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98 lines
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<link rel="up" href="../quaternions.html" title="Chapter 16. Quaternions">
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<div class="section">
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="math_toolkit.quat_overview"></a><a class="link" href="quat_overview.html" title="Overview">Overview</a>
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</h2></div></div></div>
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<p>
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Quaternions are a relative of complex numbers.
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</p>
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<p>
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Quaternions are in fact part of a small hierarchy of structures built upon
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the real numbers, which comprise only the set of real numbers (traditionally
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named <span class="emphasis"><em><span class="bold"><strong>R</strong></span></em></span>), the set of
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complex numbers (traditionally named <span class="emphasis"><em><span class="bold"><strong>C</strong></span></em></span>),
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the set of quaternions (traditionally named <span class="emphasis"><em><span class="bold"><strong>H</strong></span></em></span>)
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and the set of octonions (traditionally named <span class="emphasis"><em><span class="bold"><strong>O</strong></span></em></span>),
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which possess interesting mathematical properties (chief among which is the
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fact that they are <span class="emphasis"><em>division algebras</em></span>, <span class="emphasis"><em>i.e.</em></span>
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where the following property is true: if <span class="emphasis"><em><code class="literal">y</code></em></span>
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is an element of that algebra and is <span class="bold"><strong>not equal to zero</strong></span>,
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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>
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and <span class="emphasis"><em><code class="literal">x'</code></em></span> denote elements of that algebra,
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implies that <span class="emphasis"><em><code class="literal">x = x'</code></em></span>). Each member of
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the hierarchy is a super-set of the former.
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</p>
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<p>
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One of the most important aspects of quaternions is that they provide an efficient
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way to parameterize rotations in <span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span>
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(the usual three-dimensional space) and <span class="emphasis"><em><span class="bold"><strong>R<sup>4</sup></strong></span></em></span>.
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</p>
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<p>
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In practical terms, a quaternion is simply a quadruple of real numbers (α,β,γ,δ),
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which we can write in the form <span class="emphasis"><em><code class="literal">q = α + βi + γj + δk</code></em></span>,
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where <span class="emphasis"><em><code class="literal">i</code></em></span> is the same object as for complex
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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>
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are distinct objects which play essentially the same kind of role as <span class="emphasis"><em><code class="literal">i</code></em></span>.
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</p>
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<p>
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An addition and a multiplication is defined on the set of quaternions, which
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generalize their real and complex counterparts. The main novelty here is that
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<span class="bold"><strong>the multiplication is not commutative</strong></span> (i.e.
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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>
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such that <span class="emphasis"><em><code class="literal">xy ≠ yx</code></em></span>). A good mnemotechnical
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way of remembering things is by using the formula <span class="emphasis"><em><code class="literal">i*i =
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j*j = k*k = -1</code></em></span>.
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</p>
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<p>
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Quaternions (and their kin) are described in far more details in this other
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<a href="../../quaternion/TQE.pdf" target="_top">document</a> (with <a href="../../quaternion/TQE_EA.pdf" target="_top">errata
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and addenda</a>).
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</p>
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<p>
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Some traditional constructs, such as the exponential, carry over without too
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much change into the realms of quaternions, but other, such as taking a square
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root, do not.
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</p>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
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Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
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Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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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