add parametric equations for eptrochoid

docs/build/html/_modules/function.html

@@ -37,7 +37,7 @@
 <span class="c1"># -*- coding: utf-8 -*-</span>
 <span class="sd">&quot;&quot;&quot;Mathematical equations.</span>
 
-<span class="sd">:Date: 2019-11-01</span>
+<span class="sd">:Date: 2019-11-02</span>
 
 <span class="sd">.. module:: function</span>
 <span class="sd">  :platform: *nix, Windows</span>
@@ -146,6 +146,7 @@
 
 <span class="c1">#</span>
 <span class="c1"># Parametric equations</span>
+<span class="c1"># roulette</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>
@@ -195,6 +196,51 @@
   <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>
+
+<div class="viewcode-block" id="epitrochoid"><a class="viewcode-back" href="../function.html#function.epitrochoid">[docs]</a><span class="k">def</span> <span class="nf">epitrochoid</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;Epitrochoid</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 outside of a fixed</span>
+<span class="sd">  circle of radius R.</span>
+
+<span class="sd">  :param R: radius of the fixed interior circle</span>
+<span class="sd">  :type R: float</span>
+<span class="sd">  :param r: radius of the rolling circle outside of the fixed circle</span>
+<span class="sd">  :typre r: float</span>
+<span class="sd">  :param d: distance from the center of the exterior 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\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 = epitrochoid(3, 1, 0.5)[:1]</span>
+<span class="sd">  &gt;&gt;&gt; x, y, theta_end = epitrochoid(3, 1, 0.5)</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="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

@@ -37,7 +37,7 @@
 <p>Mathematical equations.</p>
 <dl class="field-list simple">
 <dt class="field-odd">Date</dt>
-<dd class="field-odd"><p>2019-11-01</p>
+<dd class="field-odd"><p>2019-11-02</p>
 </dd>
 </dl>
 <span class="target" id="module-function"></span><dl class="function">
@@ -72,6 +72,55 @@ time t</p>
 </div>
 </dd></dl>
 
+<dl class="function">
+<dt id="function.epitrochoid">
+<code class="sig-name descname">epitrochoid</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#epitrochoid"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#function.epitrochoid" title="Permalink to this definition">¶</a></dt>
+<dd><p>Epitrochoid</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 outside 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 interior circle</p></li>
+<li><p><strong>r</strong> – radius of the rolling circle outside of the fixed circle</p></li>
+<li><p><strong>d</strong> (<em>float</em>) – distance from the center of the exterior 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">
+\[\begin{split}x(\theta) = (R + r)\cos\theta - d\cos\left(\frac{R+r}{r}\theta\right) \\
+y(\theta) = (R + r)\sin\theta - d\sin\left(\frac{R+r}{r}\theta\right) \\
+\theta = \left[0, 2\pi\right]\end{split}\]</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="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>  <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">epitrochoid</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">0.5</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">epitrochoid</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
+</pre></div>
+</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>

docs/build/html/genindex.html

@@ -125,6 +125,8 @@
       <li><a href="date.html#date.easter_monday">easter_monday() (in module date)</a>
 </li>
       <li><a href="date.html#date.easter_sunday">easter_sunday() (in module date)</a>
+</li>
+      <li><a href="function.html#function.epitrochoid">epitrochoid() (in module function)</a>
 </li>
   </ul></td>
 </tr></table>

pylib/function.py

@@ -2,7 +2,7 @@
 # -*- coding: utf-8 -*-
 """Mathematical equations.
 
-:Date: 2019-11-01
+:Date: 2019-11-02
 
 .. module:: function
   :platform: *nix, Windows
@@ -111,6 +111,7 @@ def cosine_wave(A=1, k=1, f=1, phi=0, D=0, degree=False):
 
 #
 # Parametric equations
+# roulette
 #
 
 def hypotrochoid(R, r, d):
@@ -160,3 +161,48 @@ def hypotrochoid(R, r, d):
   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
+
+def epitrochoid(R, r, d):
+  r"""Epitrochoid
+
+  A point is attached with a distance d from the center of a circle
+  of radius r. The circle is rolling around the outside of a fixed
+  circle of radius R.
+
+  :param R: radius of the fixed interior circle
+  :type R: float
+  :param r: radius of the rolling circle outside of the fixed circle
+  :typre r: float
+  :param d: distance from the center of the exterior 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\right]
+
+  ::
+
+           *  *  *
+        * R         *
+      *               *
+     *                 *     *  *
+    *                   * * r      *
+    *                   ** ....     *
+    *                   **   d      *
+    *                   * *        *
+     *                 *     *  *
+      *               *
+        *          *
+           *  *  *
+
+  >>> x, y = epitrochoid(3, 1, 0.5)[:1]
+  >>> x, y, theta_end = epitrochoid(3, 1, 0.5)
+  """
+  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
+  return x, y, theta_end