1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Spherical Harmonics</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="../sf_poly.html" title="Polynomials"> 9<link rel="prev" href="chebyshev.html" title="Chebyshev Polynomials"> 10<link rel="next" href="cardinal_b_splines.html" title="Cardinal B-splines"> 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="chebyshev.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_poly.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="cardinal_b_splines.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h3 class="title"> 27<a name="math_toolkit.sf_poly.sph_harm"></a><a class="link" href="sph_harm.html" title="Spherical Harmonics">Spherical Harmonics</a> 28</h3></div></div></div> 29<h5> 30<a name="math_toolkit.sf_poly.sph_harm.h0"></a> 31 <span class="phrase"><a name="math_toolkit.sf_poly.sph_harm.synopsis"></a></span><a class="link" href="sph_harm.html#math_toolkit.sf_poly.sph_harm.synopsis">Synopsis</a> 32 </h5> 33<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">spherical_harmonic</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 34</pre> 35<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> 36 37<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span> 38<span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special"><</span><a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a><span class="special">></span> <span class="identifier">spherical_harmonic</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">);</span> 39 40<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 41<span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special"><</span><a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a><span class="special">></span> <span class="identifier">spherical_harmonic</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 42 43<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span> 44<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">spherical_harmonic_r</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">);</span> 45 46<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 47<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">spherical_harmonic_r</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 48 49<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span> 50<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">spherical_harmonic_i</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">);</span> 51 52<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 53<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">spherical_harmonic_i</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 54 55<span class="special">}}</span> <span class="comment">// namespaces</span> 56</pre> 57<h5> 58<a name="math_toolkit.sf_poly.sph_harm.h1"></a> 59 <span class="phrase"><a name="math_toolkit.sf_poly.sph_harm.description"></a></span><a class="link" href="sph_harm.html#math_toolkit.sf_poly.sph_harm.description">Description</a> 60 </h5> 61<p> 62 The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result 63 type calculation rules</em></span></a> when T1 and T2 are different types. 64 </p> 65<p> 66 The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can 67 be used to control the behaviour of the function: how it handles errors, 68 what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy 69 documentation for more details</a>. 70 </p> 71<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span> 72<span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special"><</span><a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a><span class="special">></span> <span class="identifier">spherical_harmonic</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">);</span> 73 74<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 75<span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special"><</span><a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a><span class="special">></span> <span class="identifier">spherical_harmonic</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 76</pre> 77<p> 78 Returns the value of the Spherical Harmonic Y<sub>n</sub><sup>m</sup>(theta, phi): 79 </p> 80<div class="blockquote"><blockquote class="blockquote"><p> 81 <span class="inlinemediaobject"><img src="../../../equations/spherical_0.svg"></span> 82 83 </p></blockquote></div> 84<p> 85 The spherical harmonics Y<sub>n</sub><sup>m</sup>(theta, phi) are the angular portion of the solution 86 to Laplace's equation in spherical coordinates where azimuthal symmetry is 87 not present. 88 </p> 89<div class="caution"><table border="0" summary="Caution"> 90<tr> 91<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td> 92<th align="left">Caution</th> 93</tr> 94<tr><td align="left" valign="top"> 95<p> 96 Care must be taken in correctly identifying the arguments to this function: 97 θ is taken as the polar (colatitudinal) coordinate with θ in [0, π], and φas 98 the azimuthal (longitudinal) coordinate with φin [0,2π). This is the convention 99 used in Physics, and matches the definition used by <a href="http://documents.wolfram.com/mathematica/functions/SphericalHarmonicY" target="_top">Mathematica 100 in the function SpericalHarmonicY</a>, but is opposite to the usual 101 mathematical conventions. 102 </p> 103<p> 104 Some other sources include an additional Condon-Shortley phase term of 105 (-1)<sup>m</sup> in the definition of this function: note however that our definition 106 of the associated Legendre polynomial already includes this term. 107 </p> 108<p> 109 This implementation returns zero for m > n 110 </p> 111<p> 112 For θ outside [0, π] and φ outside [0, 2π] this implementation follows the convention 113 used by Mathematica: the function is periodic with period π in θ and 2π in φ. 114 Please note that this is not the behaviour one would get from a casual 115 application of the function's definition. Cautious users should keep θ and 116 φ to the range [0, π] and [0, 2π] respectively. 117 </p> 118<p> 119 See: <a href="http://mathworld.wolfram.com/SphericalHarmonic.html" target="_top">Weisstein, 120 Eric W. "Spherical Harmonic." From MathWorld--A Wolfram Web Resource</a>. 121 </p> 122</td></tr> 123</table></div> 124<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span> 125<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">spherical_harmonic_r</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">);</span> 126 127<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 128<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">spherical_harmonic_r</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 129</pre> 130<p> 131 Returns the real part of Y<sub>n</sub><sup>m</sup>(theta, phi): 132 </p> 133<div class="blockquote"><blockquote class="blockquote"><p> 134 <span class="inlinemediaobject"><img src="../../../equations/spherical_1.svg"></span> 135 136 </p></blockquote></div> 137<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span> 138<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">spherical_harmonic_i</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">);</span> 139 140<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 141<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">spherical_harmonic_i</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">theta</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 142</pre> 143<p> 144 Returns the imaginary part of Y<sub>n</sub><sup>m</sup>(theta, phi): 145 </p> 146<div class="blockquote"><blockquote class="blockquote"><p> 147 <span class="inlinemediaobject"><img src="../../../equations/spherical_2.svg"></span> 148 149 </p></blockquote></div> 150<h5> 151<a name="math_toolkit.sf_poly.sph_harm.h2"></a> 152 <span class="phrase"><a name="math_toolkit.sf_poly.sph_harm.accuracy"></a></span><a class="link" href="sph_harm.html#math_toolkit.sf_poly.sph_harm.accuracy">Accuracy</a> 153 </h5> 154<p> 155 The following table shows peak errors for various domains of input arguments. 156 Note that only results for the widest floating point type on the system are 157 given as narrower types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively 158 zero error</a>. Peak errors are the same for both the real and imaginary 159 parts, as the error is dominated by calculation of the associated Legendre 160 polynomials: especially near the roots of the associated Legendre function. 161 </p> 162<p> 163 All values are in units of epsilon. 164 </p> 165<div class="table"> 166<a name="math_toolkit.sf_poly.sph_harm.table_spherical_harmonic_r"></a><p class="title"><b>Table 8.38. Error rates for spherical_harmonic_r</b></p> 167<div class="table-contents"><table class="table" summary="Error rates for spherical_harmonic_r"> 168<colgroup> 169<col> 170<col> 171<col> 172<col> 173<col> 174</colgroup> 175<thead><tr> 176<th> 177 </th> 178<th> 179 <p> 180 GNU C++ version 7.1.0<br> linux<br> double 181 </p> 182 </th> 183<th> 184 <p> 185 GNU C++ version 7.1.0<br> linux<br> long double 186 </p> 187 </th> 188<th> 189 <p> 190 Sun compiler version 0x5150<br> Sun Solaris<br> long double 191 </p> 192 </th> 193<th> 194 <p> 195 Microsoft Visual C++ version 14.1<br> Win32<br> double 196 </p> 197 </th> 198</tr></thead> 199<tbody><tr> 200<td> 201 <p> 202 Spherical Harmonics 203 </p> 204 </td> 205<td> 206 <p> 207 <span class="blue">Max = 1.58ε (Mean = 0.0707ε)</span> 208 </p> 209 </td> 210<td> 211 <p> 212 <span class="blue">Max = 2.89e+03ε (Mean = 108ε)</span> 213 </p> 214 </td> 215<td> 216 <p> 217 <span class="blue">Max = 1.03e+04ε (Mean = 327ε)</span> 218 </p> 219 </td> 220<td> 221 <p> 222 <span class="blue">Max = 2.27e+04ε (Mean = 725ε)</span> 223 </p> 224 </td> 225</tr></tbody> 226</table></div> 227</div> 228<br class="table-break"><div class="table"> 229<a name="math_toolkit.sf_poly.sph_harm.table_spherical_harmonic_i"></a><p class="title"><b>Table 8.39. Error rates for spherical_harmonic_i</b></p> 230<div class="table-contents"><table class="table" summary="Error rates for spherical_harmonic_i"> 231<colgroup> 232<col> 233<col> 234<col> 235<col> 236<col> 237</colgroup> 238<thead><tr> 239<th> 240 </th> 241<th> 242 <p> 243 GNU C++ version 7.1.0<br> linux<br> double 244 </p> 245 </th> 246<th> 247 <p> 248 GNU C++ version 7.1.0<br> linux<br> long double 249 </p> 250 </th> 251<th> 252 <p> 253 Sun compiler version 0x5150<br> Sun Solaris<br> long double 254 </p> 255 </th> 256<th> 257 <p> 258 Microsoft Visual C++ version 14.1<br> Win32<br> double 259 </p> 260 </th> 261</tr></thead> 262<tbody><tr> 263<td> 264 <p> 265 Spherical Harmonics 266 </p> 267 </td> 268<td> 269 <p> 270 <span class="blue">Max = 1.36ε (Mean = 0.0765ε)</span> 271 </p> 272 </td> 273<td> 274 <p> 275 <span class="blue">Max = 2.89e+03ε (Mean = 108ε)</span> 276 </p> 277 </td> 278<td> 279 <p> 280 <span class="blue">Max = 1.03e+04ε (Mean = 327ε)</span> 281 </p> 282 </td> 283<td> 284 <p> 285 <span class="blue">Max = 2.27e+04ε (Mean = 725ε)</span> 286 </p> 287 </td> 288</tr></tbody> 289</table></div> 290</div> 291<br class="table-break"><p> 292 Note that the worst errors occur when the degree increases, values greater 293 than ~120 are very unlikely to produce sensible results, especially when 294 the order is also large. Further the relative errors are likely to grow arbitrarily 295 large when the function is very close to a root. 296 </p> 297<h5> 298<a name="math_toolkit.sf_poly.sph_harm.h3"></a> 299 <span class="phrase"><a name="math_toolkit.sf_poly.sph_harm.testing"></a></span><a class="link" href="sph_harm.html#math_toolkit.sf_poly.sph_harm.testing">Testing</a> 300 </h5> 301<p> 302 A mixture of spot tests of values calculated using functions.wolfram.com, 303 and randomly generated test data are used: the test data was computed using 304 <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a> at 1000-bit 305 precision. 306 </p> 307<h5> 308<a name="math_toolkit.sf_poly.sph_harm.h4"></a> 309 <span class="phrase"><a name="math_toolkit.sf_poly.sph_harm.implementation"></a></span><a class="link" href="sph_harm.html#math_toolkit.sf_poly.sph_harm.implementation">Implementation</a> 310 </h5> 311<p> 312 These functions are implemented fairly naively using the formulae given above. 313 Some extra care is taken to prevent roundoff error when converting from polar 314 coordinates (so for example the <span class="emphasis"><em>1-x<sup>2</sup></em></span> term used by the 315 associated Legendre functions is calculated without roundoff error using 316 <span class="emphasis"><em>x = cos(theta)</em></span>, and <span class="emphasis"><em>1-x<sup>2</sup> = sin<sup>2</sup>(theta)</em></span>). 317 The limiting factor in the error rates for these functions is the need to 318 calculate values near the roots of the associated Legendre functions. 319 </p> 320</div> 321<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 322<td align="left"></td> 323<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 324 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 325 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 326 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 327 Daryle Walker and Xiaogang Zhang<p> 328 Distributed under the Boost Software License, Version 1.0. (See accompanying 329 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>) 330 </p> 331</div></td> 332</tr></table> 333<hr> 334<div class="spirit-nav"> 335<a accesskey="p" href="chebyshev.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_poly.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="cardinal_b_splines.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 336</div> 337</body> 338</html> 339
