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| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="boost_quaternions.quaternions.overview"></a><a class="link" href="overview.html" title="Overview">Overview</a> |
| </h3></div></div></div> |
| <p> |
| Quaternions are a relative of complex numbers. |
| </p> |
| <p> |
| Quaternions are in fact part of a small hierarchy of structures built upon |
| the real numbers, which comprise only the set of real numbers (traditionally |
| named <span class="emphasis"><em><span class="bold"><strong>R</strong></span></em></span>), the set of |
| complex numbers (traditionally named <span class="emphasis"><em><span class="bold"><strong>C</strong></span></em></span>), |
| the set of quaternions (traditionally named <span class="emphasis"><em><span class="bold"><strong>H</strong></span></em></span>) |
| and the set of octonions (traditionally named <span class="emphasis"><em><span class="bold"><strong>O</strong></span></em></span>), |
| which possess interesting mathematical properties (chief among which is the |
| fact that they are <span class="emphasis"><em>division algebras</em></span>, <span class="emphasis"><em>i.e.</em></span> |
| where the following property is true: if <span class="emphasis"><em><code class="literal">y</code></em></span> |
| is an element of that algebra and is <span class="bold"><strong>not equal to zero</strong></span>, |
| 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> |
| and <span class="emphasis"><em><code class="literal">x'</code></em></span> denote elements of that algebra, |
| implies that <span class="emphasis"><em><code class="literal">x = x'</code></em></span>). Each member |
| of the hierarchy is a super-set of the former. |
| </p> |
| <p> |
| One of the most important aspects of quaternions is that they provide an |
| efficient way to parameterize rotations in <span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span> |
| (the usual three-dimensional space) and <span class="emphasis"><em><span class="bold"><strong>R<sup>4</sup></strong></span></em></span>. |
| </p> |
| <p> |
| In practical terms, a quaternion is simply a quadruple of real numbers (α,β,γ,δ), |
| which we can write in the form <span class="emphasis"><em><code class="literal">q = α + βi + γj + δk</code></em></span>, |
| where <span class="emphasis"><em><code class="literal">i</code></em></span> is the same object as for |
| complex 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> |
| are distinct objects which play essentially the same kind of role as <span class="emphasis"><em><code class="literal">i</code></em></span>. |
| </p> |
| <p> |
| An addition and a multiplication is defined on the set of quaternions, which |
| generalize their real and complex counterparts. The main novelty here is |
| that <span class="bold"><strong>the multiplication is not commutative</strong></span> |
| (i.e. 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> such that <span class="emphasis"><em><code class="literal">xy |
| ≠ yx</code></em></span>). A good mnemotechnical way of remembering things |
| is by using the formula <span class="emphasis"><em><code class="literal">i*i = j*j = k*k = -1</code></em></span>. |
| </p> |
| <p> |
| Quaternions (and their kin) are described in far more details in this other |
| <a href="../../../../../quaternion/TQE.pdf" target="_top">document</a> (with <a href="../../../../../quaternion/TQE_EA.pdf" target="_top">errata |
| and addenda</a>). |
| </p> |
| <p> |
| Some traditional constructs, such as the exponential, carry over without |
| too much change into the realms of quaternions, but other, such as taking |
| a square root, do not. |
| </p> |
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| <td align="right"><div class="copyright-footer">Copyright © 2001 -2003 Hubert Holin<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| 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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