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<h1>TransverseMercatorExact.cpp</h1> </div>
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<a href="TransverseMercatorExact_8cpp.html">Go to the documentation of this file.</a><div class="fragment"><pre class="fragment"><a name="l00001"></a>00001 <span class="comment">/**</span>
<a name="l00002"></a>00002 <span class="comment"> * \file TransverseMercatorExact.cpp</span>
<a name="l00003"></a>00003 <span class="comment"> * \brief Implementation for GeographicLib::TransverseMercatorExact class</span>
<a name="l00004"></a>00004 <span class="comment"> *</span>
<a name="l00005"></a>00005 <span class="comment"> * Copyright (c) Charles Karney (2008, 2009, 2010, 2011) <charles@karney.com></span>
<a name="l00006"></a>00006 <span class="comment"> * and licensed under the LGPL. For more information, see</span>
<a name="l00007"></a>00007 <span class="comment"> * http://geographiclib.sourceforge.net/</span>
<a name="l00008"></a>00008 <span class="comment"> *</span>
<a name="l00009"></a>00009 <span class="comment"> * The relevant section of Lee's paper is part V, pp 67&ndash;101,</span>
<a name="l00010"></a>00010 <span class="comment"> * <a href="http://dx.doi.org/10.3138/X687-1574-4325-WM62">Conformal</span>
<a name="l00011"></a>00011 <span class="comment"> * Projections Based On Jacobian Elliptic Functions</a>.</span>
<a name="l00012"></a>00012 <span class="comment"> *</span>
<a name="l00013"></a>00013 <span class="comment"> * The method entails using the Thompson Transverse Mercator as an</span>
<a name="l00014"></a>00014 <span class="comment"> * intermediate projection. The projections from the intermediate</span>
<a name="l00015"></a>00015 <span class="comment"> * coordinates to [\e phi, \e lam] and [\e x, \e y] are given by elliptic</span>
<a name="l00016"></a>00016 <span class="comment"> * functions. The inverse of these projections are found by Newton's method</span>
<a name="l00017"></a>00017 <span class="comment"> * with a suitable starting guess.</span>
<a name="l00018"></a>00018 <span class="comment"> *</span>
<a name="l00019"></a>00019 <span class="comment"> * This implementation and notation closely follows Lee, with the following</span>
<a name="l00020"></a>00020 <span class="comment"> * exceptions:</span>
<a name="l00021"></a>00021 <span class="comment"> * <center><table></span>
<a name="l00022"></a>00022 <span class="comment"> * <tr><th>Lee <th>here <th>Description</span>
<a name="l00023"></a>00023 <span class="comment"> * <tr><td>x/a <td>xi <td>Northing (unit Earth)</span>
<a name="l00024"></a>00024 <span class="comment"> * <tr><td>y/a <td>eta <td>Easting (unit Earth)</span>
<a name="l00025"></a>00025 <span class="comment"> * <tr><td>s/a <td>sigma <td>xi + i * eta</span>
<a name="l00026"></a>00026 <span class="comment"> * <tr><td>y <td>x <td>Easting</span>
<a name="l00027"></a>00027 <span class="comment"> * <tr><td>x <td>y <td>Northing</span>
<a name="l00028"></a>00028 <span class="comment"> * <tr><td>k <td>e <td>eccentricity</span>
<a name="l00029"></a>00029 <span class="comment"> * <tr><td>k^2 <td>mu <td>elliptic function parameter</span>
<a name="l00030"></a>00030 <span class="comment"> * <tr><td>k'^2 <td>mv <td>elliptic function complementary parameter</span>
<a name="l00031"></a>00031 <span class="comment"> * <tr><td>m <td>k <td>scale</span>
<a name="l00032"></a>00032 <span class="comment"> * <tr><td>zeta <td>zeta <td>complex longitude = Mercator = chi in paper</span>
<a name="l00033"></a>00033 <span class="comment"> * <tr><td>s <td>sigma <td>complex GK = zeta in paper</span>
<a name="l00034"></a>00034 <span class="comment"> * </table></center></span>
<a name="l00035"></a>00035 <span class="comment"> *</span>
<a name="l00036"></a>00036 <span class="comment"> * Minor alterations have been made in some of Lee's expressions in an</span>
<a name="l00037"></a>00037 <span class="comment"> * attempt to control round-off. For example atanh(sin(phi)) is replaced by</span>
<a name="l00038"></a>00038 <span class="comment"> * asinh(tan(phi)) which maintains accuracy near phi = pi/2. Such changes</span>
<a name="l00039"></a>00039 <span class="comment"> * are noted in the code.</span>
<a name="l00040"></a>00040 <span class="comment"> **********************************************************************/</span>
<a name="l00041"></a>00041
<a name="l00042"></a>00042 <span class="preprocessor">#include "<a class="code" href="TransverseMercatorExact_8hpp.html" title="Header for GeographicLib::TransverseMercatorExact class.">GeographicLib/TransverseMercatorExact.hpp</a>"</span>
<a name="l00043"></a>00043
<a name="l00044"></a><a class="code" href="TransverseMercatorExact_8cpp.html#a019dc2a5f6ac9aa9ba03ad8dd974e438">00044</a> <span class="preprocessor">#define GEOGRAPHICLIB_TRANSVERSEMERCATOREXACT_CPP "$Id: TransverseMercatorExact.cpp 6937 2011-02-01 20:17:13Z karney $"</span>
<a name="l00045"></a>00045 <span class="preprocessor"></span>
<a name="l00046"></a>00046 <a class="code" href="Constants_8hpp.html#af90fa899707a2ac513d5e4c76853bbf5">RCSID_DECL</a>(<a class="code" href="TransverseMercatorExact_8cpp.html#a019dc2a5f6ac9aa9ba03ad8dd974e438">GEOGRAPHICLIB_TRANSVERSEMERCATOREXACT_CPP</a>)
<a name="l00047"></a>00047 <a class="code" href="Constants_8hpp.html#af90fa899707a2ac513d5e4c76853bbf5">RCSID_DECL</a>(<a class="code" href="TransverseMercatorExact_8hpp.html#a33c80ffd2f74ea9ed9dd4c11bf9e1198">GEOGRAPHICLIB_TRANSVERSEMERCATOREXACT_HPP</a>)
<a name="l00048"></a>00048
<a name="l00049"></a>00049 namespace GeographicLib {
<a name="l00050"></a>00050
<a name="l00051"></a>00051 <span class="keyword">using namespace </span>std;
<a name="l00052"></a>00052
<a name="l00053"></a>00053 <span class="keyword">const</span> <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">Math::real</a> TransverseMercatorExact::tol =
<a name="l00054"></a>00054 numeric_limits<real>::epsilon();
<a name="l00055"></a>00055 <span class="keyword">const</span> <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">Math::real</a> TransverseMercatorExact::tol1 = <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>(0.1) * sqrt(tol);
<a name="l00056"></a>00056 <span class="keyword">const</span> <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">Math::real</a> TransverseMercatorExact::tol2 = <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>(0.1) * tol;
<a name="l00057"></a>00057 <span class="keyword">const</span> <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">Math::real</a> TransverseMercatorExact::taytol = pow(tol, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>(0.6));
<a name="l00058"></a>00058 <span class="comment">// Overflow value s.t. atan(overflow) = pi/2</span>
<a name="l00059"></a>00059 <span class="keyword">const</span> <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">Math::real</a> TransverseMercatorExact::overflow = 1 / sq(tol);
<a name="l00060"></a>00060
<a name="l00061"></a><a class="code" href="classGeographicLib_1_1TransverseMercatorExact.html#ad029ecd780bafca1ade9c7d7df46da14">00061</a> TransverseMercatorExact::TransverseMercatorExact(real a, real r, real k0,
<a name="l00062"></a>00062 <span class="keywordtype">bool</span> extendp)
<a name="l00063"></a>00063 : _a(a)
<a name="l00064"></a>00064 , _r(r)
<a name="l00065"></a>00065 , _f(1 / _r)
<a name="l00066"></a>00066 , _k0(k0)
<a name="l00067"></a>00067 , _mu(_f * (2 - _f)) <span class="comment">// e^2</span>
<a name="l00068"></a>00068 , _mv(1 - _mu) <span class="comment">// 1 - e^2</span>
<a name="l00069"></a>00069 , _e(sqrt(_mu))
<a name="l00070"></a>00070 , _ep2(_mu / _mv) <span class="comment">// e^2 / (1 - e^2)</span>
<a name="l00071"></a>00071 , _extendp(extendp)
<a name="l00072"></a>00072 , _Eu(_mu)
<a name="l00073"></a>00073 , _Ev(_mv)
<a name="l00074"></a>00074 {
<a name="l00075"></a>00075 <span class="keywordflow">if</span> (!(_a > 0))
<a name="l00076"></a>00076 <span class="keywordflow">throw</span> <a class="code" href="classGeographicLib_1_1GeographicErr.html" title="Exception handling for GeographicLib">GeographicErr</a>(<span class="stringliteral">"Major radius is not positive"</span>);
<a name="l00077"></a>00077 <span class="keywordflow">if</span> (!(_r > 0))
<a name="l00078"></a>00078 <span class="keywordflow">throw</span> <a class="code" href="classGeographicLib_1_1GeographicErr.html" title="Exception handling for GeographicLib">GeographicErr</a>(<span class="stringliteral">"Inverse flattening is not positive"</span>);
<a name="l00079"></a>00079 <span class="keywordflow">if</span> (!(_f < 1))
<a name="l00080"></a>00080 <span class="keywordflow">throw</span> <a class="code" href="classGeographicLib_1_1GeographicErr.html" title="Exception handling for GeographicLib">GeographicErr</a>(<span class="stringliteral">"Minor radius is not positive"</span>);
<a name="l00081"></a>00081 <span class="keywordflow">if</span> (!(_k0 > 0))
<a name="l00082"></a>00082 <span class="keywordflow">throw</span> <a class="code" href="classGeographicLib_1_1GeographicErr.html" title="Exception handling for GeographicLib">GeographicErr</a>(<span class="stringliteral">"Scale is not positive"</span>);
<a name="l00083"></a>00083 }
<a name="l00084"></a>00084
<a name="l00085"></a>00085 <span class="keyword">const</span> <a class="code" href="classGeographicLib_1_1TransverseMercatorExact.html" title="An exact implementation of the Transverse Mercator Projection.">TransverseMercatorExact</a>
<a name="l00086"></a>00086 <a class="code" href="classGeographicLib_1_1TransverseMercatorExact.html#a68668637e754201897bddb82baaf634b">TransverseMercatorExact::UTM</a>(Constants::WGS84_a<real>(),
<a name="l00087"></a>00087 Constants::WGS84_r<real>(),
<a name="l00088"></a>00088 Constants::UTM_k0<real>());
<a name="l00089"></a>00089
<a name="l00090"></a>00090 <span class="comment">// tau = tan(phi), taup = sinh(psi)</span>
<a name="l00091"></a>00091 <a class="code" href="classGeographicLib_1_1Math.html#aeee4778d7cf2f9fb9648efe4911da59d">Math::real</a> TransverseMercatorExact::taup(<a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> tau) <span class="keyword">const</span> <span class="keywordflow">throw</span>() {
<a name="l00092"></a>00092 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>
<a name="l00093"></a>00093 tau1 = <a class="code" href="classGeographicLib_1_1Math.html#a0d422863198d4bec2aae6b187a60760c">Math::hypot</a>(<a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>(1), tau),
<a name="l00094"></a>00094 sig = sinh( _e * <a class="code" href="classGeographicLib_1_1Math.html#a62ae372f983cb076561204be8de263bd">Math::atanh</a>(_e * tau / tau1) );
<a name="l00095"></a>00095 <span class="keywordflow">return</span> <a class="code" href="classGeographicLib_1_1Math.html#a0d422863198d4bec2aae6b187a60760c">Math::hypot</a>(<a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>(1), sig) * tau - sig * tau1;
<a name="l00096"></a>00096 }
<a name="l00097"></a>00097
<a name="l00098"></a>00098 <a class="code" href="classGeographicLib_1_1Math.html#aeee4778d7cf2f9fb9648efe4911da59d">Math::real</a> TransverseMercatorExact::taupinv(<a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> taup) <span class="keyword">const</span> <span class="keywordflow">throw</span>() {
<a name="l00099"></a>00099 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>
<a name="l00100"></a>00100 tau = taup,
<a name="l00101"></a>00101 stol = tol * max(<a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>(1), abs(taup));
<a name="l00102"></a>00102 <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i < numit; ++i) {
<a name="l00103"></a>00103 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>
<a name="l00104"></a>00104 tau1 = <a class="code" href="classGeographicLib_1_1Math.html#a0d422863198d4bec2aae6b187a60760c">Math::hypot</a>(<a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>(1), tau),
<a name="l00105"></a>00105 sig = sinh( _e * <a class="code" href="classGeographicLib_1_1Math.html#a62ae372f983cb076561204be8de263bd">Math::atanh</a>(_e * tau / tau1 ) ),
<a name="l00106"></a>00106 taupa = <a class="code" href="classGeographicLib_1_1Math.html#a0d422863198d4bec2aae6b187a60760c">Math::hypot</a>(<a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>(1), sig) * tau - sig * tau1,
<a name="l00107"></a>00107 dtau = (taup - taupa) * (1 + _mv * sq(tau)) /
<a name="l00108"></a>00108 ( _mv * tau1 * <a class="code" href="classGeographicLib_1_1Math.html#a0d422863198d4bec2aae6b187a60760c">Math::hypot</a>(<a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>(1), taupa) );
<a name="l00109"></a>00109 tau += dtau;
<a name="l00110"></a>00110 <span class="keywordflow">if</span> (!(abs(dtau) >= stol))
<a name="l00111"></a>00111 <span class="keywordflow">break</span>;
<a name="l00112"></a>00112 }
<a name="l00113"></a>00113 <span class="keywordflow">return</span> tau;
<a name="l00114"></a>00114 }
<a name="l00115"></a>00115
<a name="l00116"></a>00116 <span class="keywordtype">void</span> TransverseMercatorExact::zeta(<a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> u, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> snu, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> cnu, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> dnu,
<a name="l00117"></a>00117 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> v, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> snv, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> cnv, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> dnv,
<a name="l00118"></a>00118 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>& taup, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>& lam) <span class="keyword">const</span> <span class="keywordflow">throw</span>() {
<a name="l00119"></a>00119 <span class="comment">// Lee 54.17 but write</span>
<a name="l00120"></a>00120 <span class="comment">// atanh(snu * dnv) = asinh(snu * dnv / sqrt(cnu^2 + _mv * snu^2 * snv^2))</span>
<a name="l00121"></a>00121 <span class="comment">// atanh(_e * snu / dnv) =</span>
<a name="l00122"></a>00122 <span class="comment">// asinh(_e * snu / sqrt(_mu * cnu^2 + _mv * cnv^2))</span>
<a name="l00123"></a>00123 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>
<a name="l00124"></a>00124 d1 = sqrt(sq(cnu) + _mv * sq(snu * snv)),
<a name="l00125"></a>00125 d2 = sqrt(_mu * sq(cnu) + _mv * sq(cnv)),
<a name="l00126"></a>00126 t1 = (d1 ? snu * dnv / d1 : snu < 0 ? -overflow : overflow),
<a name="l00127"></a>00127 t2 = (d2 ? sinh( _e * <a class="code" href="classGeographicLib_1_1Math.html#ab0998a80c8946d1c016c1bc4810a0698">Math::asinh</a>(_e * snu / d2) ) :
<a name="l00128"></a>00128 snu < 0 ? -overflow : overflow);
<a name="l00129"></a>00129 <span class="comment">// psi = asinh(t1) - asinh(t2)</span>
<a name="l00130"></a>00130 <span class="comment">// taup = sinh(psi)</span>
<a name="l00131"></a>00131 taup = t1 * <a class="code" href="classGeographicLib_1_1Math.html#a0d422863198d4bec2aae6b187a60760c">Math::hypot</a>(<a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>(1), t2) - t2 * <a class="code" href="classGeographicLib_1_1Math.html#a0d422863198d4bec2aae6b187a60760c">Math::hypot</a>(<a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>(1), t1);
<a name="l00132"></a>00132 lam = (d1 != 0 && d2 != 0) ?
<a name="l00133"></a>00133 atan2(dnu * snv, cnu * cnv) - _e * atan2(_e * cnu * snv, dnu * cnv) :
<a name="l00134"></a>00134 0;
<a name="l00135"></a>00135 }
<a name="l00136"></a>00136
<a name="l00137"></a>00137 <span class="keywordtype">void</span> TransverseMercatorExact::dwdzeta(<a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> u, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> snu, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> cnu, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> dnu,
<a name="l00138"></a>00138 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> v, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> snv, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> cnv, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> dnv,
<a name="l00139"></a>00139 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>& du, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>& dv) <span class="keyword">const</span> <span class="keywordflow">throw</span>() {
<a name="l00140"></a>00140 <span class="comment">// Lee 54.21 but write (1 - dnu^2 * snv^2) = (cnv^2 + _mu * snu^2 * snv^2)</span>
<a name="l00141"></a>00141 <span class="comment">// (see A+S 16.21.4)</span>
<a name="l00142"></a>00142 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> d = _mv * sq(sq(cnv) + _mu * sq(snu * snv));
<a name="l00143"></a>00143 du = cnu * dnu * dnv * (sq(cnv) - _mu * sq(snu * snv)) / d;
<a name="l00144"></a>00144 dv = -snu * snv * cnv * (sq(dnu * dnv) + _mu * sq(cnu)) / d;
<a name="l00145"></a>00145 }
<a name="l00146"></a>00146
<a name="l00147"></a>00147 <span class="comment">// Starting point for zetainv</span>
<a name="l00148"></a>00148 <span class="keywordtype">bool</span> TransverseMercatorExact::zetainv0(<a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> psi, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> lam, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>& u, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>& v)
<a name="l00149"></a>00149 <span class="keyword">const</span> <span class="keywordflow">throw</span>() {
<a name="l00150"></a>00150 <span class="keywordtype">bool</span> retval = <span class="keyword">false</span>;
<a name="l00151"></a>00151 <span class="keywordflow">if</span> (psi < -_e * Math::pi<real>()/4 &&
<a name="l00152"></a>00152 lam > (1 - 2 * _e) * Math::pi<real>()/2 &&
<a name="l00153"></a>00153 psi < lam - (1 - _e) * Math::pi<real>()/2) {
<a name="l00154"></a>00154 <span class="comment">// N.B. this branch is normally not taken because psi < 0 is converted</span>
<a name="l00155"></a>00155 <span class="comment">// psi > 0 by Forward.</span>
<a name="l00156"></a>00156 <span class="comment">//</span>
<a name="l00157"></a>00157 <span class="comment">// There's a log singularity at w = w0 = Eu.K() + i * Ev.K(),</span>
<a name="l00158"></a>00158 <span class="comment">// corresponding to the south pole, where we have, approximately</span>
<a name="l00159"></a>00159 <span class="comment">//</span>
<a name="l00160"></a>00160 <span class="comment">// psi = _e + i * pi/2 - _e * atanh(cos(i * (w - w0)/(1 + _mu/2)))</span>
<a name="l00161"></a>00161 <span class="comment">//</span>
<a name="l00162"></a>00162 <span class="comment">// Inverting this gives:</span>
<a name="l00163"></a>00163 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>
<a name="l00164"></a>00164 psix = 1 - psi / _e,
<a name="l00165"></a>00165 lamx = (Math::pi<real>()/2 - lam) / _e;
<a name="l00166"></a>00166 u = <a class="code" href="classGeographicLib_1_1Math.html#ab0998a80c8946d1c016c1bc4810a0698">Math::asinh</a>(sin(lamx) / <a class="code" href="classGeographicLib_1_1Math.html#a0d422863198d4bec2aae6b187a60760c">Math::hypot</a>(cos(lamx), sinh(psix))) *
<a name="l00167"></a>00167 (1 + _mu/2);
<a name="l00168"></a>00168 v = atan2(cos(lamx), sinh(psix)) * (1 + _mu/2);
<a name="l00169"></a>00169 u = _Eu.K() - u;
<a name="l00170"></a>00170 v = _Ev.K() - v;
<a name="l00171"></a>00171 } <span class="keywordflow">else</span> <span class="keywordflow">if</span> (psi < _e * Math::pi<real>()/2 &&
<a name="l00172"></a>00172 lam > (1 - 2 * _e) * Math::pi<real>()/2) {
<a name="l00173"></a>00173 <span class="comment">// At w = w0 = i * Ev.K(), we have</span>
<a name="l00174"></a>00174 <span class="comment">//</span>
<a name="l00175"></a>00175 <span class="comment">// zeta = zeta0 = i * (1 - _e) * pi/2</span>
<a name="l00176"></a>00176 <span class="comment">// zeta' = zeta'' = 0</span>
<a name="l00177"></a>00177 <span class="comment">//</span>
<a name="l00178"></a>00178 <span class="comment">// including the next term in the Taylor series gives:</span>
<a name="l00179"></a>00179 <span class="comment">//</span>
<a name="l00180"></a>00180 <span class="comment">// zeta = zeta0 - (_mv * _e) / 3 * (w - w0)^3</span>
<a name="l00181"></a>00181 <span class="comment">//</span>
<a name="l00182"></a>00182 <span class="comment">// When inverting this, we map arg(w - w0) = [-90, 0] to</span>
<a name="l00183"></a>00183 <span class="comment">// arg(zeta - zeta0) = [-90, 180]</span>
<a name="l00184"></a>00184 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>
<a name="l00185"></a>00185 dlam = lam - (1 - _e) * Math::pi<real>()/2,
<a name="l00186"></a>00186 rad = <a class="code" href="classGeographicLib_1_1Math.html#a0d422863198d4bec2aae6b187a60760c">Math::hypot</a>(psi, dlam),
<a name="l00187"></a>00187 <span class="comment">// atan2(dlam-psi, psi+dlam) + 45d gives arg(zeta - zeta0) in range</span>
<a name="l00188"></a>00188 <span class="comment">// [-135, 225). Subtracting 180 (since multiplier is negative) makes</span>
<a name="l00189"></a>00189 <span class="comment">// range [-315, 45). Multiplying by 1/3 (for cube root) gives range</span>
<a name="l00190"></a>00190 <span class="comment">// [-105, 15). In particular the range [-90, 180] in zeta space maps</span>
<a name="l00191"></a>00191 <span class="comment">// to [-90, 0] in w space as required.</span>
<a name="l00192"></a>00192 ang = atan2(dlam-psi, psi+dlam) - <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>(0.75) * Math::pi<real>();
<a name="l00193"></a>00193 <span class="comment">// Error using this guess is about 0.21 * (rad/e)^(5/3)</span>
<a name="l00194"></a>00194 retval = rad < _e * taytol;
<a name="l00195"></a>00195 rad = <a class="code" href="classGeographicLib_1_1Math.html#a21b33765456b50bf9e32ed0d57b441a6">Math::cbrt</a>(3 / (_mv * _e) * rad);
<a name="l00196"></a>00196 ang /= 3;
<a name="l00197"></a>00197 u = rad * cos(ang);
<a name="l00198"></a>00198 v = rad * sin(ang) + _Ev.K();
<a name="l00199"></a>00199 } <span class="keywordflow">else</span> {
<a name="l00200"></a>00200 <span class="comment">// Use spherical TM, Lee 12.6 -- writing atanh(sin(lam) / cosh(psi)) =</span>
<a name="l00201"></a>00201 <span class="comment">// asinh(sin(lam) / hypot(cos(lam), sinh(psi))). This takes care of the</span>
<a name="l00202"></a>00202 <span class="comment">// log singularity at zeta = Eu.K() (corresponding to the north pole)</span>
<a name="l00203"></a>00203 v = <a class="code" href="classGeographicLib_1_1Math.html#ab0998a80c8946d1c016c1bc4810a0698">Math::asinh</a>(sin(lam) / <a class="code" href="classGeographicLib_1_1Math.html#a0d422863198d4bec2aae6b187a60760c">Math::hypot</a>(cos(lam), sinh(psi)));
<a name="l00204"></a>00204 u = atan2(sinh(psi), cos(lam));
<a name="l00205"></a>00205 <span class="comment">// But scale to put 90,0 on the right place</span>
<a name="l00206"></a>00206 u *= _Eu.K() / (Math::pi<real>()/2);
<a name="l00207"></a>00207 v *= _Eu.K() / (Math::pi<real>()/2);
<a name="l00208"></a>00208 }
<a name="l00209"></a>00209 <span class="keywordflow">return</span> retval;
<a name="l00210"></a>00210 }
<a name="l00211"></a>00211
<a name="l00212"></a>00212 <span class="comment">// Invert zeta using Newton's method</span>
<a name="l00213"></a>00213 <span class="keywordtype">void</span> TransverseMercatorExact::zetainv(<a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> taup, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> lam, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>& u, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>& v)
<a name="l00214"></a>00214 <span class="keyword">const</span> <span class="keywordflow">throw</span>() {
<a name="l00215"></a>00215 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>
<a name="l00216"></a>00216 psi = <a class="code" href="classGeographicLib_1_1Math.html#ab0998a80c8946d1c016c1bc4810a0698">Math::asinh</a>(taup),
<a name="l00217"></a>00217 scal = 1/<a class="code" href="classGeographicLib_1_1Math.html#a0d422863198d4bec2aae6b187a60760c">Math::hypot</a>(<a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>(1), taup);
<a name="l00218"></a>00218 <span class="keywordflow">if</span> (zetainv0(psi, lam, u, v))
<a name="l00219"></a>00219 <span class="keywordflow">return</span>;
<a name="l00220"></a>00220 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> stol2 = tol2 / sq(max(psi, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>(1)));
<a name="l00221"></a>00221 <span class="comment">// min iterations = 2, max iterations = 6; mean = 4.0</span>
<a name="l00222"></a>00222 <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0, trip = 0; i < numit; ++i) {
<a name="l00223"></a>00223 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> snu, cnu, dnu, snv, cnv, dnv;
<a name="l00224"></a>00224 _Eu.sncndn(u, snu, cnu, dnu);
<a name="l00225"></a>00225 _Ev.sncndn(v, snv, cnv, dnv);
<a name="l00226"></a>00226 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> tau1, lam1, du1, dv1;
<a name="l00227"></a>00227 zeta(u, snu, cnu, dnu, v, snv, cnv, dnv, tau1, lam1);
<a name="l00228"></a>00228 dwdzeta(u, snu, cnu, dnu, v, snv, cnv, dnv, du1, dv1);
<a name="l00229"></a>00229 tau1 -= taup;
<a name="l00230"></a>00230 lam1 -= lam;
<a name="l00231"></a>00231 tau1 *= scal;
<a name="l00232"></a>00232 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>
<a name="l00233"></a>00233 delu = tau1 * du1 - lam1 * dv1,
<a name="l00234"></a>00234 delv = tau1 * dv1 + lam1 * du1;
<a name="l00235"></a>00235 u -= delu;
<a name="l00236"></a>00236 v -= delv;
<a name="l00237"></a>00237 <span class="keywordflow">if</span> (trip)
<a name="l00238"></a>00238 <span class="keywordflow">break</span>;
<a name="l00239"></a>00239 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> delw2 = sq(delu) + sq(delv);
<a name="l00240"></a>00240 <span class="keywordflow">if</span> (!(delw2 >= stol2))
<a name="l00241"></a>00241 ++trip;
<a name="l00242"></a>00242 }
<a name="l00243"></a>00243 }
<a name="l00244"></a>00244
<a name="l00245"></a>00245 <span class="keywordtype">void</span> TransverseMercatorExact::sigma(<a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> u, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> snu, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> cnu, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> dnu,
<a name="l00246"></a>00246 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> v, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> snv, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> cnv, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> dnv,
<a name="l00247"></a>00247 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>& xi, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>& eta) <span class="keyword">const</span> <span class="keywordflow">throw</span>() {
<a name="l00248"></a>00248 <span class="comment">// Lee 55.4 writing</span>
<a name="l00249"></a>00249 <span class="comment">// dnu^2 + dnv^2 - 1 = _mu * cnu^2 + _mv * cnv^2</span>
<a name="l00250"></a>00250 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> d = _mu * sq(cnu) + _mv * sq(cnv);
<a name="l00251"></a>00251 xi = _Eu.E(snu, cnu, dnu) - _mu * snu * cnu * dnu / d;
<a name="l00252"></a>00252 eta = v - _Ev.E(snv, cnv, dnv) + _mv * snv * cnv * dnv / d;
<a name="l00253"></a>00253 }
<a name="l00254"></a>00254
<a name="l00255"></a>00255 <span class="keywordtype">void</span> TransverseMercatorExact::dwdsigma(<a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> u, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> snu, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> cnu, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> dnu,
<a name="l00256"></a>00256 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> v, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> snv, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> cnv, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> dnv,
<a name="l00257"></a>00257 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>& du, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>& dv) <span class="keyword">const</span> <span class="keywordflow">throw</span>() {
<a name="l00258"></a>00258 <span class="comment">// Reciprocal of 55.9: dw/ds = dn(w)^2/_mv, expanding complex dn(w) using</span>
<a name="l00259"></a>00259 <span class="comment">// A+S 16.21.4</span>
<a name="l00260"></a>00260 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> d = _mv * sq(sq(cnv) + _mu * sq(snu * snv));
<a name="l00261"></a>00261 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>
<a name="l00262"></a>00262 dnr = dnu * cnv * dnv,
<a name="l00263"></a>00263 dni = - _mu * snu * cnu * snv;
<a name="l00264"></a>00264 du = (sq(dnr) - sq(dni)) / d;
<a name="l00265"></a>00265 dv = 2 * dnr * dni / d;
<a name="l00266"></a>00266 }
<a name="l00267"></a>00267
<a name="l00268"></a>00268 <span class="comment">// Starting point for sigmainv</span>
<a name="l00269"></a>00269 <span class="keywordtype">bool</span> TransverseMercatorExact::sigmainv0(<a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> xi, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> eta, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>& u, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>& v)
<a name="l00270"></a>00270 <span class="keyword">const</span> <span class="keywordflow">throw</span>() {
<a name="l00271"></a>00271 <span class="keywordtype">bool</span> retval = <span class="keyword">false</span>;
<a name="l00272"></a>00272 <span class="keywordflow">if</span> (eta > <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>(1.25) * _Ev.KE() ||
<a name="l00273"></a>00273 (xi < -<a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>(0.25) * _Eu.E() && xi < eta - _Ev.KE())) {
<a name="l00274"></a>00274 <span class="comment">// sigma as a simple pole at w = w0 = Eu.K() + i * Ev.K() and sigma is</span>
<a name="l00275"></a>00275 <span class="comment">// approximated by</span>
<a name="l00276"></a>00276 <span class="comment">//</span>
<a name="l00277"></a>00277 <span class="comment">// sigma = (Eu.E() + i * Ev.KE()) + 1/(w - w0)</span>
<a name="l00278"></a>00278 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>
<a name="l00279"></a>00279 x = xi - _Eu.E(),
<a name="l00280"></a>00280 y = eta - _Ev.KE(),
<a name="l00281"></a>00281 r2 = sq(x) + sq(y);
<a name="l00282"></a>00282 u = _Eu.K() + x/r2;
<a name="l00283"></a>00283 v = _Ev.K() - y/r2;
<a name="l00284"></a>00284 } <span class="keywordflow">else</span> <span class="keywordflow">if</span> ((eta > <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>(0.75) * _Ev.KE() && xi < <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>(0.25) * _Eu.E())
<a name="l00285"></a>00285 || eta > _Ev.KE()) {
<a name="l00286"></a>00286 <span class="comment">// At w = w0 = i * Ev.K(), we have</span>
<a name="l00287"></a>00287 <span class="comment">//</span>
<a name="l00288"></a>00288 <span class="comment">// sigma = sigma0 = i * Ev.KE()</span>
<a name="l00289"></a>00289 <span class="comment">// sigma' = sigma'' = 0</span>
<a name="l00290"></a>00290 <span class="comment">//</span>
<a name="l00291"></a>00291 <span class="comment">// including the next term in the Taylor series gives:</span>
<a name="l00292"></a>00292 <span class="comment">//</span>
<a name="l00293"></a>00293 <span class="comment">// sigma = sigma0 - _mv / 3 * (w - w0)^3</span>
<a name="l00294"></a>00294 <span class="comment">//</span>
<a name="l00295"></a>00295 <span class="comment">// When inverting this, we map arg(w - w0) = [-pi/2, -pi/6] to</span>
<a name="l00296"></a>00296 <span class="comment">// arg(sigma - sigma0) = [-pi/2, pi/2]</span>
<a name="l00297"></a>00297 <span class="comment">// mapping arg = [-pi/2, -pi/6] to [-pi/2, pi/2]</span>
<a name="l00298"></a>00298 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>
<a name="l00299"></a>00299 deta = eta - _Ev.KE(),
<a name="l00300"></a>00300 rad = <a class="code" href="classGeographicLib_1_1Math.html#a0d422863198d4bec2aae6b187a60760c">Math::hypot</a>(xi, deta),
<a name="l00301"></a>00301 <span class="comment">// Map the range [-90, 180] in sigma space to [-90, 0] in w space. See</span>
<a name="l00302"></a>00302 <span class="comment">// discussion in zetainv0 on the cut for ang.</span>
<a name="l00303"></a>00303 ang = atan2(deta-xi, xi+deta) - <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>(0.75) * Math::pi<real>();
<a name="l00304"></a>00304 <span class="comment">// Error using this guess is about 0.068 * rad^(5/3)</span>
<a name="l00305"></a>00305 retval = rad < 2 * taytol;
<a name="l00306"></a>00306 rad = <a class="code" href="classGeographicLib_1_1Math.html#a21b33765456b50bf9e32ed0d57b441a6">Math::cbrt</a>(3 / _mv * rad);
<a name="l00307"></a>00307 ang /= 3;
<a name="l00308"></a>00308 u = rad * cos(ang);
<a name="l00309"></a>00309 v = rad * sin(ang) + _Ev.K();
<a name="l00310"></a>00310 } <span class="keywordflow">else</span> {
<a name="l00311"></a>00311 <span class="comment">// Else use w = sigma * Eu.K/Eu.E (which is correct in the limit _e -> 0)</span>
<a name="l00312"></a>00312 u = xi * _Eu.K()/_Eu.E();
<a name="l00313"></a>00313 v = eta * _Eu.K()/_Eu.E();
<a name="l00314"></a>00314 }
<a name="l00315"></a>00315 <span class="keywordflow">return</span> retval;
<a name="l00316"></a>00316 }
<a name="l00317"></a>00317
<a name="l00318"></a>00318 <span class="comment">// Invert sigma using Newton's method</span>
<a name="l00319"></a>00319 <span class="keywordtype">void</span> TransverseMercatorExact::sigmainv(<a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> xi, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> eta, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>& u, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>& v)
<a name="l00320"></a>00320 <span class="keyword">const</span> <span class="keywordflow">throw</span>() {
<a name="l00321"></a>00321 <span class="keywordflow">if</span> (sigmainv0(xi, eta, u, v))
<a name="l00322"></a>00322 <span class="keywordflow">return</span>;
<a name="l00323"></a>00323 <span class="comment">// min iterations = 2, max iterations = 7; mean = 3.9</span>
<a name="l00324"></a>00324 <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0, trip = 0; i < numit; ++i) {
<a name="l00325"></a>00325 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> snu, cnu, dnu, snv, cnv, dnv;
<a name="l00326"></a>00326 _Eu.sncndn(u, snu, cnu, dnu);
<a name="l00327"></a>00327 _Ev.sncndn(v, snv, cnv, dnv);
<a name="l00328"></a>00328 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> xi1, eta1, du1, dv1;
<a name="l00329"></a>00329 sigma(u, snu, cnu, dnu, v, snv, cnv, dnv, xi1, eta1);
<a name="l00330"></a>00330 dwdsigma(u, snu, cnu, dnu, v, snv, cnv, dnv, du1, dv1);
<a name="l00331"></a>00331 xi1 -= xi;
<a name="l00332"></a>00332 eta1 -= eta;
<a name="l00333"></a>00333 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>
<a name="l00334"></a>00334 delu = xi1 * du1 - eta1 * dv1,
<a name="l00335"></a>00335 delv = xi1 * dv1 + eta1 * du1;
<a name="l00336"></a>00336 u -= delu;
<a name="l00337"></a>00337 v -= delv;
<a name="l00338"></a>00338 <span class="keywordflow">if</span> (trip)
<a name="l00339"></a>00339 <span class="keywordflow">break</span>;
<a name="l00340"></a>00340 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> delw2 = sq(delu) + sq(delv);
<a name="l00341"></a>00341 <span class="keywordflow">if</span> (!(delw2 >= tol2))
<a name="l00342"></a>00342 ++trip;
<a name="l00343"></a>00343 }
<a name="l00344"></a>00344 }
<a name="l00345"></a>00345
<a name="l00346"></a>00346 <span class="keywordtype">void</span> TransverseMercatorExact::Scale(<a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> tau, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> lam,
<a name="l00347"></a>00347 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> snu, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> cnu, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> dnu,
<a name="l00348"></a>00348 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> snv, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> cnv, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> dnv,
<a name="l00349"></a>00349 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>& gamma, <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a>& k) <span class="keyword">const</span> <span class="keywordflow">throw</span>() {
<a name="l00350"></a>00350 <a class="code" href="Geod_8cpp.html#a5caf95d46b184d9ca1d3764b3781b3c9">real</a> sec2 = 1 + sq(tau); <span class="comment">// sec(phi)^2</span>
<a name="l00351"></a>00351 <span class="comment">// Lee 55.12 -- negated for our sign convention. gamma gives the bearing</span>
<a name="l00352"></a>00352 <span class="comment">// (clockwise from true north) of grid north</span>
<a name="l00353"></a>00353 gamma = atan2(_mv * snu * snv * cnv, cnu * dnu * dnv);
<a name="l00354"></a>00354 <span class="comment">// Lee 55.13 with nu given by Lee 9.1 -- in sqrt change the numerator</span>
<a name="l00355"></a>00355 <span class="comment">// from</span>
<a name="l00356"></a>00356 <span class="comment">//</span>
<a name="l00357"></a>00357 <span class="comment">// (1 - snu^2 * dnv^2) to (_mv * snv^2 + cnu^2 * dnv^2)</span>
<a name="l00358"></a>00358 <span class="comment">//</span>
<a name="l00359"></a>00359 <span class="comment">// to maintain accuracy near phi = 90 and change the denomintor from</span>
<a name="l00360"></a>00360 <span class="comment">//</span>
<a name="l00361"></a>00361 <span class="comment">// (dnu^2 + dnv^2 - 1) to (_mu * cnu^2 + _mv * cnv^2)</span>
<a name="l00362"></a>00362 <span class="comment">//</span>
<a name="l00363"></a>00363 <span class="comment">// to maintain accuracy near phi = 0, lam = 90 * (1 - e). Similarly</span>
<a name="l00364"></a>00364 <span class="comment">// rewrite sqrt term in 9.1 as</span>
<a name="l00365"></a>00365 <span class="comment">//</span>
<a name="l00366"></a>00366 <span class="comment">// _mv + _mu * c^2 instead of 1 - _mu * sin(phi)^2</span>
<a name="l00367"></a>00367 k = sqrt(_mv + _mu / sec2) * sqrt(sec2) *
<a name="l00368"></a>00368 sqrt( (_mv * sq(snv) + sq(cnu * dnv)) /
<a name="l00369"></a>00369 (_mu * sq(cnu) + _mv * sq(cnv)) );
<a name="l00370"></a>00370 }
<a name="l00371"></a>00371
<a name="l00372"></a><a class="code" href="classGeographicLib_1_1TransverseMercatorExact.html#a0dab8d1a275df9e4796d2d858f42f7e3">00372</a> <span class="keywordtype">void</span> <a class="code" href="classGeographicLib_1_1TransverseMercatorExact.html#a0dab8d1a275df9e4796d2d858f42f7e3">TransverseMercatorExact::Forward</a>(real lon0, real lat, real lon,
<a name="l00373"></a>00373 real& x, real& y, real& gamma, real& k)
<a name="l00374"></a>00374 <span class="keyword">const</span> <span class="keywordflow">throw</span>() {
<a name="l00375"></a>00375 <span class="comment">// Avoid losing a bit of accuracy in lon (assuming lon0 is an integer)</span>
<a name="l00376"></a>00376 <span class="keywordflow">if</span> (lon - lon0 > 180)
<a name="l00377"></a>00377 lon -= lon0 + 360;
<a name="l00378"></a>00378 <span class="keywordflow">else</span> <span class="keywordflow">if</span> (lon - lon0 <= -180)
<a name="l00379"></a>00379 lon -= lon0 - 360;
<a name="l00380"></a>00380 <span class="keywordflow">else</span>
<a name="l00381"></a>00381 lon -= lon0;
<a name="l00382"></a>00382 <span class="comment">// Now lon in (-180, 180]</span>
<a name="l00383"></a>00383 <span class="comment">// Explicitly enforce the parity</span>
<a name="l00384"></a>00384 <span class="keywordtype">int</span>
<a name="l00385"></a>00385 latsign = !_extendp && lat < 0 ? -1 : 1,
<a name="l00386"></a>00386 lonsign = !_extendp && lon < 0 ? -1 : 1;
<a name="l00387"></a>00387 lon *= lonsign;
<a name="l00388"></a>00388 lat *= latsign;
<a name="l00389"></a>00389 <span class="keywordtype">bool</span> backside = !_extendp && lon > 90;
<a name="l00390"></a>00390 <span class="keywordflow">if</span> (backside) {
<a name="l00391"></a>00391 <span class="keywordflow">if</span> (lat == 0)
<a name="l00392"></a>00392 latsign = -1;
<a name="l00393"></a>00393 lon = 180 - lon;
<a name="l00394"></a>00394 }
<a name="l00395"></a>00395 real
<a name="l00396"></a>00396 phi = lat * Math::degree<real>(),
<a name="l00397"></a>00397 lam = lon * Math::degree<real>(),
<a name="l00398"></a>00398 tau = tanx(phi);
<a name="l00399"></a>00399
<a name="l00400"></a>00400 <span class="comment">// u,v = coordinates for the Thompson TM, Lee 54</span>
<a name="l00401"></a>00401 real u, v;
<a name="l00402"></a>00402 <span class="keywordflow">if</span> (lat == 90) {
<a name="l00403"></a>00403 u = _Eu.K();
<a name="l00404"></a>00404 v = 0;
<a name="l00405"></a>00405 } <span class="keywordflow">else</span> <span class="keywordflow">if</span> (lat == 0 && lon == 90 * (1 - _e)) {
<a name="l00406"></a>00406 u = 0;
<a name="l00407"></a>00407 v = _Ev.K();
<a name="l00408"></a>00408 } <span class="keywordflow">else</span>
<a name="l00409"></a>00409 zetainv(taup(tau), lam, u, v);
<a name="l00410"></a>00410
<a name="l00411"></a>00411 real snu, cnu, dnu, snv, cnv, dnv;
<a name="l00412"></a>00412 _Eu.sncndn(u, snu, cnu, dnu);
<a name="l00413"></a>00413 _Ev.sncndn(v, snv, cnv, dnv);
<a name="l00414"></a>00414
<a name="l00415"></a>00415 real xi, eta;
<a name="l00416"></a>00416 sigma(u, snu, cnu, dnu, v, snv, cnv, dnv, xi, eta);
<a name="l00417"></a>00417 <span class="keywordflow">if</span> (backside)
<a name="l00418"></a>00418 xi = 2 * _Eu.E() - xi;
<a name="l00419"></a>00419 y = xi * _a * _k0 * latsign;
<a name="l00420"></a>00420 x = eta * _a * _k0 * lonsign;
<a name="l00421"></a>00421
<a name="l00422"></a>00422 <span class="comment">// Recompute (tau, lam) from (u, v) to improve accuracy of Scale</span>
<a name="l00423"></a>00423 zeta(u, snu, cnu, dnu, v, snv, cnv, dnv, tau, lam);
<a name="l00424"></a>00424 tau=taupinv(tau);
<a name="l00425"></a>00425 Scale(tau, lam, snu, cnu, dnu, snv, cnv, dnv, gamma, k);
<a name="l00426"></a>00426 gamma /= Math::degree<real>();
<a name="l00427"></a>00427 <span class="keywordflow">if</span> (backside)
<a name="l00428"></a>00428 gamma = 180 - gamma;
<a name="l00429"></a>00429 gamma *= latsign * lonsign;
<a name="l00430"></a>00430 k *= _k0;
<a name="l00431"></a>00431 }
<a name="l00432"></a>00432
<a name="l00433"></a><a class="code" href="classGeographicLib_1_1TransverseMercatorExact.html#a350070780708de5a622d4f4c4757104c">00433</a> <span class="keywordtype">void</span> <a class="code" href="classGeographicLib_1_1TransverseMercatorExact.html#a350070780708de5a622d4f4c4757104c">TransverseMercatorExact::Reverse</a>(real lon0, real x, real y,
<a name="l00434"></a>00434 real& lat, real& lon,
<a name="l00435"></a>00435 real& gamma, real& k)
<a name="l00436"></a>00436 <span class="keyword">const</span> <span class="keywordflow">throw</span>() {
<a name="l00437"></a>00437 <span class="comment">// This undoes the steps in Forward.</span>
<a name="l00438"></a>00438 real
<a name="l00439"></a>00439 xi = y / (_a * _k0),
<a name="l00440"></a>00440 eta = x / (_a * _k0);
<a name="l00441"></a>00441 <span class="comment">// Explicitly enforce the parity</span>
<a name="l00442"></a>00442 <span class="keywordtype">int</span>
<a name="l00443"></a>00443 latsign = !_extendp && y < 0 ? -1 : 1,
<a name="l00444"></a>00444 lonsign = !_extendp && x < 0 ? -1 : 1;
<a name="l00445"></a>00445 xi *= latsign;
<a name="l00446"></a>00446 eta *= lonsign;
<a name="l00447"></a>00447 <span class="keywordtype">bool</span> backside = !_extendp && xi > _Eu.E();
<a name="l00448"></a>00448 <span class="keywordflow">if</span> (backside)
<a name="l00449"></a>00449 xi = 2 * _Eu.E()- xi;
<a name="l00450"></a>00450
<a name="l00451"></a>00451 <span class="comment">// u,v = coordinates for the Thompson TM, Lee 54</span>
<a name="l00452"></a>00452 real u, v;
<a name="l00453"></a>00453 <span class="keywordflow">if</span> (xi == 0 && eta == _Ev.KE()) {
<a name="l00454"></a>00454 u = 0;
<a name="l00455"></a>00455 v = _Ev.K();
<a name="l00456"></a>00456 } <span class="keywordflow">else</span>
<a name="l00457"></a>00457 sigmainv(xi, eta, u, v);
<a name="l00458"></a>00458
<a name="l00459"></a>00459 real snu, cnu, dnu, snv, cnv, dnv;
<a name="l00460"></a>00460 _Eu.sncndn(u, snu, cnu, dnu);
<a name="l00461"></a>00461 _Ev.sncndn(v, snv, cnv, dnv);
<a name="l00462"></a>00462 real phi, lam, tau;
<a name="l00463"></a>00463 <span class="keywordflow">if</span> (v != 0 || u != _Eu.K()) {
<a name="l00464"></a>00464 zeta(u, snu, cnu, dnu, v, snv, cnv, dnv, tau, lam);
<a name="l00465"></a>00465 tau = taupinv(tau);
<a name="l00466"></a>00466 phi = atan(tau);
<a name="l00467"></a>00467 lat = phi / Math::degree<real>();
<a name="l00468"></a>00468 lon = lam / Math::degree<real>();
<a name="l00469"></a>00469 } <span class="keywordflow">else</span> {
<a name="l00470"></a>00470 tau = overflow;
<a name="l00471"></a>00471 phi = Math::pi<real>()/2;
<a name="l00472"></a>00472 lat = 90;
<a name="l00473"></a>00473 lon = lam = 0;
<a name="l00474"></a>00474 }
<a name="l00475"></a>00475 Scale(tau, lam, snu, cnu, dnu, snv, cnv, dnv, gamma, k);
<a name="l00476"></a>00476 gamma /= Math::degree<real>();
<a name="l00477"></a>00477 <span class="keywordflow">if</span> (backside)
<a name="l00478"></a>00478 lon = 180 - lon;
<a name="l00479"></a>00479 lon *= lonsign;
<a name="l00480"></a>00480 <span class="comment">// Avoid losing a bit of accuracy in lon (assuming lon0 is an integer)</span>
<a name="l00481"></a>00481 <span class="keywordflow">if</span> (lon + lon0 >= 180)
<a name="l00482"></a>00482 lon += lon0 - 360;
<a name="l00483"></a>00483 <span class="keywordflow">else</span> <span class="keywordflow">if</span> (lon + lon0 < -180)
<a name="l00484"></a>00484 lon += lon0 + 360;
<a name="l00485"></a>00485 <span class="keywordflow">else</span>
<a name="l00486"></a>00486 lon += lon0;
<a name="l00487"></a>00487 lat *= latsign;
<a name="l00488"></a>00488 <span class="keywordflow">if</span> (backside)
<a name="l00489"></a>00489 y = 2 * _Eu.E() - y;
<a name="l00490"></a>00490 y *= _a * _k0 * latsign;
<a name="l00491"></a>00491 x *= _a * _k0 * lonsign;
<a name="l00492"></a>00492 <span class="keywordflow">if</span> (backside)
<a name="l00493"></a>00493 gamma = 180 - gamma;
<a name="l00494"></a>00494 gamma *= latsign * lonsign;
<a name="l00495"></a>00495 k *= _k0;
<a name="l00496"></a>00496 }
<a name="l00497"></a>00497
<a name="l00498"></a>00498 } <span class="comment">// namespace GeographicLib</span>
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