mirror of
https://github.com/saitohirga/WSJT-X.git
synced 2025-10-30 20:40:28 -04:00
122 lines
15 KiB
HTML
122 lines
15 KiB
HTML
|
<html>
|
||
|
|
<head>
|
||
|
|
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||
|
|
<title>Quaternion Creation Functions</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="value_op.html" title="Quaternion Value Operations">
|
||
|
|
<link rel="next" href="trans.html" title="Quaternion Transcendentals">
|
||
|
|
</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="value_op.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="trans.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.create"></a><a class="link" href="create.html" title="Quaternion Creation Functions">Quaternion Creation Functions</a>
|
||
|
|
</h2></div></div></div>
|
||
|
|
<pre class="programlisting"><span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">></span> <span class="identifier">quaternion</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">spherical</span><span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">rho</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">phi1</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">phi2</span><span class="special">);</span>
|
||
|
|
<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">></span> <span class="identifier">quaternion</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">semipolar</span><span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">rho</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">alpha</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">theta1</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">theta2</span><span class="special">);</span>
|
||
|
|
<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">></span> <span class="identifier">quaternion</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">multipolar</span><span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">rho1</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">theta1</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">rho2</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">theta2</span><span class="special">);</span>
|
||
|
|
<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">></span> <span class="identifier">quaternion</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">cylindrospherical</span><span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">t</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">radius</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">longitude</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">latitude</span><span class="special">);</span>
|
||
|
|
<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">></span> <span class="identifier">quaternion</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">cylindrical</span><span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">r</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">angle</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">h1</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">h2</span><span class="special">);</span>
|
||
|
|
</pre>
|
||
|
|
<p>
|
||
|
|
These build quaternions in a way similar to the way polar builds complex numbers,
|
||
|
|
as there is no strict equivalent to polar coordinates for quaternions.
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<a name="math_quaternions.creation_spherical"></a><code class="computeroutput"><span class="identifier">spherical</span></code>
|
||
|
|
is a simple transposition of <code class="computeroutput"><span class="identifier">polar</span></code>,
|
||
|
|
it takes as inputs a (positive) magnitude and a point on the hypersphere, given
|
||
|
|
by three angles. The first of these, <code class="computeroutput"><span class="identifier">theta</span></code>
|
||
|
|
has a natural range of <code class="computeroutput"><span class="special">-</span><span class="identifier">pi</span></code>
|
||
|
|
to <code class="computeroutput"><span class="special">+</span><span class="identifier">pi</span></code>,
|
||
|
|
and the other two have natural ranges of <code class="computeroutput"><span class="special">-</span><span class="identifier">pi</span><span class="special">/</span><span class="number">2</span></code>
|
||
|
|
to <code class="computeroutput"><span class="special">+</span><span class="identifier">pi</span><span class="special">/</span><span class="number">2</span></code> (as is the
|
||
|
|
case with the usual spherical coordinates in <span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span>).
|
||
|
|
Due to the many symmetries and periodicities, nothing untoward happens if the
|
||
|
|
magnitude is negative or the angles are outside their natural ranges. The expected
|
||
|
|
degeneracies (a magnitude of zero ignores the angles settings...) do happen
|
||
|
|
however.
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<a name="math_quaternions.creation_cylindrical"></a><code class="computeroutput"><span class="identifier">cylindrical</span></code>
|
||
|
|
is likewise a simple transposition of the usual cylindrical coordinates in
|
||
|
|
<span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span>, which in turn is another
|
||
|
|
derivative of planar polar coordinates. The first two inputs are the polar
|
||
|
|
coordinates of the first <span class="emphasis"><em><span class="bold"><strong>C</strong></span></em></span>
|
||
|
|
component of the quaternion. The third and fourth inputs are placed into the
|
||
|
|
third and fourth <span class="emphasis"><em><span class="bold"><strong>R</strong></span></em></span> components
|
||
|
|
of the quaternion, respectively.
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<a name="math_quaternions.creation_multipolar"></a><code class="computeroutput"><span class="identifier">multipolar</span></code>
|
||
|
|
is yet another simple generalization of polar coordinates. This time, both
|
||
|
|
<span class="emphasis"><em><span class="bold"><strong>C</strong></span></em></span> components of the quaternion
|
||
|
|
are given in polar coordinates.
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<a name="math_quaternions.creation_cylindrospherical"></a><code class="computeroutput"><span class="identifier">cylindrospherical</span></code>
|
||
|
|
is specific to quaternions. It is often interesting to consider <span class="emphasis"><em><span class="bold"><strong>H</strong></span></em></span> as the cartesian product of <span class="emphasis"><em><span class="bold"><strong>R</strong></span></em></span> by <span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span>
|
||
|
|
(the quaternionic multiplication as then a special form, as given here). This
|
||
|
|
function therefore builds a quaternion from this representation, with the
|
||
|
|
<span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span> component given in usual
|
||
|
|
<span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span> spherical coordinates.
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<a name="math_quaternions.creation_semipolar"></a><code class="computeroutput"><span class="identifier">semipolar</span></code>
|
||
|
|
is another generator which is specific to quaternions. It takes as a first
|
||
|
|
input the magnitude of the quaternion, as a second input an angle in the range
|
||
|
|
<code class="computeroutput"><span class="number">0</span></code> to <code class="computeroutput"><span class="special">+</span><span class="identifier">pi</span><span class="special">/</span><span class="number">2</span></code>
|
||
|
|
such that magnitudes of the first two <span class="emphasis"><em><span class="bold"><strong>C</strong></span></em></span>
|
||
|
|
components of the quaternion are the product of the first input and the sine
|
||
|
|
and cosine of this angle, respectively, and finally as third and fourth inputs
|
||
|
|
angles in the range <code class="computeroutput"><span class="special">-</span><span class="identifier">pi</span><span class="special">/</span><span class="number">2</span></code> to <code class="computeroutput"><span class="special">+</span><span class="identifier">pi</span><span class="special">/</span><span class="number">2</span></code> which represent the arguments of the first
|
||
|
|
and second <span class="emphasis"><em><span class="bold"><strong>C</strong></span></em></span> components
|
||
|
|
of the quaternion, respectively. As usual, nothing untoward happens if what
|
||
|
|
should be magnitudes are negative numbers or angles are out of their natural
|
||
|
|
ranges, as symmetries and periodicities kick in.
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
In this version of our implementation of quaternions, there is no analogue
|
||
|
|
of the complex value operation <code class="computeroutput"><span class="identifier">arg</span></code>
|
||
|
|
as the situation is somewhat more complicated. Unit quaternions are linked
|
||
|
|
both to rotations in <span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span>
|
||
|
|
and in <span class="emphasis"><em><span class="bold"><strong>R<sup>4</sup></strong></span></em></span>, and the correspondences
|
||
|
|
are not too complicated, but there is currently a lack of standard (de facto
|
||
|
|
or de jure) matrix library with which the conversions could work. This should
|
||
|
|
be remedied in a further revision. In the mean time, an example of how this
|
||
|
|
could be done is presented here for <a href="../../../example/HSO3.hpp" target="_top"><span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span></a>, and here for <a href="../../../example/HSO4.hpp" target="_top"><span class="emphasis"><em><span class="bold"><strong>R<sup>4</sup></strong></span></em></span></a> (<a href="../../../example/HSO3SO4.cpp" target="_top">example
|
||
|
|
test file</a>).
|
||
|
|
</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="value_op.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="trans.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
|
||
|
|
</div>
|
||
|
|
</body>
|
||
|
|
</html>
|