mirror of
https://github.com/saitohirga/WSJT-X.git
synced 2025-10-25 10:00:23 -04:00
97 lines
6.8 KiB
HTML
97 lines
6.8 KiB
HTML
<html>
|
|
<head>
|
|
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
|
<title>Overview</title>
|
|
<link rel="stylesheet" href="../math.css" type="text/css">
|
|
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
|
|
<link rel="home" href="../index.html" title="Math Toolkit 2.5.1">
|
|
<link rel="up" href="../quaternions.html" title="Chapter 9. Quaternions">
|
|
<link rel="prev" href="../quaternions.html" title="Chapter 9. Quaternions">
|
|
<link rel="next" href="quat_header.html" title="Header File">
|
|
</head>
|
|
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
|
|
<table cellpadding="2" width="100%"><tr>
|
|
<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../boost.png"></td>
|
|
<td align="center"><a href="../../../../../index.html">Home</a></td>
|
|
<td align="center"><a href="../../../../../libs/libraries.htm">Libraries</a></td>
|
|
<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
|
|
<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
|
|
<td align="center"><a href="../../../../../more/index.htm">More</a></td>
|
|
</tr></table>
|
|
<hr>
|
|
<div class="spirit-nav">
|
|
<a accesskey="p" href="../quaternions.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../quaternions.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="quat_header.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
|
|
</div>
|
|
<div class="section">
|
|
<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>
|
|
</h2></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>
|
|
</div>
|
|
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
|
<td align="left"></td>
|
|
<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal,
|
|
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
|
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
|
|
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<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>)
|
|
</p>
|
|
</div></td>
|
|
</tr></table>
|
|
<hr>
|
|
<div class="spirit-nav">
|
|
<a accesskey="p" href="../quaternions.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../quaternions.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="quat_header.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
|
|
</div>
|
|
</body>
|
|
</html>
|