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Daniel Weschke f47fa19cd2 add geometry functions to translate and rotate lists and define lines and cubic lines 
add data function to fold a list and geometry_plot functions to plot lines and tests for the geometry functions
2019-08-28 11:39:50 +02:00

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<h1>Source code for numerical.ode_model</h1><div class="highlight"><pre>
<span></span><span class="ch">#!/usr/bin/env python</span>
<span class="c1"># -*- coding: utf-8 -*-</span>
<span class="sd">&quot;&quot;&quot;Mathmatical models governed by ordinary differential equations.</span>
<span class="sd">Describes initial value problems as systems of first order ordinary differential</span>
<span class="sd">equations.</span>
<span class="sd">:Date: 2019-05-25</span>
<span class="sd">.. module:: ode_model</span>
<span class="sd"> :platform: *nix, Windows</span>
<span class="sd"> :synopsis: Models of ordinary differential equations.</span>
<span class="sd">.. moduleauthor:: Daniel Weschke &lt;daniel.weschke@directbox.de&gt;</span>
<span class="sd">&quot;&quot;&quot;</span>
<span class="kn">from</span> <span class="nn">__future__</span> <span class="k">import</span> <span class="n">division</span><span class="p">,</span> <span class="n">print_function</span>
<span class="kn">from</span> <span class="nn">numpy</span> <span class="k">import</span> <span class="n">array</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">dot</span><span class="p">,</span> <span class="n">square</span>
<span class="kn">from</span> <span class="nn">numpy.linalg</span> <span class="k">import</span> <span class="n">inv</span>
<div class="viewcode-block" id="disk"><a class="viewcode-back" href="../../numerical.html#numerical.ode_model.disk">[docs]</a><span class="k">def</span> <span class="nf">disk</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="o">*</span><span class="n">p</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;Rotation of an eccentric disk.</span>
<span class="sd"> :param x: values of the function</span>
<span class="sd"> :type x: list</span>
<span class="sd"> :param t: time</span>
<span class="sd"> :type t: list</span>
<span class="sd"> :param `*p`: parameters of the function</span>
<span class="sd"> * diameter</span>
<span class="sd"> * eccentricity</span>
<span class="sd"> * torque</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="n">qp1</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span>
<span class="n">qp2</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="mi">4</span><span class="p">]</span>
<span class="n">qp3</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span>
<span class="n">M</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span><span class="o">+</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">-</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]],</span> \
<span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">4</span><span class="p">]</span><span class="o">+</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">-</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]],</span> \
<span class="p">[</span><span class="n">p</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">+</span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">])]])</span>
<span class="n">qp46</span> <span class="o">=</span> <span class="n">dot</span><span class="p">(</span><span class="n">inv</span><span class="p">(</span><span class="n">M</span><span class="p">),</span> <span class="n">y</span><span class="p">)</span>
<span class="n">qp4</span><span class="p">,</span> <span class="n">qp5</span><span class="p">,</span> <span class="n">qp6</span> <span class="o">=</span> <span class="n">qp46</span><span class="o">.</span><span class="n">reshape</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">tolist</span><span class="p">()</span> <span class="c1"># 2d array to 1d array to list</span>
<span class="k">return</span> <span class="n">qp1</span><span class="p">,</span> <span class="n">qp2</span><span class="p">,</span> <span class="n">qp3</span><span class="p">,</span> <span class="n">qp4</span><span class="p">,</span> <span class="n">qp5</span><span class="p">,</span> <span class="n">qp6</span></div>
<div class="viewcode-block" id="disk_nm"><a class="viewcode-back" href="../../numerical.html#numerical.ode_model.disk_nm">[docs]</a><span class="k">def</span> <span class="nf">disk_nm</span><span class="p">(</span><span class="n">xn</span><span class="p">,</span> <span class="n">xpn</span><span class="p">,</span> <span class="n">xppn</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="o">*</span><span class="n">p</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;Rotation of an eccentric disk.</span>
<span class="sd"> :param xn: values of the function</span>
<span class="sd"> :type xn: list</span>
<span class="sd"> :param xpn: first derivative values of the function</span>
<span class="sd"> :type xpn: list</span>
<span class="sd"> :param xppn: second derivative values of the function</span>
<span class="sd"> :type xppn: list</span>
<span class="sd"> :param t: time</span>
<span class="sd"> :type t: list</span>
<span class="sd"> :param `*p`: parameters of the function</span>
<span class="sd"> * diameter</span>
<span class="sd"> * eccentricity</span>
<span class="sd"> * torque</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="n">N</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="n">xppn</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xppn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">square</span><span class="p">(</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">+</span><span class="n">xn</span><span class="p">[</span><span class="mi">0</span><span class="p">]],</span>
<span class="p">[</span><span class="n">xppn</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xppn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">square</span><span class="p">(</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">+</span><span class="n">xn</span><span class="p">[</span><span class="mi">1</span><span class="p">]],</span>
<span class="p">[</span><span class="n">xppn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">+</span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xn</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xn</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span><span class="o">-</span><span class="n">p</span><span class="p">[</span><span class="mi">2</span><span class="p">]]])</span>
<span class="n">dN</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">*</span><span class="n">xppn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">square</span><span class="p">(</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</span>
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">*</span><span class="n">xppn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">+</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">square</span><span class="p">(</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</span>
<span class="p">[</span><span class="o">-</span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">]),</span> <span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">]),</span> <span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xn</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xn</span><span class="p">[</span><span class="mi">1</span><span class="p">])]])</span>
<span class="n">dNp</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">]],</span>
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">]],</span>
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
<span class="n">dNpp</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</span>
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</span>
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
<span class="k">return</span> <span class="n">N</span><span class="p">,</span> <span class="n">dN</span><span class="p">,</span> <span class="n">dNp</span><span class="p">,</span> <span class="n">dNpp</span></div>
<div class="viewcode-block" id="disk_nmmdk"><a class="viewcode-back" href="../../numerical.html#numerical.ode_model.disk_nmmdk">[docs]</a><span class="k">def</span> <span class="nf">disk_nmmdk</span><span class="p">(</span><span class="n">xn</span><span class="p">,</span> <span class="n">xpn</span><span class="p">,</span> <span class="n">xppn</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="o">*</span><span class="n">p</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;Rotation of an eccentric disk.</span>
<span class="sd"> :param xn: values of the function</span>
<span class="sd"> :type xn: list</span>
<span class="sd"> :param xpn: derivative values of the function</span>
<span class="sd"> :type xpn: list</span>
<span class="sd"> :param xppn: second derivative values of the function</span>
<span class="sd"> :type xppn: list</span>
<span class="sd"> :param t: time</span>
<span class="sd"> :type t: list</span>
<span class="sd"> :param `*p`: parameters of the function</span>
<span class="sd"> * diameter</span>
<span class="sd"> * eccentricity</span>
<span class="sd"> * torque</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="n">rm</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="n">xppn</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xppn</span><span class="p">[</span><span class="mi">2</span><span class="p">]],</span>
<span class="p">[</span><span class="n">xppn</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xppn</span><span class="p">[</span><span class="mi">2</span><span class="p">]],</span>
<span class="p">[</span><span class="n">xppn</span><span class="p">[</span><span class="mi">2</span><span class="p">]]])</span>
<span class="n">rmx</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">*</span><span class="n">xppn</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</span>
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">*</span><span class="n">xppn</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</span>
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
<span class="n">rmxpp</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</span>
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</span>
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
<span class="n">rd</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">square</span><span class="p">(</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</span>
<span class="p">[</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">square</span><span class="p">(</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</span>
<span class="p">[</span><span class="mi">0</span><span class="p">]])</span>
<span class="n">rdx</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">square</span><span class="p">(</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</span>
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">+</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">square</span><span class="p">(</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</span>
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
<span class="n">rdxp</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">]],</span>
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">]],</span>
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
<span class="n">rk</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="n">xn</span><span class="p">[</span><span class="mi">0</span><span class="p">]],</span>
<span class="p">[</span><span class="n">xn</span><span class="p">[</span><span class="mi">1</span><span class="p">]],</span>
<span class="p">[</span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xn</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xn</span><span class="p">[</span><span class="mi">1</span><span class="p">])]])</span>
<span class="n">rkx</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<span class="p">[</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<span class="p">[</span><span class="o">-</span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">]),</span> <span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">]),</span> <span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xn</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xn</span><span class="p">[</span><span class="mi">1</span><span class="p">])]])</span>
<span class="n">f</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="n">p</span><span class="p">[</span><span class="mi">2</span><span class="p">]]])</span>
<span class="k">return</span> <span class="n">rm</span><span class="p">,</span> <span class="n">rmx</span><span class="p">,</span> <span class="n">rmxpp</span><span class="p">,</span> <span class="n">rd</span><span class="p">,</span> <span class="n">rdx</span><span class="p">,</span> <span class="n">rdxp</span><span class="p">,</span> <span class="n">rk</span><span class="p">,</span> <span class="n">rkx</span><span class="p">,</span> <span class="n">f</span></div>
<span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s1">&#39;__main__&#39;</span><span class="p">:</span>
<span class="kc">True</span>
</pre></div>
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