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<title>GeographicLib: GeographicLib::EllipticFunction Class Reference</title>
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<a href="#pub-methods">Public Member Functions</a> </div>
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<h1>GeographicLib::EllipticFunction Class Reference</h1> </div>
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<!-- doxytag: class="GeographicLib::EllipticFunction" -->
<p>Elliptic functions needed for <a class="el" href="classGeographicLib_1_1TransverseMercatorExact.html" title="An exact implementation of the Transverse Mercator Projection.">TransverseMercatorExact</a>.
<a href="#_details">More...</a></p>
<p><code>#include <<a class="el" href="EllipticFunction_8hpp_source.html">GeographicLib/EllipticFunction.hpp</a>></code></p>
<p><a href="classGeographicLib_1_1EllipticFunction-members.html">List of all members.</a></p>
<table class="memberdecls">
<tr><td colspan="2"><h2><a name="pub-methods"></a>
Public Member Functions</h2></td></tr>
<tr><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classGeographicLib_1_1EllipticFunction.html#ada122111b55eb0ab9c7b9fafb0a4b1e6">EllipticFunction</a> (real m) throw ()</td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="el" href="classGeographicLib_1_1Math.html#aeee4778d7cf2f9fb9648efe4911da59d">Math::real</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="classGeographicLib_1_1EllipticFunction.html#aba29c348874db7880ab7b1065f03ff7c">m</a> () const throw ()</td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="el" href="classGeographicLib_1_1Math.html#aeee4778d7cf2f9fb9648efe4911da59d">Math::real</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="classGeographicLib_1_1EllipticFunction.html#ad1fbe801f6731285462d35d874a149ba">m1</a> () const throw ()</td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="el" href="classGeographicLib_1_1Math.html#aeee4778d7cf2f9fb9648efe4911da59d">Math::real</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="classGeographicLib_1_1EllipticFunction.html#a6c6c8029d93946b32184ee964525938c">K</a> () const throw ()</td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="el" href="classGeographicLib_1_1Math.html#aeee4778d7cf2f9fb9648efe4911da59d">Math::real</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="classGeographicLib_1_1EllipticFunction.html#a184174d29953891896b41c597c34eb9a">E</a> () const throw ()</td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="el" href="classGeographicLib_1_1Math.html#aeee4778d7cf2f9fb9648efe4911da59d">Math::real</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="classGeographicLib_1_1EllipticFunction.html#a12bcdf2c3fc7bcb165318822171328f7">KE</a> () const throw ()</td></tr>
<tr><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="classGeographicLib_1_1EllipticFunction.html#a0b8d63970c4276503d03d4133ee6c321">sncndn</a> (real x, real &sn, real &cn, real &dn) const throw ()</td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="el" href="classGeographicLib_1_1Math.html#aeee4778d7cf2f9fb9648efe4911da59d">Math::real</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="classGeographicLib_1_1EllipticFunction.html#a8be312b5da54b01f2976416aa82e7d08">E</a> (real phi) const throw ()</td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="el" href="classGeographicLib_1_1Math.html#aeee4778d7cf2f9fb9648efe4911da59d">Math::real</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="classGeographicLib_1_1EllipticFunction.html#af1b3b2265bca5d2b52c17ca44dabffa8">E</a> (real sn, real cn, real dn) const throw ()</td></tr>
</table>
<hr/><a name="_details"></a><h2>Detailed Description</h2>
<p>Elliptic functions needed for <a class="el" href="classGeographicLib_1_1TransverseMercatorExact.html" title="An exact implementation of the Transverse Mercator Projection.">TransverseMercatorExact</a>. </p>
<p>This provides the subset of elliptic functions needed for <a class="el" href="classGeographicLib_1_1TransverseMercatorExact.html" title="An exact implementation of the Transverse Mercator Projection.">TransverseMercatorExact</a>. For a given ellipsoid, only parameters <em>e<sup>2</sup></em> and 1 - <em>e<sup>2</sup></em> are needed. This class taken the parameter as a constructor parameters and caches the values of the required complete integrals. A method is provided for Jacobi elliptic functions and for the incomplete elliptic integral of the second kind in terms of the amplitude.</p>
<p>The computation of the elliptic integrals uses the algorithms given in</p>
<ul>
<li>B. C. Carlson, <a href="http://dx.doi.org/10.1007/BF02198293">Computation of elliptic integrals</a>, Numerical Algorithms 10, 13–26 (1995).</li>
</ul>
<p>The computation of the Jacobi elliptic functions uses the algorithm given in</p>
<ul>
<li>R. Bulirsch, <a href="http://dx.doi.org/10.1007/BF01397975">Numerical Calculation of Elliptic Integrals and Elliptic Functions</a>, Numericshe Mathematik 7, 78–90 (1965).</li>
</ul>
<p>The notation follows Abramowitz and Stegun, Chapters 16 and 17. </p>
<p>Definition at line <a class="el" href="EllipticFunction_8hpp_source.html#l00042">42</a> of file <a class="el" href="EllipticFunction_8hpp_source.html">EllipticFunction.hpp</a>.</p>
<hr/><h2>Constructor & Destructor Documentation</h2>
<a class="anchor" id="ada122111b55eb0ab9c7b9fafb0a4b1e6"></a><!-- doxytag: member="GeographicLib::EllipticFunction::EllipticFunction" ref="ada122111b55eb0ab9c7b9fafb0a4b1e6" args="(real m)" -->
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<td class="memname">GeographicLib::EllipticFunction::EllipticFunction </td>
<td>(</td>
<td class="paramtype">real </td>
<td class="paramname"> <em>m</em></td>
<td> ) </td>
<td> throw ()<code> [explicit]</code></td>
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<p>Constructor.</p>
<dl><dt><b>Parameters:</b></dt><dd>
<table border="0" cellspacing="2" cellpadding="0">
<tr><td valign="top"><tt>[in]</tt> </td><td valign="top"><em>m</em> </td><td>the parameter which must lie in [0, 1]. (No checking is done.) </td></tr>
</table>
</dd>
</dl>
<p>Definition at line <a class="el" href="EllipticFunction_8cpp_source.html#l00124">124</a> of file <a class="el" href="EllipticFunction_8cpp_source.html">EllipticFunction.cpp</a>.</p>
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<hr/><h2>Member Function Documentation</h2>
<a class="anchor" id="aba29c348874db7880ab7b1065f03ff7c"></a><!-- doxytag: member="GeographicLib::EllipticFunction::m" ref="aba29c348874db7880ab7b1065f03ff7c" args="() const " -->
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<td class="memname"><a class="el" href="classGeographicLib_1_1Math.html#aeee4778d7cf2f9fb9648efe4911da59d">Math::real</a> GeographicLib::EllipticFunction::m </td>
<td>(</td>
<td class="paramname"></td>
<td> ) </td>
<td> const throw ()<code> [inline]</code></td>
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<dl class="return"><dt><b>Returns:</b></dt><dd>the parameter <em>m</em>. </dd></dl>
<p>Definition at line <a class="el" href="EllipticFunction_8hpp_source.html#l00067">67</a> of file <a class="el" href="EllipticFunction_8hpp_source.html">EllipticFunction.hpp</a>.</p>
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<a class="anchor" id="ad1fbe801f6731285462d35d874a149ba"></a><!-- doxytag: member="GeographicLib::EllipticFunction::m1" ref="ad1fbe801f6731285462d35d874a149ba" args="() const " -->
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<td class="memname"><a class="el" href="classGeographicLib_1_1Math.html#aeee4778d7cf2f9fb9648efe4911da59d">Math::real</a> GeographicLib::EllipticFunction::m1 </td>
<td>(</td>
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<td> const throw ()<code> [inline]</code></td>
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<dl class="return"><dt><b>Returns:</b></dt><dd>the complementary parameter <em>m'</em> = (1 - <em>m</em>). </dd></dl>
<p>Definition at line <a class="el" href="EllipticFunction_8hpp_source.html#l00072">72</a> of file <a class="el" href="EllipticFunction_8hpp_source.html">EllipticFunction.hpp</a>.</p>
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<a class="anchor" id="a6c6c8029d93946b32184ee964525938c"></a><!-- doxytag: member="GeographicLib::EllipticFunction::K" ref="a6c6c8029d93946b32184ee964525938c" args="() const " -->
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<td class="memname"><a class="el" href="classGeographicLib_1_1Math.html#aeee4778d7cf2f9fb9648efe4911da59d">Math::real</a> GeographicLib::EllipticFunction::K </td>
<td>(</td>
<td class="paramname"></td>
<td> ) </td>
<td> const throw ()<code> [inline]</code></td>
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<dl class="return"><dt><b>Returns:</b></dt><dd>the complete integral of first kind, <em>K</em>(<em>m</em>). </dd></dl>
<p>Definition at line <a class="el" href="EllipticFunction_8hpp_source.html#l00077">77</a> of file <a class="el" href="EllipticFunction_8hpp_source.html">EllipticFunction.hpp</a>.</p>
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<a class="anchor" id="a184174d29953891896b41c597c34eb9a"></a><!-- doxytag: member="GeographicLib::EllipticFunction::E" ref="a184174d29953891896b41c597c34eb9a" args="() const " -->
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<td class="memname"><a class="el" href="classGeographicLib_1_1Math.html#aeee4778d7cf2f9fb9648efe4911da59d">Math::real</a> GeographicLib::EllipticFunction::E </td>
<td>(</td>
<td class="paramname"></td>
<td> ) </td>
<td> const throw ()<code> [inline]</code></td>
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<dl class="return"><dt><b>Returns:</b></dt><dd>the complete integral of second kind, <em>E</em>(<em>m</em>). </dd></dl>
<p>Definition at line <a class="el" href="EllipticFunction_8hpp_source.html#l00082">82</a> of file <a class="el" href="EllipticFunction_8hpp_source.html">EllipticFunction.hpp</a>.</p>
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<a class="anchor" id="a12bcdf2c3fc7bcb165318822171328f7"></a><!-- doxytag: member="GeographicLib::EllipticFunction::KE" ref="a12bcdf2c3fc7bcb165318822171328f7" args="() const " -->
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<td class="memname"><a class="el" href="classGeographicLib_1_1Math.html#aeee4778d7cf2f9fb9648efe4911da59d">Math::real</a> GeographicLib::EllipticFunction::KE </td>
<td>(</td>
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<td> const throw ()<code> [inline]</code></td>
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<dl class="return"><dt><b>Returns:</b></dt><dd>the difference <em>K</em>(<em>m</em>) - <em>E</em>(<em>m</em>) (which can be computed directly). </dd></dl>
<p>Definition at line <a class="el" href="EllipticFunction_8hpp_source.html#l00088">88</a> of file <a class="el" href="EllipticFunction_8hpp_source.html">EllipticFunction.hpp</a>.</p>
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<a class="anchor" id="a0b8d63970c4276503d03d4133ee6c321"></a><!-- doxytag: member="GeographicLib::EllipticFunction::sncndn" ref="a0b8d63970c4276503d03d4133ee6c321" args="(real x, real &sn, real &cn, real &dn) const " -->
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<td class="memname">void GeographicLib::EllipticFunction::sncndn </td>
<td>(</td>
<td class="paramtype">real </td>
<td class="paramname"> <em>x</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">real & </td>
<td class="paramname"> <em>sn</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">real & </td>
<td class="paramname"> <em>cn</em>, </td>
</tr>
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<td class="paramkey"></td>
<td></td>
<td class="paramtype">real & </td>
<td class="paramname"> <em>dn</em></td><td> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td><td> const throw ()</td>
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<p>The Jacobi elliptic functions.</p>
<dl><dt><b>Parameters:</b></dt><dd>
<table border="0" cellspacing="2" cellpadding="0">
<tr><td valign="top"><tt>[in]</tt> </td><td valign="top"><em>x</em> </td><td>the argument. </td></tr>
<tr><td valign="top"><tt>[out]</tt> </td><td valign="top"><em>sn</em> </td><td>sn(<em>x</em>|<em>m</em>). </td></tr>
<tr><td valign="top"><tt>[out]</tt> </td><td valign="top"><em>cn</em> </td><td>cn(<em>x</em>|<em>m</em>). </td></tr>
<tr><td valign="top"><tt>[out]</tt> </td><td valign="top"><em>dn</em> </td><td>dn(<em>x</em>|<em>m</em>). </td></tr>
</table>
</dd>
</dl>
<p>Definition at line <a class="el" href="EllipticFunction_8cpp_source.html#l00150">150</a> of file <a class="el" href="EllipticFunction_8cpp_source.html">EllipticFunction.cpp</a>.</p>
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<a class="anchor" id="a8be312b5da54b01f2976416aa82e7d08"></a><!-- doxytag: member="GeographicLib::EllipticFunction::E" ref="a8be312b5da54b01f2976416aa82e7d08" args="(real phi) const " -->
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<td class="memname"><a class="el" href="classGeographicLib_1_1Math.html#aeee4778d7cf2f9fb9648efe4911da59d">Math::real</a> GeographicLib::EllipticFunction::E </td>
<td>(</td>
<td class="paramtype">real </td>
<td class="paramname"> <em>phi</em></td>
<td> ) </td>
<td> const throw ()</td>
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<p>The incomplete integral of the second kind.</p>
<dl><dt><b>Parameters:</b></dt><dd>
<table border="0" cellspacing="2" cellpadding="0">
<tr><td valign="top"><tt>[in]</tt> </td><td valign="top"><em>phi</em> </td><td></td></tr>
</table>
</dd>
</dl>
<dl class="return"><dt><b>Returns:</b></dt><dd>int sqrt(1 - <em>m</em> sin<sup>2</sup>(<em>phi</em>)) <em>dphi</em>. </dd></dl>
<p>Definition at line <a class="el" href="EllipticFunction_8cpp_source.html#l00212">212</a> of file <a class="el" href="EllipticFunction_8cpp_source.html">EllipticFunction.cpp</a>.</p>
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<a class="anchor" id="af1b3b2265bca5d2b52c17ca44dabffa8"></a><!-- doxytag: member="GeographicLib::EllipticFunction::E" ref="af1b3b2265bca5d2b52c17ca44dabffa8" args="(real sn, real cn, real dn) const " -->
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<td class="memname"><a class="el" href="classGeographicLib_1_1Math.html#aeee4778d7cf2f9fb9648efe4911da59d">Math::real</a> GeographicLib::EllipticFunction::E </td>
<td>(</td>
<td class="paramtype">real </td>
<td class="paramname"> <em>sn</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">real </td>
<td class="paramname"> <em>cn</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">real </td>
<td class="paramname"> <em>dn</em></td><td> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td><td> const throw ()</td>
</tr>
</table>
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<p>The incomplete integral of the second kind in terms of Jacobi elliptic functions</p>
<dl><dt><b>Parameters:</b></dt><dd>
<table border="0" cellspacing="2" cellpadding="0">
<tr><td valign="top"><tt>[in]</tt> </td><td valign="top"><em>sn</em> </td><td></td></tr>
<tr><td valign="top"><tt>[in]</tt> </td><td valign="top"><em>cn</em> </td><td></td></tr>
<tr><td valign="top"><tt>[in]</tt> </td><td valign="top"><em>dn</em> </td><td></td></tr>
</table>
</dd>
</dl>
<dl class="return"><dt><b>Returns:</b></dt><dd>int dn(<em>w</em>)<sup>2</sup> <em>dw</em> (A+S 17.2.10).</dd></dl>
<p>Instead of specifying the ampltiude <em>phi</em>, we provide <em>sn</em> = sin(<em>phi</em>), <em>cn</em> = cos(<em>phi</em>), <em>dn</em> = sqrt(1 - <em>m</em> sin<sup>2</sup>(<em>phi</em>)). </p>
<p>Definition at line <a class="el" href="EllipticFunction_8cpp_source.html#l00197">197</a> of file <a class="el" href="EllipticFunction_8cpp_source.html">EllipticFunction.cpp</a>.</p>
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<hr/>The documentation for this class was generated from the following files:<ul>
<li><a class="el" href="EllipticFunction_8hpp_source.html">EllipticFunction.hpp</a></li>
<li><a class="el" href="EllipticFunction_8cpp_source.html">EllipticFunction.cpp</a></li>
</ul>
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