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pylib/docs/build/html/pylib.function.html
Daniel Weschke 4fc4903dc2 change data, mathematics, function, geometry and geometry_plot_pylab. add data_step, data_step_std and helper. add an example and more documentation. 
inside the data module rename read to read_columns add add new read function to read the whole file as string. add print_list function to print one element per line. add unique_list and unique_list_hashable to reduce the list into a unique list with same order. add find_last, str_between, str_to_list functions.

inside the mathematics module for vector add normalized creating a new object (normalize will change the object), isclose and iscloseto, change ang to round internal number. for matrix improve slicing and add transposed creating a new object (transpose will change object).

inside the function module add b_spline_basis, b_spline_curve_with_knots and b_spline_knots functions. add sample_hal_open and sample_half_open_seq. add circle and ellipse.

inside the geometry module change CS init from using lists to Directions and add new constructor CS.init_xzy using lists. rename Wireframe to Polyline. add classes B_spline_curve_with_knots, Ellipse, ArcCircle, ArcEllipse, ArcBSplineCurveWithKnots. add function sample_half_open to create a list of Points.

inside the geometry_plot_pylab module change the help text.

add step_and data data_step_std module to read a step file to list and draw the content.

add helper module with context manager and decorator timeit to meassure the time for a section or function.

add example for b_spline function.
2020-01-08 21:59:53 +01:00

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<div class="section" id="module-pylib.function">
<span id="pylib-function-module"></span><h1>pylib.function module<a class="headerlink" href="#module-pylib.function" title="Permalink to this headline"></a></h1>
<p>Mathematical equations.</p>
<dl class="field-list simple">
<dt class="field-odd">Date</dt>
<dd class="field-odd"><p>2019-11-15</p>
</dd>
</dl>
<span class="target" id="module-function"></span><p>Functions returns function to apply conditions.</p>
<dl class="function">
<dt id="pylib.function.b_spline_basis">
<code class="sig-name descname">b_spline_basis</code><span class="sig-paren">(</span><em class="sig-param">knots</em>, <em class="sig-param">knot_span</em>, <em class="sig-param">degree</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/pylib/function.html#b_spline_basis"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.function.b_spline_basis" title="Permalink to this definition"></a></dt>
<dd><p>Cox-de Boor algorithm / recursion formula.</p>
<p>Calculate the i-th B-spline basis function of degree p: N_{i,p}(u)</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>knots</strong> (<em>list</em>) – Knot vector U. m + 1 non-decreasing numbers / knots,
<span class="math notranslate nohighlight">\(u_0 &lt;= u_1 &lt;= u_2 &lt;= ... &lt;= u_m\)</span></p></li>
<li><p><strong>knot_span</strong> (<em>int</em>) – i-th knot span</p></li>
<li><p><strong>degree</strong> (<em>int</em>) – degree of B-spline basis function</p></li>
</ul>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><p>B-spline basis function using variable, u in [u_0, u_m]</p>
</dd>
<dt class="field-odd">Return type</dt>
<dd class="field-odd"><p>function</p>
</dd>
</dl>
<div class="math notranslate nohighlight">
\[\begin{split}N_{i,0}(u) &amp;= \begin{cases} 1 &amp; \text{if } u_i \le u \lt u_{i+1} \\
0 &amp; \text{otherwise}\end{cases} \\
N_{i,p}(u) &amp;= \frac{u - u_i}{u_{i+p} - u_i} N_{i,p-1}(u) +
\frac{u_{i+p+1} - u}{u_{i+p+1} - u_{i+1}} N_{i+1,p-1}(u)\end{split}\]</div>
</dd></dl>
<dl class="function">
<dt id="pylib.function.b_spline_curve_with_knots">
<code class="sig-name descname">b_spline_curve_with_knots</code><span class="sig-paren">(</span><em class="sig-param">degree</em>, <em class="sig-param">control_points</em>, <em class="sig-param">knots</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/pylib/function.html#b_spline_curve_with_knots"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.function.b_spline_curve_with_knots" title="Permalink to this definition"></a></dt>
<dd><p>B-spline curve of degree p (order + 1) on a given set of knots.</p>
<p>n, m and p must satisfy m = n + p + 1.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>degree</strong> (<em>int</em>) – degree of B-spline basis functions</p></li>
<li><p><strong>control_points</strong> (<em>list</em>) – control points P, n + 1 control points</p></li>
<li><p><strong>knots</strong> (<em>list</em>) – Knot vector U. m + 1 non-decreasing numbers / knots,
<span class="math notranslate nohighlight">\(u_0 &lt;= u_1 &lt;= u_2 &lt;= ... &lt;= u_m\)</span></p></li>
</ul>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><p>B-spline curve using variable, u in [u_0, u_m]</p>
</dd>
<dt class="field-odd">Return type</dt>
<dd class="field-odd"><p>function</p>
</dd>
</dl>
<div class="math notranslate nohighlight">
\[\mathbf{C}_p(u) = \sum\limits_{i=0}^{n} N_{i,p}(u) \mathbf{P}_i\]</div>
<ul class="simple">
<li><p>open B-spline curves</p>
<ul>
<li><p>the curve will not touch the first and last legs of the
control polyline</p></li>
<li><p>the knot vector does not have any particular structure</p></li>
<li><p>for degree p, intervals [u_0, u_p) and [u_{n-p}, u_n) will not
have “full support” of basis functions and are ignored when a
B-spline curve is open. For open B-spline curves, the domain
is inteval [u_p, u_{n-p}]</p></li>
</ul>
</li>
<li><p>clamped B-spline curves, nonperiodic B-spline curves</p>
<ul>
<li><p>the curve is tangent to the first and the last legs just like
a Bézier curve</p></li>
<li><p>the first knot and the last knot must be repeated p+1 times
(i. e., of multiplicity p+1)</p></li>
</ul>
</li>
<li><p>closed B-spline curves</p>
<ul>
<li><p>the start and the end of the generated curve join together
forming a closed loop</p></li>
<li><p>repeating some knots and control points # TODO: which?</p></li>
</ul>
</li>
<li><p>uniform B-spline curves</p>
<ul>
<li><p>internal knots are equally spaced</p></li>
</ul>
</li>
<li><p>Bézier curves</p>
<ul>
<li><p>a B-spline with no internal knots.</p></li>
</ul>
</li>
</ul>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<p><a class="reference internal" href="#pylib.function.b_spline_knots" title="pylib.function.b_spline_knots"><code class="xref py py-meth docutils literal notranslate"><span class="pre">b_spline_knots()</span></code></a></p>
</div>
</dd></dl>
<dl class="function">
<dt id="pylib.function.b_spline_knots">
<code class="sig-name descname">b_spline_knots</code><span class="sig-paren">(</span><em class="sig-param">control_point_spans</em>, <em class="sig-param">degree=3</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/pylib/function.html#b_spline_knots"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.function.b_spline_knots" title="Permalink to this definition"></a></dt>
<dd><p>B-spline knots to generate a clamped uniform B-spline curve
of degree p (order + 1).</p>
<p>The internal knots are equally spaced (uniform B-spline curve)</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>control_point_spans</strong> (<em>int</em>) – number of control points + 1</p></li>
<li><p><strong>degree</strong> (<em>int</em>) – degree of B-spline basis functions (default = 3)</p></li>
</ul>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><p>knot vector</p>
</dd>
<dt class="field-odd">Return type</dt>
<dd class="field-odd"><p>tuple</p>
</dd>
</dl>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<p><a class="reference internal" href="#pylib.function.b_spline_curve_with_knots" title="pylib.function.b_spline_curve_with_knots"><code class="xref py py-meth docutils literal notranslate"><span class="pre">b_spline_curve_with_knots()</span></code></a></p>
</div>
</dd></dl>
<dl class="function">
<dt id="pylib.function.circle">
<code class="sig-name descname">circle</code><span class="sig-paren">(</span><em class="sig-param">r</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/pylib/function.html#circle"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.function.circle" title="Permalink to this definition"></a></dt>
<dd><p>Circle</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><p><strong>r</strong> (<em>float</em>) – radius of the circle</p>
</dd>
<dt class="field-even">Results</dt>
<dd class="field-even"><p>functions for x of theta and y of theta and the interval</p>
</dd>
<dt class="field-odd">Return type</dt>
<dd class="field-odd"><p>tuple</p>
</dd>
</dl>
<div class="math notranslate nohighlight">
\[\begin{split}x(\theta) = r\cos\theta \\
y(\theta) = r\sin\theta \\
\theta = \left[0, 2\pi\right]\end{split}\]</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="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>
</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">circle</span><span class="p">(</span><span class="mi">20</span><span class="p">)[:</span><span class="mi">2</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">_</span> <span class="o">=</span> <span class="n">circle</span><span class="p">(</span><span class="mi">20</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">interval</span> <span class="o">=</span> <span class="n">circle</span><span class="p">(</span><span class="mi">20</span><span class="p">)</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="pylib.function.cosine_wave">
<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/pylib/function.html#cosine_wave"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.function.cosine_wave" title="Permalink to this definition"></a></dt>
<dd><p>A cosine wave is said to be sinusoidal, because,
<span class="math notranslate nohighlight">\(\cos(x) = \sin(x + \pi/2)\)</span>, which is also a sine wave with a
phase-shift of π/2 radians. Because of this head start, it is often
said that the cosine function leads the sine function or the sine
lags the cosine.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>A</strong> (<em>float</em><em> or </em><em>int</em>) – amplitude</p></li>
<li><p><strong>k</strong> (<em>float</em><em> or </em><em>int</em>) – (angular) wave number</p></li>
<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> (<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>
<dt class="field-odd">Return type</dt>
<dd class="field-odd"><p>function</p>
</dd>
</dl>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<p><a class="reference internal" href="#pylib.function.sine_wave" title="pylib.function.sine_wave"><code class="xref py py-meth docutils literal notranslate"><span class="pre">sine_wave()</span></code></a></p>
</div>
</dd></dl>
<dl class="function">
<dt id="pylib.function.ellipse">
<code class="sig-name descname">ellipse</code><span class="sig-paren">(</span><em class="sig-param">a</em>, <em class="sig-param">b</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/pylib/function.html#ellipse"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.function.ellipse" title="Permalink to this definition"></a></dt>
<dd><p>Ellipse</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>a</strong> (<em>float</em>) – semi-major axis</p></li>
<li><p><strong>b</strong> (<em>float</em>) – semi-minor axis</p></li>
</ul>
</dd>
<dt class="field-even">Results</dt>
<dd class="field-even"><p>functions for x of theta and y of theta and the interval</p>
</dd>
<dt class="field-odd">Return type</dt>
<dd class="field-odd"><p>tuple</p>
</dd>
</dl>
<div class="math notranslate nohighlight">
\[\begin{split}x(\theta) = a\cos\theta \\
y(\theta) = b\sin\theta \\
\theta = \left[0, 2\pi\right]\end{split}\]</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="n">b</span> <span class="p">:</span> <span class="o">*</span>
<span class="o">*</span> <span class="p">:</span><span class="o">......*</span>
<span class="o">*</span> <span class="n">a</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">ellipse</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">5</span><span class="p">)[:</span><span class="mi">2</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">_</span> <span class="o">=</span> <span class="n">ellipse</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">5</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">interval</span> <span class="o">=</span> <span class="n">ellipse</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="pylib.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/pylib/function.html#epitrochoid"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.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 and the interval</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="p">,</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">2</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">_</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="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">interval</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="pylib.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/pylib/function.html#hypotrochoid"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.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 and the interval</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="p">,</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">2</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">_</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="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">interval</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="pylib.mathematics.html#pylib.mathematics.lcm" title="pylib.mathematics.lcm"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pylib.mathematics.lcm()</span></code></a></p>
</div>
</dd></dl>
<dl class="function">
<dt id="pylib.function.sample_half_open">
<code class="sig-name descname">sample_half_open</code><span class="sig-paren">(</span><em class="sig-param">f</em>, <em class="sig-param">a</em>, <em class="sig-param">b</em>, <em class="sig-param">n=50</em>, <em class="sig-param">endpoint_epsilon=1e-07</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/pylib/function.html#sample_half_open"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.function.sample_half_open" title="Permalink to this definition"></a></dt>
<dd></dd></dl>
<dl class="function">
<dt id="pylib.function.sample_half_open_seq">
<code class="sig-name descname">sample_half_open_seq</code><span class="sig-paren">(</span><em class="sig-param">f</em>, <em class="sig-param">x</em>, <em class="sig-param">endpoint_epsilon=1e-07</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/pylib/function.html#sample_half_open_seq"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.function.sample_half_open_seq" title="Permalink to this definition"></a></dt>
<dd></dd></dl>
<dl class="function">
<dt id="pylib.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/pylib/function.html#sine_wave"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.function.sine_wave" title="Permalink to this definition"></a></dt>
<dd><p>A sine wave or sinusoid is a mathematical curve that describes a
smooth periodic oscillation.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>A</strong> (<em>float</em><em> or </em><em>int</em>) – amplitude</p></li>
<li><p><strong>k</strong> (<em>float</em><em> or </em><em>int</em>) – (angular) wave number</p></li>
<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> (<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>
<dt class="field-odd">Return type</dt>
<dd class="field-odd"><p>function</p>
</dd>
</dl>
<p>In general, the function is:</p>
<div class="math notranslate nohighlight">
\[\begin{split}y(x,t) = A\sin(kx + \omega t + \varphi) + D \\
y(x,t) = A\sin(kx + 2\pi f t + \varphi) + D\end{split}\]</div>
<p>where:</p>
<blockquote>
<div><ul class="simple">
<li><p>A, amplitude, the peak deviation of the function from zero.</p></li>
<li><p>f, ordinary frequency, the number of oscillations (cycles) that
occur each second of time.</p></li>
<li><p>ω = 2πf, angular frequency, the rate of change of the function
argument in units of radians per second. If ω &lt; 0 the wave is
moving to the right, if ω &gt; 0 the wave is moving to the left.</p></li>
<li><p>φ, phase, specifies (in radians) where in its cycle the
oscillation is at t = 0.</p></li>
<li><p>x, spatial variable that represents the position on the
dimension on which the wave propagates.</p></li>
<li><p>k, characteristic parameter called wave number (or angular wave
number), which represents the proportionality between the
angular frequency ω and the linear speed (speed of propagation)
ν.</p></li>
<li><p>D, non-zero center amplitude.</p></li>
</ul>
</div></blockquote>
<p>The wavenumber is related to the angular frequency by:</p>
<div class="math notranslate nohighlight">
\[k={\omega \over v}={2\pi f \over v}={2\pi \over \lambda }\]</div>
<p>where λ (lambda) is the wavelength, f is the frequency, and v is the
linear speed.</p>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<p><a class="reference internal" href="#pylib.function.cosine_wave" title="pylib.function.cosine_wave"><code class="xref py py-meth docutils literal notranslate"><span class="pre">cosine_wave()</span></code></a></p>
</div>
</dd></dl>
<dl class="function">
<dt id="pylib.function.to_str">
<code class="sig-name descname">to_str</code><span class="sig-paren">(</span><em class="sig-param">f</em>, <em class="sig-param">x=None</em>, <em class="sig-param">x_0=0</em>, <em class="sig-param">x_1=1</em>, <em class="sig-param">t=None</em>, <em class="sig-param">h=10</em>, <em class="sig-param">w=80</em>, <em class="sig-param">density=1</em>, <em class="sig-param">char_set='line'</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/pylib/function.html#to_str"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.function.to_str" title="Permalink to this definition"></a></dt>
<dd><p>Represent functions as string frame with a specific character set.
which are normed to the range of [0, 1] to</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>f</strong> (<em>function</em><em> or </em><em>list</em>) – function or list of functions normed to the range of [0, 1]</p></li>
<li><p><strong>h</strong> (<em>int</em>) – number of chars in vertical direction</p></li>
<li><p><strong>w</strong> (<em>int</em>) – number of chars in horizontal direction</p></li>
<li><p><strong>char_set</strong> (<em>str</em>) – either “braille” or “block”. “braille” uses Unicode
Characters in the Braille Patterns Block (first index U+2800, last
index U+28FF <a class="reference internal" href="#cudb" id="id1"><span>[CUDB]</span></a>) and the “block” uses part of the Unicode
Characters in the Block Elements Block (fisrt index U+2580, last
index U+259F <a class="reference internal" href="#cudb" id="id2"><span>[CUDB]</span></a>). Alias for braille is line and alias for
block is histogram</p></li>
</ul>
</dd>
</dl>
<dl class="simple">
<dt>Usage:</dt><dd><ul class="simple">
<li><p>case dependent arguments</p>
<ul>
<li><p>1 quasi line plot (braille characters)
f = lambda function (lambda x, t=0: …)
x_1 = max x value
x_0 = min x value
t = time (animation)</p></li>
<li><p>2 histogram (block characters)
f = lambda function (lambda x, t=0: …)
x_1 = max x value
x_0 = min x value
t = time (animation)
chart=”histogram”</p></li>
</ul>
</li>
<li><p>general arguments
w = window width in number of chars
density = number of data point per pixel (for line chart higher density makes thicker lines)
chart = either “line” or “histogram”</p></li>
</ul>
</dd>
</dl>
<p class="rubric">Braille Patterns Block</p>
<ul class="simple">
<li><p>Dots or pixels per character in vertical direction: 6</p></li>
<li><p>Dots or pixels per character in horizontal direction: 2</p></li>
</ul>
<p>First dot (bottom left) is the zero (0) position of the
function. So, a function value of zero does not mean to have zero
dots but one.
Example of 3 columns and 3 rows (upside down view of ‘normal’ braille/drawille position)</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span> | ^ y axis
| |
| ,_____,,_____,,_____,
7 + | * *|| * *|| * *|
6 + | * *|| * *|| * *|
5 + | * *|| * *|| * *|
4 + | * *|| * *|| * *|
| ;=====;;=====;;=====;
3 + | * *|| * *|| * *|
2 + | * *|| * *|| * *|
1 + | * *|| * *|| * *|
0 + - | * *|| * *|| * *| ---&gt; x axis
| ;=====;;=====;;=====;
-1 + | * *|| * *|| * *|
-2 + | * *|| * *|| * *|
-3 + | * *|| * *|| * *|
-4 + | * *|| * *|| * *|
| `````````````````````
| |
`-----+--+---+--+---+--+-------------
-2 -1 0 1 2 3
</pre></div>
</div>
<p class="rubric">Block Elements Block</p>
<ul class="simple">
<li><p>Dots or pixels per character in vertical direction: 8</p></li>
<li><p>Dots or pixels per character in horizontal direction: 1</p></li>
</ul>
<p>Example of 3 columns and 3 rows</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span> | ^ y axis
| |
| ,_____,,_____,,_____,
15 + | --- || --- || --- |
14 + | --- || --- || --- |
13 + | --- || --- || --- |
12 + | --- || --- || --- |
11 + | --- || --- || --- |
10 + | --- || --- || --- |
9 + | --- || --- || --- |
8 + | --- || --- || --- |
| ;=====;;=====;;=====;
7 + | --- || --- || --- |
6 + | --- || --- || --- |
5 + | --- || --- || --- |
4 + | --- || --- || --- |
3 + | --- || --- || --- |
2 + | --- || --- || --- |
1 + | --- || --- || --- |
0 + - | --- || --- || --- | ---&gt; x axis
| ;=====;;=====;;=====;
-1 + | --- || --- || --- |
-2 + | --- || --- || --- |
-3 + | --- || --- || --- |
-4 + | --- || --- || --- |
-5 + | --- || --- || --- |
-6 + | --- || --- || --- |
-7 + | --- || --- || --- |
-8 + | --- || --- || --- |
| `````````````````````
| |
`------+------+------+---------------
-1 0 1
</pre></div>
</div>
<p class="rubric">References</p>
<dl class="citation">
<dt class="label" id="cudb"><span class="brackets">CUDB</span><span class="fn-backref">(<a href="#id1">1</a>,<a href="#id2">2</a>)</span></dt>
<dd><p><a class="reference external" href="ftp://ftp.unicode.org/Public/UNIDATA/Blocks.txt">Unicode Database - Blocks</a></p>
</dd>
</dl>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<p><a class="reference internal" href="#pylib.function.transformation" title="pylib.function.transformation"><code class="xref py py-meth docutils literal notranslate"><span class="pre">transformation()</span></code></a></p>
</div>
</dd></dl>
<dl class="function">
<dt id="pylib.function.transformation">
<code class="sig-name descname">transformation</code><span class="sig-paren">(</span><em class="sig-param">f</em>, <em class="sig-param">scale_vertical=1</em>, <em class="sig-param">scale_horizontal=1</em>, <em class="sig-param">shift_horizontal=0</em>, <em class="sig-param">shift_vertical=0</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/pylib/function.html#transformation"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.function.transformation" title="Permalink to this definition"></a></dt>
<dd><p>Transform functions.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>f</strong> (<em>function</em><em> or </em><em>list</em>) – function or list of functions</p></li>
<li><p><strong>scale_vertical</strong> – “a” scale factor in vertical direction (default
= 1)</p></li>
<li><p><strong>scale_horizontal</strong> – “b” scale factor in horizontal direction
(default = 1)</p></li>
<li><p><strong>shift_horizontal</strong> – “c” shift factor in horizontal direction
(default = 0)</p></li>
<li><p><strong>shift_vertical</strong> – “d” shift factor in vertical direction (default
= 0)</p></li>
</ul>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><p>transformed function or list of transformed functions</p>
</dd>
<dt class="field-odd">Return type</dt>
<dd class="field-odd"><p>function or list</p>
</dd>
</dl>
<div class="math notranslate nohighlight">
\[y = a \, f(b\,(x-c)) + d\]</div>
</dd></dl>
</div>
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