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| <div class="titlepage"><div><div><h2 class="title" style="clear: both"> |
| <a name="math_toolkit.quat_overview"></a><a class="link" href="quat_overview.html" title="Overview">Overview</a> |
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| <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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