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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"><</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 © 2006-2010, 2012-2014 Nikhar Agrawal, | |
108 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
109 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
110 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
111 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
112 | 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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