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  25. <div class="section">
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  26. <div class="titlepage"><div><div><h2 class="title" style="clear: both">
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  27. <a name="math_toolkit.oct_create"></a><a class="link" href="oct_create.html" title="Octonion Creation Functions">Octonion Creation Functions</a>
    28. 

  28. </h2></div></div></div>
    29. 

  29. <pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">spherical</span><span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">rho</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">phi1</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">phi2</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">phi3</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">phi4</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">phi5</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">phi6</span><span class="special">);</span>
    30. 

  30. <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">multipolar</span><span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">rho1</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">theta1</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">rho2</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">theta2</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">rho3</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">theta3</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">rho4</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">theta4</span><span class="special">);</span>
    31. 

  31. <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">cylindrical</span><span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">r</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">angle</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">h1</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">h2</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">h3</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">h4</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">h5</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">h6</span><span class="special">);</span>
    32. 

  32. </pre>
    33. 

  33. <p>
    34. 

  34.       These build octonions in a way similar to the way polar builds complex numbers,
    35. 

  35.       as there is no strict equivalent to polar coordinates for octonions.
    36. 

  36.     </p>
    37. 

  37. <p>
    38. 

  38.       <code class="computeroutput"><span class="identifier">spherical</span></code> is a simple transposition
    39. 

  39.       of <code class="computeroutput"><span class="identifier">polar</span></code>, it takes as inputs
    40. 

  40.       a (positive) magnitude and a point on the hypersphere, given by three angles.
    41. 

  41.       The first of these, <span class="emphasis"><em>theta</em></span> has a natural range of -pi to
    42. 

  42.       +pi, and the other two have natural ranges of -pi/2 to +pi/2 (as is the case
    43. 

  43.       with the usual spherical coordinates in <span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span>).
    44. 

  44.       Due to the many symmetries and periodicities, nothing untoward happens if the
    45. 

  45.       magnitude is negative or the angles are outside their natural ranges. The expected
    46. 

  46.       degeneracies (a magnitude of zero ignores the angles settings...) do happen
    47. 

  47.       however.
    48. 

  48.     </p>
    49. 

  49. <p>
    50. 

  50.       <code class="computeroutput"><span class="identifier">cylindrical</span></code> is likewise a simple
    51. 

  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. 

  52.       which in turn is another derivative of planar polar coordinates. The first
    53. 

  53.       two inputs are the polar coordinates of the first <span class="emphasis"><em><span class="bold"><strong>C</strong></span></em></span>
    54. 

  54.       component of the octonion. The third and fourth inputs are placed into the
    55. 

  55.       third and fourth <span class="emphasis"><em><span class="bold"><strong>R</strong></span></em></span> components
    56. 

  56.       of the octonion, respectively.
    57. 

  57.     </p>
    58. 

  58. <p>
    59. 

  59.       <code class="computeroutput"><span class="identifier">multipolar</span></code> is yet another simple
    60. 

  60.       generalization of polar coordinates. This time, both <span class="emphasis"><em><span class="bold"><strong>C</strong></span></em></span>
    61. 

  61.       components of the octonion are given in polar coordinates.
    62. 

  62.     </p>
    63. 

  63. <p>
    64. 

  64.       In this version of our implementation of octonions, there is no analogue of
    65. 

  65.       the complex value operation arg as the situation is somewhat more complicated.
    66. 

  66.     </p>
    67. 

  67. </div>
    68. 

  68. <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
    69. 

  69. <td align="left"></td>
    70. 

  70. <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
    71. 

  71.       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
    72. 

  72.       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
    73. 

  73.       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
    74. 

  74.       Daryle Walker and Xiaogang Zhang<p>
    75. 

  75.         Distributed under the Boost Software License, Version 1.0. (See accompanying
    76. 

  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. 

  77.       </p>
    78. 

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    79. 

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