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24 | </div> | |
25 | <div class="section"> | |
26 | <div class="titlepage"><div><div><h2 class="title" style="clear: both"> | |
27 | <a name="math_toolkit.quat_overview"></a><a class="link" href="quat_overview.html" title="Overview">Overview</a> | |
28 | </h2></div></div></div> | |
29 | <p> | |
30 | Quaternions are a relative of complex numbers. | |
31 | </p> | |
32 | <p> | |
33 | Quaternions are in fact part of a small hierarchy of structures built upon | |
34 | the real numbers, which comprise only the set of real numbers (traditionally | |
35 | named <span class="emphasis"><em><span class="bold"><strong>R</strong></span></em></span>), the set of | |
36 | complex numbers (traditionally named <span class="emphasis"><em><span class="bold"><strong>C</strong></span></em></span>), | |
37 | the set of quaternions (traditionally named <span class="emphasis"><em><span class="bold"><strong>H</strong></span></em></span>) | |
38 | and the set of octonions (traditionally named <span class="emphasis"><em><span class="bold"><strong>O</strong></span></em></span>), | |
39 | which possess interesting mathematical properties (chief among which is the | |
40 | fact that they are <span class="emphasis"><em>division algebras</em></span>, <span class="emphasis"><em>i.e.</em></span> | |
41 | where the following property is true: if <span class="emphasis"><em><code class="literal">y</code></em></span> | |
42 | is an element of that algebra and is <span class="bold"><strong>not equal to zero</strong></span>, | |
43 | then <span class="emphasis"><em><code class="literal">yx = yx'</code></em></span>, where <span class="emphasis"><em><code class="literal">x</code></em></span> | |
44 | and <span class="emphasis"><em><code class="literal">x'</code></em></span> denote elements of that algebra, | |
45 | implies that <span class="emphasis"><em><code class="literal">x = x'</code></em></span>). Each member of | |
46 | the hierarchy is a super-set of the former. | |
47 | </p> | |
48 | <p> | |
49 | One of the most important aspects of quaternions is that they provide an efficient | |
50 | way to parameterize rotations in <span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span> | |
51 | (the usual three-dimensional space) and <span class="emphasis"><em><span class="bold"><strong>R<sup>4</sup></strong></span></em></span>. | |
52 | </p> | |
53 | <p> | |
54 | In practical terms, a quaternion is simply a quadruple of real numbers (α,β,γ,δ), | |
55 | which we can write in the form <span class="emphasis"><em><code class="literal">q = α + βi + γj + δk</code></em></span>, | |
56 | where <span class="emphasis"><em><code class="literal">i</code></em></span> is the same object as for complex | |
57 | numbers, and <span class="emphasis"><em><code class="literal">j</code></em></span> and <span class="emphasis"><em><code class="literal">k</code></em></span> | |
58 | are distinct objects which play essentially the same kind of role as <span class="emphasis"><em><code class="literal">i</code></em></span>. | |
59 | </p> | |
60 | <p> | |
61 | An addition and a multiplication is defined on the set of quaternions, which | |
62 | generalize their real and complex counterparts. The main novelty here is that | |
63 | <span class="bold"><strong>the multiplication is not commutative</strong></span> (i.e. | |
64 | there are quaternions <span class="emphasis"><em><code class="literal">x</code></em></span> and <span class="emphasis"><em><code class="literal">y</code></em></span> | |
65 | such that <span class="emphasis"><em><code class="literal">xy ≠ yx</code></em></span>). A good mnemotechnical | |
66 | way of remembering things is by using the formula <span class="emphasis"><em><code class="literal">i*i = | |
67 | j*j = k*k = -1</code></em></span>. | |
68 | </p> | |
69 | <p> | |
70 | Quaternions (and their kin) are described in far more details in this other | |
71 | <a href="../../quaternion/TQE.pdf" target="_top">document</a> (with <a href="../../quaternion/TQE_EA.pdf" target="_top">errata | |
72 | and addenda</a>). | |
73 | </p> | |
74 | <p> | |
75 | Some traditional constructs, such as the exponential, carry over without too | |
76 | much change into the realms of quaternions, but other, such as taking a square | |
77 | root, do not. | |
78 | </p> | |
79 | </div> | |
80 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
81 | <td align="left"></td> | |
82 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
83 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
84 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
85 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
86 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
87 | 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>) | |
88 | </p> | |
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