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  27 <a name="math_toolkit.quat_overview"></a><a class="link" href="quat_overview.html" title="Overview">Overview</a>
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  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 (&#945;,&#946;,&#947;,&#948;),
  55       which we can write in the form <span class="emphasis"><em><code class="literal">q = &#945; + &#946;i + &#947;j + &#948;k</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 &#8800; 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>
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