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  27 <a name="math_toolkit.oct_create"></a><a class="link" href="oct_create.html" title="Octonion Creation Functions">Octonion Creation Functions</a>
  28 </h2></div></div></div>
  29 <pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">spherical</span><span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">rho</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">phi1</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">phi2</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">phi3</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">phi4</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">phi5</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">phi6</span><span class="special">);</span>
  30 <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">multipolar</span><span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">rho1</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">theta1</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">rho2</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">theta2</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">rho3</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">theta3</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">rho4</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">theta4</span><span class="special">);</span>
  31 <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">cylindrical</span><span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">r</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">angle</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">h1</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">h2</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">h3</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">h4</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">h5</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">h6</span><span class="special">);</span>
  32 </pre>
  33 <p>
  34       These build octonions in a way similar to the way polar builds complex numbers,
  35       as there is no strict equivalent to polar coordinates for octonions.
  36     </p>
  37 <p>
  38       <code class="computeroutput"><span class="identifier">spherical</span></code> is a simple transposition
  39       of <code class="computeroutput"><span class="identifier">polar</span></code>, it takes as inputs
  40       a (positive) magnitude and a point on the hypersphere, given by three angles.
  41       The first of these, <span class="emphasis"><em>theta</em></span> has a natural range of -pi to
  42       +pi, and the other two have natural ranges of -pi/2 to +pi/2 (as is the case
  43       with the usual spherical coordinates in <span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span>).
  44       Due to the many symmetries and periodicities, nothing untoward happens if the
  45       magnitude is negative or the angles are outside their natural ranges. The expected
  46       degeneracies (a magnitude of zero ignores the angles settings...) do happen
  47       however.
  48     </p>
  49 <p>
  50       <code class="computeroutput"><span class="identifier">cylindrical</span></code> is likewise a simple
  51       transposition of the usual cylindrical coordinates in <span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span>,
  52       which in turn is another derivative of planar polar coordinates. The first
  53       two inputs are the polar coordinates of the first <span class="emphasis"><em><span class="bold"><strong>C</strong></span></em></span>
  54       component of the octonion. The third and fourth inputs are placed into the
  55       third and fourth <span class="emphasis"><em><span class="bold"><strong>R</strong></span></em></span> components
  56       of the octonion, respectively.
  57     </p>
  58 <p>
  59       <code class="computeroutput"><span class="identifier">multipolar</span></code> is yet another simple
  60       generalization of polar coordinates. This time, both <span class="emphasis"><em><span class="bold"><strong>C</strong></span></em></span>
  61       components of the octonion are given in polar coordinates.
  62     </p>
  63 <p>
  64       In this version of our implementation of octonions, there is no analogue of
  65       the complex value operation arg as the situation is somewhat more complicated.
  66     </p>
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  70 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
  71       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
  72       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
  73       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
  74       Daryle Walker and Xiaogang Zhang<p>
  75         Distributed under the Boost Software License, Version 1.0. (See accompanying
  76         file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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