add parametric equations for hypotrochoid

docs/build/html/_modules/function.html

@@ -35,17 +35,18 @@
   <h1>Source code for function</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;Mathematical functions.</span>
+<span class="sd">&quot;&quot;&quot;Mathematical equations.</span>
 
-<span class="sd">:Date: 2019-10-27</span>
+<span class="sd">:Date: 2019-11-01</span>
 
 <span class="sd">.. module:: function</span>
 <span class="sd">  :platform: *nix, Windows</span>
-<span class="sd">  :synopsis: Mathematical functions.</span>
+<span class="sd">  :synopsis: Mathematical 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">import</span> <span class="nn">math</span>
+<span class="kn">from</span> <span class="nn">pylib.mathematics</span> <span class="k">import</span> <span class="n">lcm</span>
 
 <div class="viewcode-block" id="sine_wave"><a class="viewcode-back" href="../function.html#function.sine_wave">[docs]</a><span class="k">def</span> <span class="nf">sine_wave</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">k</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">f</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">phi</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">D</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
   <span class="sa">r</span><span class="sd">&quot;&quot;&quot;A sine wave or sinusoid is a mathematical curve that describes a</span>
@@ -64,6 +65,10 @@
 <span class="sd">  :param degree: boolean to switch between radians and degree. If</span>
 <span class="sd">    False phi is interpreted in radians and if True then phi is</span>
 <span class="sd">    interpreted in degrees.</span>
+<span class="sd">  :type degree: bool</span>
+
+<span class="sd">  :results: sine wave function of spatial variable x and optional</span>
+<span class="sd">    time t</span>
 
 <span class="sd">  In general, the function is:</span>
 
@@ -124,6 +129,10 @@
 <span class="sd">  :param degree: boolean to switch between radians and degree. If</span>
 <span class="sd">    False phi is interpreted in radians and if True then phi is</span>
 <span class="sd">    interpreted in degrees.</span>
+<span class="sd">  :type degree: bool</span>
+
+<span class="sd">  :results: sine wave function of spatial variable x and optional</span>
+<span class="sd">    time t</span>
 
 <span class="sd">  .. seealso::</span>
 <span class="sd">    :meth:`function_sine_wave_degree`</span>
@@ -133,6 +142,59 @@
   <span class="k">else</span><span class="p">:</span>
     <span class="n">phi</span> <span class="o">=</span> <span class="n">phi</span> <span class="o">+</span> <span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span>
   <span class="k">return</span> <span class="n">sine_wave</span><span class="p">(</span><span class="n">A</span><span class="o">=</span><span class="n">A</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="n">k</span><span class="p">,</span> <span class="n">f</span><span class="o">=</span><span class="n">f</span><span class="p">,</span> <span class="n">phi</span><span class="o">=</span><span class="n">phi</span><span class="p">,</span> <span class="n">D</span><span class="o">=</span><span class="n">D</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span></div>
+
+
+<span class="c1">#</span>
+<span class="c1"># Parametric equations</span>
+<span class="c1">#</span>
+
+<div class="viewcode-block" id="hypotrochoid"><a class="viewcode-back" href="../function.html#function.hypotrochoid">[docs]</a><span class="k">def</span> <span class="nf">hypotrochoid</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">d</span><span class="p">):</span>
+  <span class="sa">r</span><span class="sd">&quot;&quot;&quot;Hypotrochoid</span>
+
+<span class="sd">  A point is attached with a distance d from the center of a circle</span>
+<span class="sd">  of radius r. The circle is rolling around the inside of a fixed</span>
+<span class="sd">  circle of radius R.</span>
+
+<span class="sd">  :param R: radius of the fixed exterior circle</span>
+<span class="sd">  :type R: float</span>
+<span class="sd">  :param r: radius of the rolling circle inside of the fixed circle</span>
+<span class="sd">  :typre r: float</span>
+<span class="sd">  :param d: distance from the center of the interior circle</span>
+<span class="sd">  :type d: float</span>
+
+<span class="sd">  :results: functions for x of theta and y of theta</span>
+<span class="sd">  :rtype: tuple</span>
+
+<span class="sd">  .. math::</span>
+<span class="sd">    x(\theta) = (R - r)\cos\theta + d\cos\left(\frac{R-r}{r}\theta\right) \\</span>
+<span class="sd">    y(\theta) = (R - r)\sin\theta - d\sin\left(\frac{R-r}{r}\theta\right) \\</span>
+<span class="sd">    \theta = \left[0, 2\pi\frac{\mathrm{lcm}(r, R)}{R}\right]</span>
+
+<span class="sd">  ::</span>
+
+<span class="sd">           *  *  *</span>
+<span class="sd">        * R         *</span>
+<span class="sd">      *               *</span>
+<span class="sd">     *           *  *  *</span>
+<span class="sd">    *         * r      **</span>
+<span class="sd">    *        *     .... *</span>
+<span class="sd">    *        *      d   *</span>
+<span class="sd">    *         *        **</span>
+<span class="sd">     *           *  *  *</span>
+<span class="sd">      *               *</span>
+<span class="sd">        *          *</span>
+<span class="sd">           *  *  *</span>
+
+<span class="sd">  &gt;&gt;&gt; x, y = hyotrochoid(20, 6, 6)[:1]</span>
+<span class="sd">  &gt;&gt;&gt; x, y, theta_end = hyotrochoid(20, 6, 6)</span>
+
+<span class="sd">  .. seealso::</span>
+<span class="sd">    :meth:`mathematics.lcm`</span>
+<span class="sd">  &quot;&quot;&quot;</span>
+  <span class="n">x</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">theta</span><span class="p">:</span> <span class="p">(</span><span class="n">R</span><span class="o">-</span><span class="n">r</span><span class="p">)</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span> <span class="o">+</span> <span class="n">d</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">cos</span><span class="p">((</span><span class="n">R</span><span class="o">-</span><span class="n">r</span><span class="p">)</span><span class="o">/</span><span class="n">r</span> <span class="o">*</span> <span class="n">theta</span><span class="p">)</span>
+  <span class="n">y</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">theta</span><span class="p">:</span> <span class="p">(</span><span class="n">R</span><span class="o">-</span><span class="n">r</span><span class="p">)</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span> <span class="o">-</span> <span class="n">d</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">((</span><span class="n">R</span><span class="o">-</span><span class="n">r</span><span class="p">)</span><span class="o">/</span><span class="n">r</span> <span class="o">*</span> <span class="n">theta</span><span class="p">)</span>
+  <span class="n">theta_end</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">lcm</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">R</span><span class="p">)</span><span class="o">/</span><span class="n">R</span>
+  <span class="k">return</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">theta_end</span></div>
 </pre></div>
 
           </div>

docs/build/html/function.html

@@ -34,10 +34,10 @@
             
   <div class="section" id="module-function">
 <span id="function-module"></span><h1>function module<a class="headerlink" href="#module-function" title="Permalink to this headline">¶</a></h1>
-<p>Mathematical functions.</p>
+<p>Mathematical equations.</p>
 <dl class="field-list simple">
 <dt class="field-odd">Date</dt>
-<dd class="field-odd"><p>2019-10-27</p>
+<dd class="field-odd"><p>2019-11-01</p>
 </dd>
 </dl>
 <span class="target" id="module-function"></span><dl class="function">
@@ -56,11 +56,15 @@ lags the cosine.</p>
 <li><p><strong>f</strong> (<em>float</em><em> or </em><em>int</em>) – ordinary frequency</p></li>
 <li><p><strong>phi</strong> (<em>float</em><em> or </em><em>int</em>) – phase</p></li>
 <li><p><strong>D</strong> (<em>float</em><em> or </em><em>int</em>) – non-zero center amplitude</p></li>
-<li><p><strong>degree</strong> – boolean to switch between radians and degree. If
+<li><p><strong>degree</strong> (<em>bool</em>) – boolean to switch between radians and degree. If
 False phi is interpreted in radians and if True then phi is
 interpreted in degrees.</p></li>
 </ul>
 </dd>
+<dt class="field-even">Results</dt>
+<dd class="field-even"><p>sine wave function of spatial variable x and optional
+time t</p>
+</dd>
 </dl>
 <div class="admonition seealso">
 <p class="admonition-title">See also</p>
@@ -68,6 +72,59 @@ interpreted in degrees.</p></li>
 </div>
 </dd></dl>
 
+<dl class="function">
+<dt id="function.hypotrochoid">
+<code class="sig-name descname">hypotrochoid</code><span class="sig-paren">(</span><em class="sig-param">R</em>, <em class="sig-param">r</em>, <em class="sig-param">d</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/function.html#hypotrochoid"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#function.hypotrochoid" title="Permalink to this definition">¶</a></dt>
+<dd><p>Hypotrochoid</p>
+<p>A point is attached with a distance d from the center of a circle
+of radius r. The circle is rolling around the inside of a fixed
+circle of radius R.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters</dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>R</strong> (<em>float</em>) – radius of the fixed exterior circle</p></li>
+<li><p><strong>r</strong> – radius of the rolling circle inside of the fixed circle</p></li>
+<li><p><strong>d</strong> (<em>float</em>) – distance from the center of the interior circle</p></li>
+</ul>
+</dd>
+<dt class="field-even">Typre r</dt>
+<dd class="field-even"><p>float</p>
+</dd>
+<dt class="field-odd">Results</dt>
+<dd class="field-odd"><p>functions for x of theta and y of theta</p>
+</dd>
+<dt class="field-even">Return type</dt>
+<dd class="field-even"><p>tuple</p>
+</dd>
+</dl>
+<div class="math notranslate nohighlight">
+\[\begin{split}x(\theta) = (R - r)\cos\theta + d\cos\left(\frac{R-r}{r}\theta\right) \\
+y(\theta) = (R - r)\sin\theta - d\sin\left(\frac{R-r}{r}\theta\right) \\
+\theta = \left[0, 2\pi\frac{\mathrm{lcm}(r, R)}{R}\right]\end{split}\]</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span>       <span class="o">*</span>  <span class="o">*</span>  <span class="o">*</span>
+    <span class="o">*</span> <span class="n">R</span>         <span class="o">*</span>
+  <span class="o">*</span>               <span class="o">*</span>
+ <span class="o">*</span>           <span class="o">*</span>  <span class="o">*</span>  <span class="o">*</span>
+<span class="o">*</span>         <span class="o">*</span> <span class="n">r</span>      <span class="o">**</span>
+<span class="o">*</span>        <span class="o">*</span>     <span class="o">....</span> <span class="o">*</span>
+<span class="o">*</span>        <span class="o">*</span>      <span class="n">d</span>   <span class="o">*</span>
+<span class="o">*</span>         <span class="o">*</span>        <span class="o">**</span>
+ <span class="o">*</span>           <span class="o">*</span>  <span class="o">*</span>  <span class="o">*</span>
+  <span class="o">*</span>               <span class="o">*</span>
+    <span class="o">*</span>          <span class="o">*</span>
+       <span class="o">*</span>  <span class="o">*</span>  <span class="o">*</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">hyotrochoid</span><span class="p">(</span><span class="mi">20</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">6</span><span class="p">)[:</span><span class="mi">1</span><span class="p">]</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">theta_end</span> <span class="o">=</span> <span class="n">hyotrochoid</span><span class="p">(</span><span class="mi">20</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="admonition seealso">
+<p class="admonition-title">See also</p>
+<p><a class="reference internal" href="mathematics.html#mathematics.lcm" title="mathematics.lcm"><code class="xref py py-meth docutils literal notranslate"><span class="pre">mathematics.lcm()</span></code></a></p>
+</div>
+</dd></dl>
+
 <dl class="function">
 <dt id="function.sine_wave">
 <code class="sig-name descname">sine_wave</code><span class="sig-paren">(</span><em class="sig-param">A=1</em>, <em class="sig-param">k=1</em>, <em class="sig-param">f=1</em>, <em class="sig-param">phi=0</em>, <em class="sig-param">D=0</em>, <em class="sig-param">degree=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/function.html#sine_wave"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#function.sine_wave" title="Permalink to this definition">¶</a></dt>
@@ -81,11 +138,15 @@ smooth periodic oscillation.</p>
 <li><p><strong>f</strong> (<em>float</em><em> or </em><em>int</em>) – ordinary frequency</p></li>
 <li><p><strong>phi</strong> (<em>float</em><em> or </em><em>int</em>) – phase</p></li>
 <li><p><strong>D</strong> (<em>float</em><em> or </em><em>int</em>) – non-zero center amplitude</p></li>
-<li><p><strong>degree</strong> – boolean to switch between radians and degree. If
+<li><p><strong>degree</strong> (<em>bool</em>) – boolean to switch between radians and degree. If
 False phi is interpreted in radians and if True then phi is
 interpreted in degrees.</p></li>
 </ul>
 </dd>
+<dt class="field-even">Results</dt>
+<dd class="field-even"><p>sine wave function of spatial variable x and optional
+time t</p>
+</dd>
 </dl>
 <p>In general, the function is:</p>
 <div class="math notranslate nohighlight">

docs/build/html/genindex.html

@@ -173,6 +173,8 @@
   </ul></td>
   <td style="width: 33%; vertical-align: top;"><ul>
       <li><a href="time_of_day.html#time_of_day.hours_norm">hours_norm() (in module time_of_day)</a>
+</li>
+      <li><a href="function.html#function.hypotrochoid">hypotrochoid() (in module function)</a>
 </li>
   </ul></td>
 </tr></table>

docs/build/html/py-modindex.html

@@ -79,7 +79,7 @@
        <td></td>
        <td>
        <a href="function.html#module-function"><code class="xref">function</code></a> <em>(*nix, Windows)</em></td><td>
-       <em>Mathematical functions.</em></td></tr>
+       <em>Mathematical equations.</em></td></tr>
      <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
      <tr class="cap" id="cap-g"><td></td><td>
        <strong>g</strong></td><td></td></tr>

pylib/function.py

@@ -1,16 +1,17 @@
 #!/usr/bin/env python
 # -*- coding: utf-8 -*-
-"""Mathematical functions.
+"""Mathematical equations.
 
-:Date: 2019-10-27
+:Date: 2019-11-01
 
 .. module:: function
   :platform: *nix, Windows
-  :synopsis: Mathematical functions.
+  :synopsis: Mathematical equations.
 
 .. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de>
 """
 import math
+from pylib.mathematics import lcm
 
 def sine_wave(A=1, k=1, f=1, phi=0, D=0, degree=False):
   r"""A sine wave or sinusoid is a mathematical curve that describes a
@@ -29,6 +30,10 @@ def sine_wave(A=1, k=1, f=1, phi=0, D=0, degree=False):
   :param degree: boolean to switch between radians and degree. If
     False phi is interpreted in radians and if True then phi is
     interpreted in degrees.
+  :type degree: bool
+
+  :results: sine wave function of spatial variable x and optional
+    time t
 
   In general, the function is:
 
@@ -89,6 +94,10 @@ def cosine_wave(A=1, k=1, f=1, phi=0, D=0, degree=False):
   :param degree: boolean to switch between radians and degree. If
     False phi is interpreted in radians and if True then phi is
     interpreted in degrees.
+  :type degree: bool
+
+  :results: sine wave function of spatial variable x and optional
+    time t
 
   .. seealso::
     :meth:`function_sine_wave_degree`
@@ -98,3 +107,56 @@ def cosine_wave(A=1, k=1, f=1, phi=0, D=0, degree=False):
   else:
     phi = phi + math.pi/2
   return sine_wave(A=A, k=k, f=f, phi=phi, D=D, degree=degree)
+
+
+#
+# Parametric equations
+#
+
+def hypotrochoid(R, r, d):
+  r"""Hypotrochoid
+
+  A point is attached with a distance d from the center of a circle
+  of radius r. The circle is rolling around the inside of a fixed
+  circle of radius R.
+
+  :param R: radius of the fixed exterior circle
+  :type R: float
+  :param r: radius of the rolling circle inside of the fixed circle
+  :typre r: float
+  :param d: distance from the center of the interior circle
+  :type d: float
+
+  :results: functions for x of theta and y of theta
+  :rtype: tuple
+
+  .. math::
+    x(\theta) = (R - r)\cos\theta + d\cos\left(\frac{R-r}{r}\theta\right) \\
+    y(\theta) = (R - r)\sin\theta - d\sin\left(\frac{R-r}{r}\theta\right) \\
+    \theta = \left[0, 2\pi\frac{\mathrm{lcm}(r, R)}{R}\right]
+
+  ::
+
+           *  *  *
+        * R         *
+      *               *
+     *           *  *  *
+    *         * r      **
+    *        *     .... *
+    *        *      d   *
+    *         *        **
+     *           *  *  *
+      *               *
+        *          *
+           *  *  *
+
+  >>> x, y = hyotrochoid(20, 6, 6)[:1]
+  >>> x, y, theta_end = hyotrochoid(20, 6, 6)
+
+  .. seealso::
+    :meth:`mathematics.lcm`
+  """
+  x = lambda theta: (R-r)*math.cos(theta) + d*math.cos((R-r)/r * theta)
+  y = lambda theta: (R-r)*math.sin(theta) - d*math.sin((R-r)/r * theta)
+  theta_end = 2*math.pi*lcm(r, R)/R
+  return x, y, theta_end