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  27 <a name="math_toolkit.create"></a><a class="link" href="create.html" title="Quaternion Creation Functions">Quaternion 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">quaternion</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>
  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">quaternion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">semipolar</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">alpha</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">theta2</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">quaternion</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>
  32 <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">quaternion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">cylindrospherical</span><span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">t</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">radius</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">longitude</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">latitude</span><span class="special">);</span>
  33 <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">quaternion</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>
  34 </pre>
  35 <p>
  36       These build quaternions in a way similar to the way polar builds complex numbers,
  37       as there is no strict equivalent to polar coordinates for quaternions.
  38     </p>
  39 <p>
  40       <a name="math_quaternions.creation_spherical"></a><code class="computeroutput"><span class="identifier">spherical</span></code>
  41       is a simple transposition of <code class="computeroutput"><span class="identifier">polar</span></code>,
  42       it takes as inputs a (positive) magnitude and a point on the hypersphere, given
  43       by three angles. The first of these, <code class="computeroutput"><span class="identifier">theta</span></code>
  44       has a natural range of <code class="computeroutput"><span class="special">-</span><span class="identifier">pi</span></code>
  45       to <code class="computeroutput"><span class="special">+</span><span class="identifier">pi</span></code>,
  46       and the other two have natural ranges of <code class="computeroutput"><span class="special">-</span><span class="identifier">pi</span><span class="special">/</span><span class="number">2</span></code>
  47       to <code class="computeroutput"><span class="special">+</span><span class="identifier">pi</span><span class="special">/</span><span class="number">2</span></code> (as is the
  48       case with the usual spherical coordinates in <span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span>).
  49       Due to the many symmetries and periodicities, nothing untoward happens if the
  50       magnitude is negative or the angles are outside their natural ranges. The expected
  51       degeneracies (a magnitude of zero ignores the angles settings...) do happen
  52       however.
  53     </p>
  54 <p>
  55       <a name="math_quaternions.creation_cylindrical"></a><code class="computeroutput"><span class="identifier">cylindrical</span></code>
  56       is likewise a simple transposition of the usual cylindrical coordinates in
  57       <span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span>, which in turn is another
  58       derivative of planar polar coordinates. The first two inputs are the polar
  59       coordinates of the first <span class="emphasis"><em><span class="bold"><strong>C</strong></span></em></span>
  60       component of the quaternion. The third and fourth inputs are placed into the
  61       third and fourth <span class="emphasis"><em><span class="bold"><strong>R</strong></span></em></span> components
  62       of the quaternion, respectively.
  63     </p>
  64 <p>
  65       <a name="math_quaternions.creation_multipolar"></a><code class="computeroutput"><span class="identifier">multipolar</span></code>
  66       is yet another simple generalization of polar coordinates. This time, both
  67       <span class="emphasis"><em><span class="bold"><strong>C</strong></span></em></span> components of the quaternion
  68       are given in polar coordinates.
  69     </p>
  70 <p>
  71       <a name="math_quaternions.creation_cylindrospherical"></a><code class="computeroutput"><span class="identifier">cylindrospherical</span></code>
  72       is specific to quaternions. It is often interesting to consider <span class="emphasis"><em><span class="bold"><strong>H</strong></span></em></span> as the cartesian product of <span class="emphasis"><em><span class="bold"><strong>R</strong></span></em></span> by <span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span>
  73       (the quaternionic multiplication as then a special form, as given here). This
  74       function therefore builds a quaternion from this representation, with the
  75       <span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span> component given in usual
  76       <span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span> spherical coordinates.
  77     </p>
  78 <p>
  79       <a name="math_quaternions.creation_semipolar"></a><code class="computeroutput"><span class="identifier">semipolar</span></code>
  80       is another generator which is specific to quaternions. It takes as a first
  81       input the magnitude of the quaternion, as a second input an angle in the range
  82       <code class="computeroutput"><span class="number">0</span></code> to <code class="computeroutput"><span class="special">+</span><span class="identifier">pi</span><span class="special">/</span><span class="number">2</span></code>
  83       such that magnitudes of the first two <span class="emphasis"><em><span class="bold"><strong>C</strong></span></em></span>
  84       components of the quaternion are the product of the first input and the sine
  85       and cosine of this angle, respectively, and finally as third and fourth inputs
  86       angles in the range <code class="computeroutput"><span class="special">-</span><span class="identifier">pi</span><span class="special">/</span><span class="number">2</span></code> to <code class="computeroutput"><span class="special">+</span><span class="identifier">pi</span><span class="special">/</span><span class="number">2</span></code> which represent the arguments of the first
  87       and second <span class="emphasis"><em><span class="bold"><strong>C</strong></span></em></span> components
  88       of the quaternion, respectively. As usual, nothing untoward happens if what
  89       should be magnitudes are negative numbers or angles are out of their natural
  90       ranges, as symmetries and periodicities kick in.
  91     </p>
  92 <p>
  93       In this version of our implementation of quaternions, there is no analogue
  94       of the complex value operation <code class="computeroutput"><span class="identifier">arg</span></code>
  95       as the situation is somewhat more complicated. Unit quaternions are linked
  96       both to rotations in <span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span>
  97       and in <span class="emphasis"><em><span class="bold"><strong>R<sup>4</sup></strong></span></em></span>, and the correspondences
  98       are not too complicated, but there is currently a lack of standard (de facto
  99       or de jure) matrix library with which the conversions could work. This should
 100       be remedied in a further revision. In the mean time, an example of how this
 101       could be done is presented here for <a href="../../../example/HSO3.hpp" target="_top"><span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span></a>, and here for <a href="../../../example/HSO4.hpp" target="_top"><span class="emphasis"><em><span class="bold"><strong>R<sup>4</sup></strong></span></em></span></a> (<a href="../../../example/HSO3SO4.cpp" target="_top">example
 102       test file</a>).
 103     </p>
 104 </div>
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 106 <td align="left"></td>
 107 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 108       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 109       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 110       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 111       Daryle Walker and Xiaogang Zhang<p>
 112         Distributed under the Boost Software License, Version 1.0. (See accompanying
 113         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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