1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Overview</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="../quaternions.html" title="Chapter 16. Quaternions"> 10<link rel="next" href="quat_header.html" title="Header File"> 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="../quaternions.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="quat_header.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.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 © 2006-2019 Nikhar 83 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 84 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 85 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 86 Daryle Walker and Xiaogang Zhang<p> 87 Distributed under the Boost Software License, Version 1.0. (See accompanying 88 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>) 89 </p> 90</div></td> 91</tr></table> 92<hr> 93<div class="spirit-nav"> 94<a accesskey="p" href="../quaternions.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="quat_header.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 95</div> 96</body> 97</html> 98
