GeographicLib  1.21
TransverseMercatorExact.hpp
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00001 /**
00002  * \file TransverseMercatorExact.hpp
00003  * \brief Header for GeographicLib::TransverseMercatorExact class
00004  *
00005  * Copyright (c) Charles Karney (2008-2011) <charles@karney.com> and licensed
00006  * under the MIT/X11 License.  For more information, see
00007  * http://geographiclib.sourceforge.net/
00008  **********************************************************************/
00009 
00010 #if !defined(GEOGRAPHICLIB_TRANSVERSEMERCATOREXACT_HPP)
00011 #define GEOGRAPHICLIB_TRANSVERSEMERCATOREXACT_HPP \
00012   "$Id: bd96340b9dc3e7bfd09d4374296a75f4c6e00fc3 $"
00013 
00014 #include <GeographicLib/Constants.hpp>
00015 #include <GeographicLib/EllipticFunction.hpp>
00016 
00017 namespace GeographicLib {
00018 
00019   /**
00020    * \brief An exact implementation of the Transverse Mercator Projection
00021    *
00022    * Implementation of the Transverse Mercator Projection given in
00023    *  - L. P. Lee,
00024    *    <a href="http://dx.doi.org/10.3138/X687-1574-4325-WM62"> Conformal
00025    *    Projections Based On Jacobian Elliptic Functions</a>, Part V of
00026    *    Conformal Projections Based on Elliptic Functions,
00027    *    (B. V. Gutsell, Toronto, 1976), 128pp.,
00028    *    ISBN: 0919870163
00029    *    (also appeared as:
00030    *    Monograph 16, Suppl. No. 1 to Canadian Cartographer, Vol 13).
00031    *  - C. F. F. Karney,
00032    *    <a href="http://dx.doi.org/10.1007/s00190-011-0445-3">
00033    *    Transverse Mercator with an accuracy of a few nanometers,</a>
00034    *    J. Geodesy 85(8), 475-485 (Aug. 2011);
00035    *    preprint
00036    *    <a href="http://arxiv.org/abs/1002.1417">arXiv:1002.1417</a>.
00037    *
00038    * Lee gives the correct results for forward and reverse transformations
00039    * subject to the branch cut rules (see the description of the \e extendp
00040    * argument to the constructor).  The maximum error is about 8 nm (8
00041    * nanometers), ground distance, for the forward and reverse transformations.
00042    * The error in the convergence is 2e-15&quot;, the relative error in the
00043    * scale is 7e-12%%.  See Sec. 3 of
00044    * <a href="http://arxiv.org/abs/1002.1417">arXiv:1002.1417</a> for details.
00045    * The method is "exact" in the sense that the errors are close to the
00046    * round-off limit and that no changes are needed in the algorithms for them
00047    * to be used with reals of a higher precision.  Thus the errors using long
00048    * double (with a 64-bit fraction) are about 2000 times smaller than using
00049    * double (with a 53-bit fraction).
00050    *
00051    * This algorithm is about 4.5 times slower than the 6th-order Kr&uuml;ger
00052    * method, TransverseMercator, taking about 11 us for a combined forward and
00053    * reverse projection on a 2.66 GHz Intel machine (g++, version 4.3.0, -O3).
00054    *
00055    * The ellipsoid parameters and the central scale are set in the constructor.
00056    * The central meridian (which is a trivial shift of the longitude) is
00057    * specified as the \e lon0 argument of the TransverseMercatorExact::Forward
00058    * and TransverseMercatorExact::Reverse functions.  The latitude of origin is
00059    * taken to be the equator.  See the documentation on TransverseMercator for
00060    * how to include a false easting, false northing, or a latitude of origin.
00061    *
00062    * See <a href="http://geographiclib.sourceforge.net/tm-grid.kmz"
00063    * type="application/vnd.google-earth.kmz"> tm-grid.kmz</a>, for an
00064    * illustration of the transverse Mercator grid in Google Earth.
00065    *
00066    * See TransverseMercatorExact.cpp for more information on the
00067    * implementation.
00068    *
00069    * See \ref transversemercator for a discussion of this projection.
00070    *
00071    * Example of use:
00072    * \include example-TransverseMercatorExact.cpp
00073    *
00074    * <a href="TransverseMercatorProj.1.html">TransverseMercatorProj</a> is a
00075    * command-line utility providing access to the functionality of
00076    * TransverseMercator and TransverseMercatorExact.
00077    **********************************************************************/
00078 
00079   class GEOGRAPHIC_EXPORT TransverseMercatorExact {
00080   private:
00081     typedef Math::real real;
00082     static const real tol_;
00083     static const real tol1_;
00084     static const real tol2_;
00085     static const real taytol_;
00086     static const real overflow_;
00087     static const int numit_ = 10;
00088     real _a, _f, _k0, _mu, _mv, _e, _ep2;
00089     bool _extendp;
00090     EllipticFunction _Eu, _Ev;
00091     // tan(x) for x in [-pi/2, pi/2] ensuring that the sign is right
00092     static inline real tanx(real x) throw() {
00093       real t = std::tan(x);
00094       // Write the tests this way to ensure that tanx(NaN()) is NaN()
00095       return x >= 0 ? (!(t < 0) ? t : overflow_) : (!(t >= 0) ? t : -overflow_);
00096     }
00097 
00098     real taup(real tau) const throw();
00099     real taupinv(real taup) const throw();
00100 
00101     void zeta(real u, real snu, real cnu, real dnu,
00102               real v, real snv, real cnv, real dnv,
00103               real& taup, real& lam) const throw();
00104 
00105     void dwdzeta(real u, real snu, real cnu, real dnu,
00106                  real v, real snv, real cnv, real dnv,
00107                  real& du, real& dv) const throw();
00108 
00109     bool zetainv0(real psi, real lam, real& u, real& v) const throw();
00110     void zetainv(real taup, real lam, real& u, real& v) const throw();
00111 
00112     void sigma(real u, real snu, real cnu, real dnu,
00113                real v, real snv, real cnv, real dnv,
00114                real& xi, real& eta) const throw();
00115 
00116     void dwdsigma(real u, real snu, real cnu, real dnu,
00117                   real v, real snv, real cnv, real dnv,
00118                   real& du, real& dv) const throw();
00119 
00120     bool sigmainv0(real xi, real eta, real& u, real& v) const throw();
00121     void sigmainv(real xi, real eta, real& u, real& v) const throw();
00122 
00123     void Scale(real tau, real lam,
00124                real snu, real cnu, real dnu,
00125                real snv, real cnv, real dnv,
00126                real& gamma, real& k) const throw();
00127 
00128   public:
00129 
00130     /**
00131      * Constructor for a ellipsoid with
00132      *
00133      * @param[in] a equatorial radius (meters).
00134      * @param[in] f flattening of ellipsoid.  If \e f > 1, set flattening
00135      *   to 1/\e f.
00136      * @param[in] k0 central scale factor.
00137      * @param[in] extendp use extended domain.
00138      *
00139      * The transverse Mercator projection has a branch point singularity at \e
00140      * lat = 0 and \e lon - \e lon0 = 90 (1 - \e e) or (for
00141      * TransverseMercatorExact::UTM) x = 18381 km, y = 0m.  The \e extendp
00142      * argument governs where the branch cut is placed.  With \e extendp =
00143      * false, the "standard" convention is followed, namely the cut is placed
00144      * along x > 18381 km, y = 0m.  Forward can be called with any \e lat and
00145      * \e lon then produces the transformation shown in Lee, Fig 46.  Reverse
00146      * analytically continues this in the +/- \e x direction.  As a
00147      * consequence, Reverse may map multiple points to the same geographic
00148      * location; for example, for TransverseMercatorExact::UTM, \e x =
00149      * 22051449.037349 m, \e y = -7131237.022729 m and \e x = 29735142.378357
00150      * m, \e y = 4235043.607933 m both map to \e lat = -2 deg, \e lon = 88 deg.
00151      *
00152      * With \e extendp = true, the branch cut is moved to the lower left
00153      * quadrant.  The various symmetries of the transverse Mercator projection
00154      * can be used to explore the projection on any sheet.  In this mode the
00155      * domains of \e lat, \e lon, \e x, and \e y are restricted to
00156      * - the union of
00157      *   - \e lat in [0, 90] and \e lon - \e lon0 in [0, 90]
00158      *   - \e lat in (-90, 0] and \e lon - \e lon0 in [90 (1 - \e e), 90]
00159      * - the union of
00160      *   - <i>x</i>/(\e k0 \e a) in [0, inf) and
00161      *     <i>y</i>/(\e k0 \e a) in [0, E(<i>e</i><sup>2</sup>)]
00162      *   - <i>x</i>/(\e k0 \e a) in [K(1 - <i>e</i><sup>2</sup>) - E(1 -
00163      *     <i>e</i><sup>2</sup>), inf) and <i>y</i>/(\e k0 \e a) in (-inf, 0]
00164      * .
00165      * See Sec. 5 of
00166      * <a href="http://arxiv.org/abs/1002.1417">arXiv:1002.1417</a> for a full
00167      * discussion of the treatment of the branch cut.
00168      *
00169      * The method will work for all ellipsoids used in terrestrial geodesy.
00170      * The method cannot be applied directly to the case of a sphere (\e f = 0)
00171      * because some the constants characterizing this method diverge in that
00172      * limit, and in practice, \e f should be larger than about numeric_limits<
00173      * real >::%epsilon().  However, TransverseMercator treats the sphere
00174      * exactly.  An exception is thrown if either axis of the ellipsoid or \e
00175      * k0 is not positive or if \e f <= 0.
00176      **********************************************************************/
00177     TransverseMercatorExact(real a, real f, real k0, bool extendp = false);
00178 
00179     /**
00180      * Forward projection, from geographic to transverse Mercator.
00181      *
00182      * @param[in] lon0 central meridian of the projection (degrees).
00183      * @param[in] lat latitude of point (degrees).
00184      * @param[in] lon longitude of point (degrees).
00185      * @param[out] x easting of point (meters).
00186      * @param[out] y northing of point (meters).
00187      * @param[out] gamma meridian convergence at point (degrees).
00188      * @param[out] k scale of projection at point.
00189      *
00190      * No false easting or northing is added. \e lat should be in the range
00191      * [-90, 90]; \e lon and \e lon0 should be in the range [-180, 360].
00192      **********************************************************************/
00193     void Forward(real lon0, real lat, real lon,
00194                  real& x, real& y, real& gamma, real& k) const throw();
00195 
00196     /**
00197      * Reverse projection, from transverse Mercator to geographic.
00198      *
00199      * @param[in] lon0 central meridian of the projection (degrees).
00200      * @param[in] x easting of point (meters).
00201      * @param[in] y northing of point (meters).
00202      * @param[out] lat latitude of point (degrees).
00203      * @param[out] lon longitude of point (degrees).
00204      * @param[out] gamma meridian convergence at point (degrees).
00205      * @param[out] k scale of projection at point.
00206      *
00207      * No false easting or northing is added.  \e lon0 should be in the range
00208      * [-180, 360].  The value of \e lon returned is in the range [-180, 180).
00209      **********************************************************************/
00210     void Reverse(real lon0, real x, real y,
00211                  real& lat, real& lon, real& gamma, real& k) const throw();
00212 
00213     /**
00214      * TransverseMercatorExact::Forward without returning the convergence and
00215      * scale.
00216      **********************************************************************/
00217     void Forward(real lon0, real lat, real lon,
00218                  real& x, real& y) const throw() {
00219       real gamma, k;
00220       Forward(lon0, lat, lon, x, y, gamma, k);
00221     }
00222 
00223     /**
00224      * TransverseMercatorExact::Reverse without returning the convergence and
00225      * scale.
00226      **********************************************************************/
00227     void Reverse(real lon0, real x, real y,
00228                  real& lat, real& lon) const throw() {
00229       real gamma, k;
00230       Reverse(lon0, x, y, lat, lon, gamma, k);
00231     }
00232 
00233     /** \name Inspector functions
00234      **********************************************************************/
00235     ///@{
00236     /**
00237      * @return \e a the equatorial radius of the ellipsoid (meters).  This is
00238      *   the value used in the constructor.
00239      **********************************************************************/
00240     Math::real MajorRadius() const throw() { return _a; }
00241 
00242     /**
00243      * @return \e f the flattening of the ellipsoid.  This is the value used in
00244      *   the constructor.
00245      **********************************************************************/
00246     Math::real Flattening() const throw() { return _f; }
00247 
00248     /// \cond SKIP
00249     /**
00250      * <b>DEPRECATED</b>
00251      * @return \e r the inverse flattening of the ellipsoid.
00252      **********************************************************************/
00253     Math::real InverseFlattening() const throw() { return 1/_f; }
00254     /// \endcond
00255 
00256     /**
00257      * @return \e k0 central scale for the projection.  This is the value of \e
00258      *   k0 used in the constructor and is the scale on the central meridian.
00259      **********************************************************************/
00260     Math::real CentralScale() const throw() { return _k0; }
00261     ///@}
00262 
00263     /**
00264      * A global instantiation of TransverseMercatorExact with the WGS84
00265      * ellipsoid and the UTM scale factor.  However, unlike UTM, no false
00266      * easting or northing is added.
00267      **********************************************************************/
00268     static const TransverseMercatorExact UTM;
00269   };
00270 
00271 } // namespace GeographicLib
00272 
00273 #endif  // GEOGRAPHICLIB_TRANSVERSEMERCATOREXACT_HPP