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<title>function module — pylib 2019.5.19 documentation</title>
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<div class="section" id="module-function">
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<span id="function-module"></span><h1>function module<a class="headerlink" href="#module-function" title="Permalink to this headline">¶</a></h1>
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<p>Mathematical equations.</p>
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<dl class="field-list simple">
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<dt class="field-odd">Date</dt>
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<dd class="field-odd"><p>2019-11-02</p>
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</dd>
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</dl>
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<span class="target" id="module-function"></span><dl class="function">
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<dt id="function.cosine_wave">
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<code class="sig-name descname">cosine_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#cosine_wave"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#function.cosine_wave" title="Permalink to this definition">¶</a></dt>
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<dd><p>A cosine wave is said to be sinusoidal, because,
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<span class="math notranslate nohighlight">\(\cos(x) = \sin(x + \pi/2)\)</span>, which is also a sine wave with a
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phase-shift of π/2 radians. Because of this head start, it is often
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said that the cosine function leads the sine function or the sine
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lags the cosine.</p>
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<dl class="field-list simple">
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<dt class="field-odd">Parameters</dt>
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<dd class="field-odd"><ul class="simple">
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<li><p><strong>A</strong> (<em>float</em><em> or </em><em>int</em>) – amplitude</p></li>
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<li><p><strong>k</strong> (<em>float</em><em> or </em><em>int</em>) – (angular) wave number</p></li>
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<li><p><strong>f</strong> (<em>float</em><em> or </em><em>int</em>) – ordinary frequency</p></li>
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<li><p><strong>phi</strong> (<em>float</em><em> or </em><em>int</em>) – phase</p></li>
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<li><p><strong>D</strong> (<em>float</em><em> or </em><em>int</em>) – non-zero center amplitude</p></li>
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<li><p><strong>degree</strong> (<em>bool</em>) – boolean to switch between radians and degree. If
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False phi is interpreted in radians and if True then phi is
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interpreted in degrees.</p></li>
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</ul>
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</dd>
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<dt class="field-even">Results</dt>
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<dd class="field-even"><p>sine wave function of spatial variable x and optional
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time t</p>
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</dd>
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</dl>
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<div class="admonition seealso">
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<p class="admonition-title">See also</p>
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<p><code class="xref py py-meth docutils literal notranslate"><span class="pre">function_sine_wave_degree()</span></code></p>
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</div>
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</dd></dl>
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<dl class="function">
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<dt id="function.epitrochoid">
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<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>
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<dd><p>Epitrochoid</p>
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<p>A point is attached with a distance d from the center of a circle
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of radius r. The circle is rolling around the outside of a fixed
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circle of radius R.</p>
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<dl class="field-list simple">
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<dt class="field-odd">Parameters</dt>
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<dd class="field-odd"><ul class="simple">
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<li><p><strong>R</strong> (<em>float</em>) – radius of the fixed interior circle</p></li>
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<li><p><strong>r</strong> – radius of the rolling circle outside of the fixed circle</p></li>
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<li><p><strong>d</strong> (<em>float</em>) – distance from the center of the exterior circle</p></li>
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</ul>
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</dd>
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<dt class="field-even">Typre r</dt>
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<dd class="field-even"><p>float</p>
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</dd>
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<dt class="field-odd">Results</dt>
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<dd class="field-odd"><p>functions for x of theta and y of theta</p>
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</dd>
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<dt class="field-even">Return type</dt>
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<dd class="field-even"><p>tuple</p>
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</dd>
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</dl>
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<div class="math notranslate nohighlight">
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\[\begin{split}x(\theta) = (R + r)\cos\theta - d\cos\left(\frac{R+r}{r}\theta\right) \\
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y(\theta) = (R + r)\sin\theta - d\sin\left(\frac{R+r}{r}\theta\right) \\
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\theta = \left[0, 2\pi\right]\end{split}\]</div>
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<div class="highlight-default notranslate"><div class="highlight"><pre><span></span> <span class="o">*</span> <span class="o">*</span> <span class="o">*</span>
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<span class="o">*</span> <span class="n">R</span> <span class="o">*</span>
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<span class="o">*</span> <span class="o">*</span>
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<span class="o">*</span> <span class="o">*</span> <span class="o">*</span> <span class="o">*</span>
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<span class="o">*</span> <span class="o">*</span> <span class="o">*</span> <span class="n">r</span> <span class="o">*</span>
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<span class="o">*</span> <span class="o">**</span> <span class="o">....</span> <span class="o">*</span>
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<span class="o">*</span> <span class="o">**</span> <span class="n">d</span> <span class="o">*</span>
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<span class="o">*</span> <span class="o">*</span> <span class="o">*</span> <span class="o">*</span>
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<span class="o">*</span> <span class="o">*</span> <span class="o">*</span> <span class="o">*</span>
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<span class="o">*</span> <span class="o">*</span>
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<span class="o">*</span> <span class="o">*</span>
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<span class="o">*</span> <span class="o">*</span> <span class="o">*</span>
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</pre></div>
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</div>
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<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </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>
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<span class="gp">>>> </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>
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</pre></div>
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</div>
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</dd></dl>
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<dl class="function">
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<dt id="function.hypotrochoid">
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<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>
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<dd><p>Hypotrochoid</p>
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<p>A point is attached with a distance d from the center of a circle
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of radius r. The circle is rolling around the inside of a fixed
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circle of radius R.</p>
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<dl class="field-list simple">
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<dt class="field-odd">Parameters</dt>
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<dd class="field-odd"><ul class="simple">
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<li><p><strong>R</strong> (<em>float</em>) – radius of the fixed exterior circle</p></li>
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<li><p><strong>r</strong> – radius of the rolling circle inside of the fixed circle</p></li>
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<li><p><strong>d</strong> (<em>float</em>) – distance from the center of the interior circle</p></li>
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</ul>
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</dd>
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<dt class="field-even">Typre r</dt>
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<dd class="field-even"><p>float</p>
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</dd>
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<dt class="field-odd">Results</dt>
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<dd class="field-odd"><p>functions for x of theta and y of theta</p>
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</dd>
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<dt class="field-even">Return type</dt>
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<dd class="field-even"><p>tuple</p>
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</dd>
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</dl>
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<div class="math notranslate nohighlight">
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\[\begin{split}x(\theta) = (R - r)\cos\theta + d\cos\left(\frac{R-r}{r}\theta\right) \\
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y(\theta) = (R - r)\sin\theta - d\sin\left(\frac{R-r}{r}\theta\right) \\
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\theta = \left[0, 2\pi\frac{\mathrm{lcm}(r, R)}{R}\right]\end{split}\]</div>
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<div class="highlight-default notranslate"><div class="highlight"><pre><span></span> <span class="o">*</span> <span class="o">*</span> <span class="o">*</span>
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<span class="o">*</span> <span class="n">R</span> <span class="o">*</span>
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<span class="o">*</span> <span class="o">*</span>
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<span class="o">*</span> <span class="o">*</span> <span class="o">*</span> <span class="o">*</span>
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<span class="o">*</span> <span class="o">*</span> <span class="n">r</span> <span class="o">**</span>
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<span class="o">*</span> <span class="o">*</span> <span class="o">....</span> <span class="o">*</span>
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<span class="o">*</span> <span class="o">*</span> <span class="n">d</span> <span class="o">*</span>
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<span class="o">*</span> <span class="o">*</span> <span class="o">**</span>
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<span class="o">*</span> <span class="o">*</span> <span class="o">*</span> <span class="o">*</span>
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<span class="o">*</span> <span class="o">*</span>
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<span class="o">*</span> <span class="o">*</span>
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<span class="o">*</span> <span class="o">*</span> <span class="o">*</span>
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</pre></div>
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</div>
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<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </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>
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<span class="gp">>>> </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>
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</pre></div>
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</div>
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<div class="admonition seealso">
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<p class="admonition-title">See also</p>
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<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>
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</div>
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</dd></dl>
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<dl class="function">
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<dt id="function.sine_wave">
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<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>
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<dd><p>A sine wave or sinusoid is a mathematical curve that describes a
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smooth periodic oscillation.</p>
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<dl class="field-list simple">
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<dt class="field-odd">Parameters</dt>
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<dd class="field-odd"><ul class="simple">
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<li><p><strong>A</strong> (<em>float</em><em> or </em><em>int</em>) – amplitude</p></li>
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<li><p><strong>k</strong> (<em>float</em><em> or </em><em>int</em>) – (angular) wave number</p></li>
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<li><p><strong>f</strong> (<em>float</em><em> or </em><em>int</em>) – ordinary frequency</p></li>
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<li><p><strong>phi</strong> (<em>float</em><em> or </em><em>int</em>) – phase</p></li>
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<li><p><strong>D</strong> (<em>float</em><em> or </em><em>int</em>) – non-zero center amplitude</p></li>
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<li><p><strong>degree</strong> (<em>bool</em>) – boolean to switch between radians and degree. If
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False phi is interpreted in radians and if True then phi is
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interpreted in degrees.</p></li>
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</ul>
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</dd>
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<dt class="field-even">Results</dt>
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<dd class="field-even"><p>sine wave function of spatial variable x and optional
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time t</p>
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</dd>
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</dl>
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<p>In general, the function is:</p>
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<div class="math notranslate nohighlight">
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\[\begin{split}y(x,t) = A\sin(kx + 2\pi f t + \varphi) + D \\
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y(x,t) = A\sin(kx + \omega t + \varphi) + D\end{split}\]</div>
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<p>where:</p>
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<blockquote>
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<div><ul class="simple">
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<li><p>A, amplitude, the peak deviation of the function from zero.</p></li>
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<li><p>f, ordinary frequency, the number of oscillations (cycles) that
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occur each second of time.</p></li>
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<li><p>ω = 2πf, angular frequency, the rate of change of the function
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argument in units of radians per second. If ω < 0 the wave is
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moving to the right, if ω > 0 the wave is moving to the left.</p></li>
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<li><p>φ, phase, specifies (in radians) where in its cycle the
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oscillation is at t = 0.</p></li>
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<li><p>x, spatial variable that represents the position on the
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dimension on which the wave propagates.</p></li>
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<li><p>k, characteristic parameter called wave number (or angular wave
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number), which represents the proportionality between the
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angular frequency ω and the linear speed (speed of propagation)
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ν.</p></li>
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<li><p>D, non-zero center amplitude.</p></li>
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</ul>
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</div></blockquote>
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<p>The wavenumber is related to the angular frequency by:</p>
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<div class="math notranslate nohighlight">
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\[k={\omega \over v}={2\pi f \over v}={2\pi \over \lambda }\]</div>
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<p>where λ (lambda) is the wavelength, f is the frequency, and v is the
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linear speed.</p>
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<div class="admonition seealso">
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<p class="admonition-title">See also</p>
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<p><code class="xref py py-meth docutils literal notranslate"><span class="pre">function_cosine_wave_degree()</span></code></p>
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</div>
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</dd></dl>
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