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| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="boost_quaternions.quaternions.create"></a><a class="link" href="create.html" title="Quaternion Creation Functions"> Quaternion Creation |
| Functions</a> |
| </h3></div></div></div> |
| <pre class="programlisting"><span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">></span> <span class="identifier">quaternion</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">spherical</span><span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">rho</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">phi1</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">phi2</span><span class="special">);</span> |
| <span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">></span> <span class="identifier">quaternion</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">semipolar</span><span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">rho</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">alpha</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">theta1</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">theta2</span><span class="special">);</span> |
| <span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">></span> <span class="identifier">quaternion</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">multipolar</span><span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">rho1</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">theta1</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">rho2</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">theta2</span><span class="special">);</span> |
| <span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">></span> <span class="identifier">quaternion</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">cylindrospherical</span><span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">t</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">radius</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">longitude</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">latitude</span><span class="special">);</span> |
| <span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">></span> <span class="identifier">quaternion</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">cylindrical</span><span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">r</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">angle</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">h1</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</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> |
| <a name="boost_quaternions.quaternions.creation_spherical"></a><p> |
| <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> |
| <a name="boost_quaternions.quaternions.creation_cylindrical"></a><p> |
| <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> |
| <a name="boost_quaternions.quaternions.creation_multipolar"></a><p> |
| <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> |
| <a name="boost_quaternions.quaternions.creation_cylindrospherical"></a><p> |
| <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> |
| <a name="boost_quaternions.quaternions.creation_semipolar"></a><p> |
| <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="../../../../../quaternion/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="../../../../../quaternion/HSO4.hpp" target="_top"><span class="emphasis"><em><span class="bold"><strong>R<sup>4</sup></strong></span></em></span></a> (<a href="../../../../../quaternion/HSO3SO4.cpp" target="_top">example |
| test file</a>). |
| </p> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2001 -2003 Hubert Holin<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| 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>) |
| </p> |
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