1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Quaternion Creation Functions</title> 5<link rel="stylesheet" href="../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../quaternions.html" title="Chapter 16. Quaternions"> 9<link rel="prev" href="value_op.html" title="Quaternion Value Operations"> 10<link rel="next" href="trans.html" title="Quaternion Transcendentals"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<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> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h2 class="title" style="clear: both"> 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"><</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> 30<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> 31<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> 32<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> 33<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> 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> 105<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 106<td align="left"></td> 107<td align="right"><div class="copyright-footer">Copyright © 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 Råde, 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>) 114 </p> 115</div></td> 116</tr></table> 117<hr> 118<div class="spirit-nav"> 119<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> 120</div> 121</body> 122</html> 123
