1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Octonion Creation Functions</title> 5<link rel="stylesheet" href="../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../octonions.html" title="Chapter 17. Octonions"> 9<link rel="prev" href="oct_value_ops.html" title="Octonion Value Operations"> 10<link rel="next" href="oct_trans.html" title="Octonions Transcendentals"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="oct_value_ops.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../octonions.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="oct_trans.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h2 class="title" style="clear: both"> 27<a name="math_toolkit.oct_create"></a><a class="link" href="oct_create.html" title="Octonion Creation Functions">Octonion Creation Functions</a> 28</h2></div></div></div> 29<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">octonion</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="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">phi3</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">phi4</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">phi5</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">phi6</span><span class="special">);</span> 30<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">octonion</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="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">rho3</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">theta3</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">rho4</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">theta4</span><span class="special">);</span> 31<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">octonion</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> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">h3</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">h4</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">h5</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">h6</span><span class="special">);</span> 32</pre> 33<p> 34 These build octonions in a way similar to the way polar builds complex numbers, 35 as there is no strict equivalent to polar coordinates for octonions. 36 </p> 37<p> 38 <code class="computeroutput"><span class="identifier">spherical</span></code> is a simple transposition 39 of <code class="computeroutput"><span class="identifier">polar</span></code>, it takes as inputs 40 a (positive) magnitude and a point on the hypersphere, given by three angles. 41 The first of these, <span class="emphasis"><em>theta</em></span> has a natural range of -pi to 42 +pi, and the other two have natural ranges of -pi/2 to +pi/2 (as is the case 43 with the usual spherical coordinates in <span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span>). 44 Due to the many symmetries and periodicities, nothing untoward happens if the 45 magnitude is negative or the angles are outside their natural ranges. The expected 46 degeneracies (a magnitude of zero ignores the angles settings...) do happen 47 however. 48 </p> 49<p> 50 <code class="computeroutput"><span class="identifier">cylindrical</span></code> is likewise a simple 51 transposition of the usual cylindrical coordinates in <span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span>, 52 which in turn is another derivative of planar polar coordinates. The first 53 two inputs are the polar coordinates of the first <span class="emphasis"><em><span class="bold"><strong>C</strong></span></em></span> 54 component of the octonion. The third and fourth inputs are placed into the 55 third and fourth <span class="emphasis"><em><span class="bold"><strong>R</strong></span></em></span> components 56 of the octonion, respectively. 57 </p> 58<p> 59 <code class="computeroutput"><span class="identifier">multipolar</span></code> is yet another simple 60 generalization of polar coordinates. This time, both <span class="emphasis"><em><span class="bold"><strong>C</strong></span></em></span> 61 components of the octonion are given in polar coordinates. 62 </p> 63<p> 64 In this version of our implementation of octonions, there is no analogue of 65 the complex value operation arg as the situation is somewhat more complicated. 66 </p> 67</div> 68<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 69<td align="left"></td> 70<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 71 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 72 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 73 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 74 Daryle Walker and Xiaogang Zhang<p> 75 Distributed under the Boost Software License, Version 1.0. (See accompanying 76 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>) 77 </p> 78</div></td> 79</tr></table> 80<hr> 81<div class="spirit-nav"> 82<a accesskey="p" href="oct_value_ops.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../octonions.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="oct_trans.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 83</div> 84</body> 85</html> 86
