<html> <head> <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Quaternion Creation Functions</title> <link rel="stylesheet" href="../../../../../../../doc/src/boostbook.css" type="text/css"> <meta name="generator" content="DocBook XSL Stylesheets V1.75.2"> <link rel="home" href="../../index.html" title="Boost.Quaternions"> <link rel="up" href="../quaternions.html" title="Quaternions"> <link rel="prev" href="value_op.html" title="Quaternion Value Operations"> <link rel="next" href="trans.html" title="Quaternion Transcendentals"> </head> <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> <table cellpadding="2" width="100%"><tr> <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../boost.png"></td> <td align="center"><a href="../../../../../../../index.html">Home</a></td> <td align="center"><a href="../../../../../../../libs/libraries.htm">Libraries</a></td> <td align="center"><a 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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> </div></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="value_op.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../quaternions.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="trans.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> </body> </html>