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<a name="math_toolkit.create"></a><a class="link" href="create.html" title="Quaternion Creation Functions">Quaternion Creation Functions</a>
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<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>
<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>
<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>
<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>
<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>
</pre>
<p>
      These build quaternions in a way similar to the way polar builds complex numbers,
      as there is no strict equivalent to polar coordinates for quaternions.
    </p>
<p>
      <a name="math_quaternions.creation_spherical"></a><code class="computeroutput"><span class="identifier">spherical</span></code>
      is a simple transposition of <code class="computeroutput"><span class="identifier">polar</span></code>,
      it takes as inputs a (positive) magnitude and a point on the hypersphere, given
      by three angles. The first of these, <code class="computeroutput"><span class="identifier">theta</span></code>
      has a natural range of <code class="computeroutput"><span class="special">-</span><span class="identifier">pi</span></code>
      to <code class="computeroutput"><span class="special">+</span><span class="identifier">pi</span></code>,
      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>
      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
      case with the usual spherical coordinates in <span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span>).
      Due to the many symmetries and periodicities, nothing untoward happens if the
      magnitude is negative or the angles are outside their natural ranges. The expected
      degeneracies (a magnitude of zero ignores the angles settings...) do happen
      however.
    </p>
<p>
      <a name="math_quaternions.creation_cylindrical"></a><code class="computeroutput"><span class="identifier">cylindrical</span></code>
      is likewise a simple transposition of the usual cylindrical coordinates in
      <span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span>, which in turn is another
      derivative of planar polar coordinates. The first two inputs are the polar
      coordinates of the first <span class="emphasis"><em><span class="bold"><strong>C</strong></span></em></span>
      component of the quaternion. The third and fourth inputs are placed into the
      third and fourth <span class="emphasis"><em><span class="bold"><strong>R</strong></span></em></span> components
      of the quaternion, respectively.
    </p>
<p>
      <a name="math_quaternions.creation_multipolar"></a><code class="computeroutput"><span class="identifier">multipolar</span></code>
      is yet another simple generalization of polar coordinates. This time, both
      <span class="emphasis"><em><span class="bold"><strong>C</strong></span></em></span> components of the quaternion
      are given in polar coordinates.
    </p>
<p>
      <a name="math_quaternions.creation_cylindrospherical"></a><code class="computeroutput"><span class="identifier">cylindrospherical</span></code>
      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>
      (the quaternionic multiplication as then a special form, as given here). This
      function therefore builds a quaternion from this representation, with the
      <span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span> component given in usual
      <span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span> spherical coordinates.
    </p>
<p>
      <a name="math_quaternions.creation_semipolar"></a><code class="computeroutput"><span class="identifier">semipolar</span></code>
      is another generator which is specific to quaternions. It takes as a first
      input the magnitude of the quaternion, as a second input an angle in the range
      <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>
      such that magnitudes of the first two <span class="emphasis"><em><span class="bold"><strong>C</strong></span></em></span>
      components of the quaternion are the product of the first input and the sine
      and cosine of this angle, respectively, and finally as third and fourth inputs
      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
      and second <span class="emphasis"><em><span class="bold"><strong>C</strong></span></em></span> components
      of the quaternion, respectively. As usual, nothing untoward happens if what
      should be magnitudes are negative numbers or angles are out of their natural
      ranges, as symmetries and periodicities kick in.
    </p>
<p>
      In this version of our implementation of quaternions, there is no analogue
      of the complex value operation <code class="computeroutput"><span class="identifier">arg</span></code>
      as the situation is somewhat more complicated. Unit quaternions are linked
      both to rotations in <span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span>
      and in <span class="emphasis"><em><span class="bold"><strong>R<sup>4</sup></strong></span></em></span>, and the correspondences
      are not too complicated, but there is currently a lack of standard (de facto
      or de jure) matrix library with which the conversions could work. This should
      be remedied in a further revision. In the mean time, an example of how this
      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
      test file</a>).
    </p>
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