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30 Quaternions are a relative of complex numbers.
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.
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>.
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>.
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>.
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
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
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82 <td align=
"right"><div class=
"copyright-footer">Copyright
© 2006-
2010,
2012-
2014 Nikhar Agrawal,
83 Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
84 Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R
åde, Gautam Sewani,
85 Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang
<p>
86 Distributed under the Boost Software License, Version
1.0. (See accompanying
87 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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