265 lines
12 KiB
C++
265 lines
12 KiB
C++
/**
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* \file TransverseMercatorExact.hpp
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* \brief Header for GeographicLib::TransverseMercatorExact class
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*
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* Copyright (c) Charles Karney (2008-2020) <charles@karney.com> and licensed
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* under the MIT/X11 License. For more information, see
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* https://geographiclib.sourceforge.io/
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**********************************************************************/
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#if !defined(GEOGRAPHICLIB_TRANSVERSEMERCATOREXACT_HPP)
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#define GEOGRAPHICLIB_TRANSVERSEMERCATOREXACT_HPP 1
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#include <GeographicLib/Constants.hpp>
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#include <GeographicLib/EllipticFunction.hpp>
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namespace GeographicLib {
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/**
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* \brief An exact implementation of the transverse Mercator projection
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*
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* Implementation of the Transverse Mercator Projection given in
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* - L. P. Lee,
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* <a href="https://doi.org/10.3138/X687-1574-4325-WM62"> Conformal
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* Projections Based On Jacobian Elliptic Functions</a>, Part V of
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* Conformal Projections Based on Elliptic Functions,
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* (B. V. Gutsell, Toronto, 1976), 128pp.,
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* ISBN: 0919870163
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* (also appeared as:
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* Monograph 16, Suppl. No. 1 to Canadian Cartographer, Vol 13).
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* - C. F. F. Karney,
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* <a href="https://doi.org/10.1007/s00190-011-0445-3">
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* Transverse Mercator with an accuracy of a few nanometers,</a>
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* J. Geodesy 85(8), 475--485 (Aug. 2011);
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* preprint
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* <a href="https://arxiv.org/abs/1002.1417">arXiv:1002.1417</a>.
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*
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* Lee gives the correct results for forward and reverse transformations
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* subject to the branch cut rules (see the description of the \e extendp
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* argument to the constructor). The maximum error is about 8 nm (8
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* nanometers), ground distance, for the forward and reverse transformations.
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* The error in the convergence is 2 × 10<sup>−15</sup>",
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* the relative error in the scale is 7 × 10<sup>−12</sup>%%.
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* See Sec. 3 of
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* <a href="https://arxiv.org/abs/1002.1417">arXiv:1002.1417</a> for details.
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* The method is "exact" in the sense that the errors are close to the
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* round-off limit and that no changes are needed in the algorithms for them
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* to be used with reals of a higher precision. Thus the errors using long
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* double (with a 64-bit fraction) are about 2000 times smaller than using
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* double (with a 53-bit fraction).
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*
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* This algorithm is about 4.5 times slower than the 6th-order Krüger
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* method, TransverseMercator, taking about 11 us for a combined forward and
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* reverse projection on a 2.66 GHz Intel machine (g++, version 4.3.0, -O3).
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*
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* The ellipsoid parameters and the central scale are set in the constructor.
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* The central meridian (which is a trivial shift of the longitude) is
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* specified as the \e lon0 argument of the TransverseMercatorExact::Forward
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* and TransverseMercatorExact::Reverse functions. The latitude of origin is
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* taken to be the equator. See the documentation on TransverseMercator for
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* how to include a false easting, false northing, or a latitude of origin.
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*
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* See <a href="https://geographiclib.sourceforge.io/tm-grid.kmz"
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* type="application/vnd.google-earth.kmz"> tm-grid.kmz</a>, for an
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* illustration of the transverse Mercator grid in Google Earth.
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*
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* This class also returns the meridian convergence \e gamma and scale \e k.
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* The meridian convergence is the bearing of grid north (the \e y axis)
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* measured clockwise from true north.
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*
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* See TransverseMercatorExact.cpp for more information on the
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* implementation.
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*
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* See \ref transversemercator for a discussion of this projection.
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*
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* Example of use:
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* \include example-TransverseMercatorExact.cpp
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*
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* <a href="TransverseMercatorProj.1.html">TransverseMercatorProj</a> is a
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* command-line utility providing access to the functionality of
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* TransverseMercator and TransverseMercatorExact.
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**********************************************************************/
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class GEOGRAPHICLIB_EXPORT TransverseMercatorExact {
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private:
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typedef Math::real real;
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static const int numit_ = 10;
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real tol_, tol2_, taytol_;
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real _a, _f, _k0, _mu, _mv, _e;
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bool _extendp;
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EllipticFunction _Eu, _Ev;
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void zeta(real u, real snu, real cnu, real dnu,
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real v, real snv, real cnv, real dnv,
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real& taup, real& lam) const;
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void dwdzeta(real u, real snu, real cnu, real dnu,
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real v, real snv, real cnv, real dnv,
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real& du, real& dv) const;
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bool zetainv0(real psi, real lam, real& u, real& v) const;
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void zetainv(real taup, real lam, real& u, real& v) const;
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void sigma(real u, real snu, real cnu, real dnu,
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real v, real snv, real cnv, real dnv,
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real& xi, real& eta) const;
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void dwdsigma(real u, real snu, real cnu, real dnu,
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real v, real snv, real cnv, real dnv,
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real& du, real& dv) const;
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bool sigmainv0(real xi, real eta, real& u, real& v) const;
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void sigmainv(real xi, real eta, real& u, real& v) const;
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void Scale(real tau, real lam,
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real snu, real cnu, real dnu,
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real snv, real cnv, real dnv,
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real& gamma, real& k) const;
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public:
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/**
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* Constructor for a ellipsoid with
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*
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* @param[in] a equatorial radius (meters).
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* @param[in] f flattening of ellipsoid.
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* @param[in] k0 central scale factor.
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* @param[in] extendp use extended domain.
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* @exception GeographicErr if \e a, \e f, or \e k0 is not positive.
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*
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* The transverse Mercator projection has a branch point singularity at \e
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* lat = 0 and \e lon − \e lon0 = 90 (1 − \e e) or (for
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* TransverseMercatorExact::UTM) x = 18381 km, y = 0m. The \e extendp
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* argument governs where the branch cut is placed. With \e extendp =
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* false, the "standard" convention is followed, namely the cut is placed
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* along \e x > 18381 km, \e y = 0m. Forward can be called with any \e lat
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* and \e lon then produces the transformation shown in Lee, Fig 46.
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* Reverse analytically continues this in the ± \e x direction. As
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* a consequence, Reverse may map multiple points to the same geographic
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* location; for example, for TransverseMercatorExact::UTM, \e x =
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* 22051449.037349 m, \e y = −7131237.022729 m and \e x =
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* 29735142.378357 m, \e y = 4235043.607933 m both map to \e lat =
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* −2°, \e lon = 88°.
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*
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* With \e extendp = true, the branch cut is moved to the lower left
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* quadrant. The various symmetries of the transverse Mercator projection
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* can be used to explore the projection on any sheet. In this mode the
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* domains of \e lat, \e lon, \e x, and \e y are restricted to
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* - the union of
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* - \e lat in [0, 90] and \e lon − \e lon0 in [0, 90]
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* - \e lat in (-90, 0] and \e lon − \e lon0 in [90 (1 − \e
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e), 90]
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* - the union of
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* - <i>x</i>/(\e k0 \e a) in [0, ∞) and
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* <i>y</i>/(\e k0 \e a) in [0, E(<i>e</i><sup>2</sup>)]
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* - <i>x</i>/(\e k0 \e a) in [K(1 − <i>e</i><sup>2</sup>) −
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* E(1 − <i>e</i><sup>2</sup>), ∞) and <i>y</i>/(\e k0 \e
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* a) in (−∞, 0]
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* .
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* See Sec. 5 of
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* <a href="https://arxiv.org/abs/1002.1417">arXiv:1002.1417</a> for a full
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* discussion of the treatment of the branch cut.
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*
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* The method will work for all ellipsoids used in terrestrial geodesy.
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* The method cannot be applied directly to the case of a sphere (\e f = 0)
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* because some the constants characterizing this method diverge in that
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* limit, and in practice, \e f should be larger than about
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* numeric_limits<real>::epsilon(). However, TransverseMercator treats the
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* sphere exactly.
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**********************************************************************/
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TransverseMercatorExact(real a, real f, real k0, bool extendp = false);
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/**
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* Forward projection, from geographic to transverse Mercator.
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*
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* @param[in] lon0 central meridian of the projection (degrees).
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* @param[in] lat latitude of point (degrees).
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* @param[in] lon longitude of point (degrees).
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* @param[out] x easting of point (meters).
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* @param[out] y northing of point (meters).
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* @param[out] gamma meridian convergence at point (degrees).
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* @param[out] k scale of projection at point.
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*
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* No false easting or northing is added. \e lat should be in the range
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* [−90°, 90°].
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**********************************************************************/
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void Forward(real lon0, real lat, real lon,
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real& x, real& y, real& gamma, real& k) const;
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/**
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* Reverse projection, from transverse Mercator to geographic.
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*
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* @param[in] lon0 central meridian of the projection (degrees).
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* @param[in] x easting of point (meters).
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* @param[in] y northing of point (meters).
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* @param[out] lat latitude of point (degrees).
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* @param[out] lon longitude of point (degrees).
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* @param[out] gamma meridian convergence at point (degrees).
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* @param[out] k scale of projection at point.
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*
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* No false easting or northing is added. The value of \e lon returned is
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* in the range [−180°, 180°].
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**********************************************************************/
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void Reverse(real lon0, real x, real y,
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real& lat, real& lon, real& gamma, real& k) const;
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/**
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* TransverseMercatorExact::Forward without returning the convergence and
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* scale.
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**********************************************************************/
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void Forward(real lon0, real lat, real lon,
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real& x, real& y) const {
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real gamma, k;
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Forward(lon0, lat, lon, x, y, gamma, k);
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}
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/**
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* TransverseMercatorExact::Reverse without returning the convergence and
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* scale.
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**********************************************************************/
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void Reverse(real lon0, real x, real y,
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real& lat, real& lon) const {
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real gamma, k;
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Reverse(lon0, x, y, lat, lon, gamma, k);
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}
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/** \name Inspector functions
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**********************************************************************/
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///@{
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/**
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* @return \e a the equatorial radius of the ellipsoid (meters). This is
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* the value used in the constructor.
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**********************************************************************/
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Math::real EquatorialRadius() const { return _a; }
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/**
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* @return \e f the flattening of the ellipsoid. This is the value used in
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* the constructor.
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**********************************************************************/
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Math::real Flattening() const { return _f; }
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/**
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* @return \e k0 central scale for the projection. This is the value of \e
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* k0 used in the constructor and is the scale on the central meridian.
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**********************************************************************/
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Math::real CentralScale() const { return _k0; }
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/**
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* \deprecated An old name for EquatorialRadius().
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**********************************************************************/
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GEOGRAPHICLIB_DEPRECATED("Use EquatorialRadius()")
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Math::real MajorRadius() const { return EquatorialRadius(); }
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///@}
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/**
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* A global instantiation of TransverseMercatorExact with the WGS84
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* ellipsoid and the UTM scale factor. However, unlike UTM, no false
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* easting or northing is added.
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**********************************************************************/
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static const TransverseMercatorExact& UTM();
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};
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} // namespace GeographicLib
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#endif // GEOGRAPHICLIB_TRANSVERSEMERCATOREXACT_HPP
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