $e_{ar,i}$</li>
<li>other projections $\rightarrow$ minimise $e_{ar,i}$, $e_{an,i}$ and/or $e_{ab,i}$</li>
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<h3 id="Exercises-in-Python">Exercises in Python<a class="anchor-link" href="#Exercises-in-Python">¶</a></h3>
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<p>Load modules</p>
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<div class=" highlight hl-ipython3"><pre><span class="o">%</span><span class="k">matplotlib</span> inline
<span class="c"># show plots inline</span>
<span class="kn">import</span> <span class="nn">os</span> <span class="c"># to access UNIX commandline</span>
<span class="kn">import</span> <span class="nn">urllib.request</span> <span class="c"># to download files</span>
<span class="kn">import</span> <span class="nn">zipfile</span> <span class="c"># to unzip files</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span> <span class="c"># to use arrays etc</span>
<span class="kn">from</span> <span class="nn">scipy</span> <span class="k">import</span> <span class="n">misc</span> <span class="c"># to read images etc</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span> <span class="c"># to plot</span>
<span class="kn">import</span> <span class="nn">matplotlib.cm</span> <span class="k">as</span> <span class="nn">cm</span> <span class="c"># to have colormaps</span>
<span class="kn">from</span> <span class="nn">ipywidgets</span> <span class="k">import</span> <span class="o">*</span> <span class="c"># to have interactive widgets</span>
<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="k">import</span> <span class="n">display</span><span class="p">,</span> <span class="n">HTML</span> <span class="c"># to have interactive widgets</span>
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<p>Define latitudes $\phi$ of cells. (From here on, we drop the index $i$)<br>
We are asked to use cells with an interval of 1° (in stead of 2.5°).</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">lat_deg</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">90</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span><span class="o">+</span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span>
<span class="n">lat</span><span class="o">=</span><span class="n">lat_deg</span><span class="o">/</span><span class="mi">180</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="c"># use radians by default</span>
<span class="nb">print</span><span class="p">(</span><span class="n">lat_deg</span><span class="p">)</span> <span class="c"># only use degrees when printing or plotting</span>
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<p>Define weight factor $P$.<br>
We will consider the entire sphere by default.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">P_ones</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">lat</span><span class="p">))</span>
<span class="n">P</span> <span class="o">=</span> <span class="n">P_ones</span>
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<p>The values <code>lat</code> and <code>P</code> will be used as global parameters</p>
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<p>Define functions for different local distortions $e_{..}$ times the local surface area $S$.<br>
They take the local values of $h$ and $k$ as input: the scale factor along the meridians and parallels.</p>
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<div class=" highlight hl-ipython3"><pre><span class="k">def</span> <span class="nf">eanS</span><span class="p">(</span><span class="n">h</span><span class="p">,</span><span class="n">k</span><span class="p">):</span>
<span class="k">global</span> <span class="n">lat</span><span class="p">,</span> <span class="n">P</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">amax</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">((</span><span class="n">h</span><span class="p">,</span><span class="n">k</span><span class="p">)),</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">amin</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">((</span><span class="n">h</span><span class="p">,</span><span class="n">k</span><span class="p">)),</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">twoOmega</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">arcsin</span><span class="p">((</span><span class="n">a</span><span class="o">-</span><span class="n">b</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="p">))</span>
<span class="n">eanS</span> <span class="o">=</span> <span class="n">twoOmega</span><span class="o">*</span><span class="n">P</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">lat</span><span class="p">)</span>
<span class="k">return</span> <span class="n">eanS</span>
<span class="k">def</span> <span class="nf">earS</span><span class="p">(</span><span class="n">h</span><span class="p">,</span><span class="n">k</span><span class="p">):</span>
<span class="k">global</span> <span class="n">lat</span><span class="p">,</span> <span class="n">P</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">amax</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">((</span><span class="n">h</span><span class="p">,</span><span class="n">k</span><span class="p">)),</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">amin</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">((</span><span class="n">h</span><span class="p">,</span><span class="n">k</span><span class="p">)),</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">sigma</span> <span class="o">=</span> <span class="n">a</span><span class="o">*</span><span class="n">b</span>
<span class="n">p</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">sigma</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="n">earS</span> <span class="o">=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">power</span><span class="p">(</span><span class="n">sigma</span><span class="p">,</span><span class="n">p</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">P</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">lat</span><span class="p">)</span>
<span class="k">return</span> <span class="n">earS</span>
<span class="k">def</span> <span class="nf">eabS</span><span class="p">(</span><span class="n">h</span><span class="p">,</span><span class="n">k</span><span class="p">):</span>
<span class="k">global</span> <span class="n">lat</span><span class="p">,</span> <span class="n">P</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">amax</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">((</span><span class="n">h</span><span class="p">,</span><span class="n">k</span><span class="p">)),</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">amin</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">((</span><span class="n">h</span><span class="p">,</span><span class="n">k</span><span class="p">)),</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">a</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="n">r</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">b</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="n">eabS</span> <span class="o">=</span> <span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">power</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">q</span><span class="p">)</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">power</span><span class="p">(</span><span class="n">b</span><span class="p">,</span><span class="n">r</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">P</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">lat</span><span class="p">)</span>
<span class="k">return</span> <span class="n">eabS</span>
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<p>The average distortion $E$ is the average of the local distortion $e$</p>
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<div class=" highlight hl-ipython3"><pre><span class="k">def</span> <span class="nf">E</span><span class="p">(</span><span class="n">eS</span><span class="p">):</span>
<span class="k">global</span> <span class="n">lat</span><span class="p">,</span> <span class="n">P</span>
<span class="n">S</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">P</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">lat</span><span class="p">))</span>
<span class="n">E</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">eS</span><span class="p">)</span><span class="o">/</span><span class="n">S</span>
<span class="k">return</span> <span class="n">E</span>
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<h3 id="Ex.-1:-Equivalent-Cylindrical-Projection-with-varying-standard-parallel">Ex. 1: Equivalent Cylindrical Projection with varying standard parallel<a class="anchor-link" href="#Ex.-1:-Equivalent-Cylindrical-Projection-with-varying-standard-parallel">¶</a></h3><p>Given the secant Equivalent Cylindrical Projection, which value for standard parallel yields the lowest angular distortion over the entire sphere?</p>
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<p>The Equivalent Cylindrical Projection:</p>
<p>$ x = R \lambda \cos \phi_0 $<br>
$ y = \frac{R}{\cos \phi_0} \sin \phi $</p>
<p>With (see course notes slide 4.23)</p>
<p>$ h = \frac{\cos \phi}{\cos \phi_0} $<br>
$ k = \frac{\cos \phi_0}{\cos \phi} $</p>
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<p>Define $h$ and $k$ for this projection, for varying $\phi_0$</p>
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<div class=" highlight hl-ipython3"><pre><span class="k">def</span> <span class="nf">h_ecp_lat0</span><span class="p">(</span><span class="n">lat0</span><span class="p">):</span>
<span class="k">global</span> <span class="n">lat</span>
<span class="n">h</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">lat</span><span class="p">)</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">lat0</span><span class="p">)</span>
<span class="k">return</span> <span class="n">h</span>
<span class="k">def</span> <span class="nf">k_ecp_lat0</span><span class="p">(</span><span class="n">lat0</span><span class="p">):</span>
<span class="k">global</span> <span class="n">lat</span>
<span class="n">k</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">lat0</span><span class="p">)</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">lat</span><span class="p">)</span>
<span class="k">return</span> <span class="n">k</span>
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<p>Plot the local angular distortion $\times$ surface: $e_{an} \, S$.</p>
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<div class=" highlight hl-ipython3"><pre><span class="k">def</span> <span class="nf">plt_eanS_ecp_lat0</span><span class="p">(</span><span class="n">lat0</span><span class="p">):</span>
<span class="k">global</span> <span class="n">lat</span><span class="p">,</span> <span class="n">P</span>
<span class="n">h</span> <span class="o">=</span> <span class="n">h_ecp_lat0</span><span class="p">(</span><span class="n">lat0</span><span class="p">)</span>
<span class="n">k</span> <span class="o">=</span> <span class="n">k_ecp_lat0</span><span class="p">(</span><span class="n">lat0</span><span class="p">)</span>
<span class="n">f</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="c"># make figure and axis</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">eanS</span><span class="p">(</span><span class="n">h</span><span class="p">,</span><span class="n">k</span><span class="p">)</span><span class="o">*</span><span class="mi">180</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span><span class="n">lat</span><span class="o">*</span><span class="mi">180</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s">'Local angular distortion $e_{an}$ (deg) * surface $S$'</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s">'Latitude $\phi$ (deg)'</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="n">xmin</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span><span class="n">xmax</span><span class="o">=</span><span class="mi">180</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="n">v</span> <span class="o">=</span> <span class="n">interactive</span><span class="p">(</span><span class="n">plt_eanS_ecp_lat0</span><span class="p">,</span> <span class="n">lat0</span> <span class="o">=</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">90</span><span class="o">/</span><span class="mi">180</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span><span class="mi">5</span><span class="o">/</span><span class="mi">180</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">));</span>
<span class="n">display</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
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"
>
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<p>Notice:</p>
<ul>
<li>Zero local angular distortion at $\phi = \frac{\pi}{2}$ because $\cos(\phi) = 0$ </li>
<li>Zero local angular distortion at $\phi = \phi_0$ because no distortion at standard parallel</li>
</ul>
<p>For which $\phi_0$ is the surface between the curve and the $y$-axis minimal?</p>
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<p>Compute and plot average angular distortion $E_{an}$ for $\phi_0$ = 0°, 5°, 10°, ... 60°.</p>
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<div class="prompt input_prompt">In [8]:</div>
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<div class=" highlight hl-ipython3"><pre><span class="n">lat0_range</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">95</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span><span class="o">/</span><span class="mi">180</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span>
</pre></div>
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<div class="prompt input_prompt">In [9]:</div>
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<div class=" highlight hl-ipython3"><pre><span class="n">Ean_ecp_lat0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="c"># create empty array</span>
<span class="k">for</span> <span class="n">lat0</span> <span class="ow">in</span> <span class="n">lat0_range</span><span class="p">:</span>
<span class="n">h</span> <span class="o">=</span> <span class="n">h_ecp_lat0</span><span class="p">(</span><span class="n">lat0</span><span class="p">)</span>
<span class="n">k</span> <span class="o">=</span> <span class="n">k_ecp_lat0</span><span class="p">(</span><span class="n">lat0</span><span class="p">)</span>
<span class="n">Ean_ecp_lat0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Ean_ecp_lat0</span><span class="p">,</span><span class="n">E</span><span class="p">(</span><span class="n">eanS</span><span class="p">(</span><span class="n">h</span><span class="p">,</span><span class="n">k</span><span class="p">)))</span>
</pre></div>
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<div class="input">
<div class="prompt input_prompt">In [10]:</div>
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<div class=" highlight hl-ipython3"><pre><span class="n">f</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="c"># make figure and axis</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Ean_ecp_lat0</span><span class="o">*</span><span class="mi">180</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span><span class="n">lat0_range</span><span class="o">*</span><span class="mi">180</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span><span class="s">'b.-'</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s">'Average angular distortion $E_{an}$ (deg)'</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s">'Latitude of standard parallel $\phi_0$ (deg)'</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="n">xmin</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
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