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TransverseMercator.hpp
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/**
* \file TransverseMercator.hpp
* \brief Header for GeographicLib::TransverseMercator class
*
* Copyright (c) Charles Karney (2008-2023) <karney@alum.mit.edu> and licensed
* under the MIT/X11 License. For more information, see
* https://geographiclib.sourceforge.io/
**********************************************************************/
#if !defined(GEOGRAPHICLIB_TRANSVERSEMERCATOR_HPP)
#define GEOGRAPHICLIB_TRANSVERSEMERCATOR_HPP 1
#include <GeographicLib/Constants.hpp>
#include <GeographicLib/TransverseMercatorExact.hpp>
#if !defined(GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER)
/**
* The order of the series approximation used in TransverseMercator.
* GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER can be set to any integer in [4, 8].
**********************************************************************/
# define GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER \
(GEOGRAPHICLIB_PRECISION == 2 ? 6 : \
(GEOGRAPHICLIB_PRECISION == 1 ? 4 : 8))
#endif
namespace GeographicLib {
/**
* \brief Transverse Mercator projection
*
* This uses Kr&uuml;ger's method which evaluates the projection and its
* inverse in terms of a series. See
* - L. Kr&uuml;ger,
* <a href="https://doi.org/10.2312/GFZ.b103-krueger28"> Konforme
* Abbildung des Erdellipsoids in der Ebene</a> (Conformal mapping of the
* ellipsoidal earth to the plane), Royal Prussian Geodetic Institute, New
* Series 52, 172 pp. (1912).
* - C. F. F. Karney,
* <a href="https://doi.org/10.1007/s00190-011-0445-3">
* Transverse Mercator with an accuracy of a few nanometers,</a>
* J. Geodesy 85(8), 475--485 (Aug. 2011);
* preprint
* <a href="https://arxiv.org/abs/1002.1417">arXiv:1002.1417</a>.
*
* Kr&uuml;ger's method has been extended from 4th to 6th order. The maximum
* error is 5 nm (5 nanometers), ground distance, for all positions within 35
* degrees of the central meridian. The error in the convergence is 2
* &times; 10<sup>&minus;15</sup>&quot; and the relative error in the scale
* is 6 &times; 10<sup>&minus;12</sup>%%. See Sec. 4 of
* <a href="https://arxiv.org/abs/1002.1417">arXiv:1002.1417</a> for details.
* The speed penalty in going to 6th order is only about 1%.
*
* There's a singularity in the projection at &phi; = 0&deg;, &lambda;
* &minus; &lambda;<sub>0</sub> = &plusmn;(1 &minus; \e e)90&deg; (&asymp;
* &plusmn;82.6&deg; for the WGS84 ellipsoid), where \e e is the
* eccentricity. Beyond this point, the series ceases to converge and the
* results from this method will be garbage. To be on the safe side, don't
* use this method if the angular distance from the central meridian exceeds
* (1 &minus; 2e)90&deg; (&asymp; 75&deg; for the WGS84 ellipsoid)
*
* TransverseMercatorExact is an alternative implementation of the projection
* using exact formulas which yield accurate (to 8 nm) results over the
* entire ellipsoid. This formulation is accessible in this class by calling
* the constructor with \e exact = true.
*
* The ellipsoid parameters and the central scale are set in the constructor.
* The central meridian (which is a trivial shift of the longitude) is
* specified as the \e lon0 argument of the TransverseMercator::Forward and
* TransverseMercator::Reverse functions. The latitude of origin is taken to
* be the equator. There is no provision in this class for specifying a
* false easting or false northing or a different latitude of origin.
* However these are can be simply included by the calling function. For
* example, the UTMUPS class applies the false easting and false northing for
* the UTM projections. A more complicated example is the British National
* Grid (<a href="https://www.spatialreference.org/ref/epsg/7405/">
* EPSG:7405</a>) which requires the use of a latitude of origin. This is
* implemented by the GeographicLib::OSGB class.
*
* This class also returns the meridian convergence \e gamma and scale \e k.
* The meridian convergence is the bearing of grid north (the \e y axis)
* measured clockwise from true north.
*
* See TransverseMercator.cpp for more information on the implementation.
*
* See \ref transversemercator for a discussion of this projection.
*
* Example of use:
* \include example-TransverseMercator.cpp
*
* <a href="TransverseMercatorProj.1.html">TransverseMercatorProj</a> is a
* command-line utility providing access to the functionality of this class.
**********************************************************************/
class GEOGRAPHICLIB_EXPORT TransverseMercator {
private:
typedef Math::real real;
static const int maxpow_ = GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER;
static const int numit_ = 5;
real _a, _f, _k0;
bool _exact;
real _e2, _es, _e2m, _c, _n;
// _alp[0] and _bet[0] unused
real _a1, _b1, _alp[maxpow_ + 1], _bet[maxpow_ + 1];
TransverseMercatorExact _tmexact;
public:
/**
* Constructor for an ellipsoid with
*
* @param[in] a equatorial radius (meters).
* @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere.
* Negative \e f gives a prolate ellipsoid.
* @param[in] k0 central scale factor.
* @param[in] exact if true use exact formulation in terms of elliptic
* functions instead of series expansions (default false).
* @param[in] extendp use extended domain (default false); should only be
* used if \e exact = true;
* @exception GeographicErr if \e a, (1 &minus; \e f) \e a, or \e k0 is
* not positive.
*
* With \e exact = true, this class delegates the calculations to the
* TransverseMercatorExact classes which compute the projection in terms of
* elliptic functions.
**********************************************************************/
TransverseMercator(real a, real f, real k0,
bool exact = false, bool extendp = false);
/**
* Forward projection, from geographic to transverse Mercator.
*
* @param[in] lon0 central meridian of the projection (degrees).
* @param[in] lat latitude of point (degrees).
* @param[in] lon longitude of point (degrees).
* @param[out] x easting of point (meters).
* @param[out] y northing of point (meters).
* @param[out] gamma meridian convergence at point (degrees).
* @param[out] k scale of projection at point.
*
* No false easting or northing is added. \e lat should be in the range
* [&minus;90&deg;, 90&deg;].
**********************************************************************/
void Forward(real lon0, real lat, real lon,
real& x, real& y, real& gamma, real& k) const;
/**
* Reverse projection, from transverse Mercator to geographic.
*
* @param[in] lon0 central meridian of the projection (degrees).
* @param[in] x easting of point (meters).
* @param[in] y northing of point (meters).
* @param[out] lat latitude of point (degrees).
* @param[out] lon longitude of point (degrees).
* @param[out] gamma meridian convergence at point (degrees).
* @param[out] k scale of projection at point.
*
* No false easting or northing is added. The value of \e lon returned is
* in the range [&minus;180&deg;, 180&deg;].
**********************************************************************/
void Reverse(real lon0, real x, real y,
real& lat, real& lon, real& gamma, real& k) const;
/**
* TransverseMercator::Forward without returning the convergence and scale.
**********************************************************************/
void Forward(real lon0, real lat, real lon,
real& x, real& y) const {
real gamma, k;
Forward(lon0, lat, lon, x, y, gamma, k);
}
/**
* TransverseMercator::Reverse without returning the convergence and scale.
**********************************************************************/
void Reverse(real lon0, real x, real y,
real& lat, real& lon) const {
real gamma, k;
Reverse(lon0, x, y, lat, lon, gamma, k);
}
/** \name Inspector functions
**********************************************************************/
///@{
/**
* @return \e a the equatorial radius of the ellipsoid (meters). This is
* the value used in the constructor.
**********************************************************************/
Math::real EquatorialRadius() const { return _a; }
/**
* @return \e f the flattening of the ellipsoid. This is the value used in
* the constructor.
**********************************************************************/
Math::real Flattening() const { return _f; }
/**
* @return \e k0 central scale for the projection. This is the value of \e
* k0 used in the constructor and is the scale on the central meridian.
**********************************************************************/
Math::real CentralScale() const { return _k0; }
/**
* @return \e exact whether the exact formulation is used. This is the
* value used in the constructor.
**********************************************************************/
bool Exact() const { return _exact; }
///@}
/**
* A global instantiation of TransverseMercator with the WGS84 ellipsoid
* and the UTM scale factor. However, unlike UTM, no false easting or
* northing is added.
**********************************************************************/
static const TransverseMercator& UTM();
};
} // namespace GeographicLib
#endif // GEOGRAPHICLIB_TRANSVERSEMERCATOR_HPP
Constants.hpp
Header for GeographicLib::Constants class.
GEOGRAPHICLIB_EXPORT
#define GEOGRAPHICLIB_EXPORT
Definition Constants.hpp:59
TransverseMercatorExact.hpp
Header for GeographicLib::TransverseMercatorExact class.
GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER
#define GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER
Definition TransverseMercator.hpp:21
GeographicLib::Math::real
double real
Definition Math.hpp:115
GeographicLib::TransverseMercatorExact
An exact implementation of the transverse Mercator projection.
Definition TransverseMercatorExact.hpp:85
GeographicLib::TransverseMercator
Transverse Mercator projection.
Definition TransverseMercator.hpp:94
GeographicLib::TransverseMercator::TransverseMercator
TransverseMercator(real a, real f, real k0, bool exact=false, bool extendp=false)
Definition TransverseMercator.cpp:44
GeographicLib::TransverseMercator::Reverse
void Reverse(real lon0, real x, real y, real &lat, real &lon, real &gamma, real &k) const
Definition TransverseMercator.cpp:523
GeographicLib::TransverseMercator::Flattening
Math::real Flattening() const
Definition TransverseMercator.hpp:193
GeographicLib::TransverseMercator::Exact
bool Exact() const
Definition TransverseMercator.hpp:205
GeographicLib::TransverseMercator::Reverse
void Reverse(real lon0, real x, real y, real &lat, real &lon) const
Definition TransverseMercator.hpp:174
GeographicLib::TransverseMercator::CentralScale
Math::real CentralScale() const
Definition TransverseMercator.hpp:199
GeographicLib::TransverseMercator::Forward
void Forward(real lon0, real lat, real lon, real &x, real &y) const
Definition TransverseMercator.hpp:165
GeographicLib::TransverseMercator::EquatorialRadius
Math::real EquatorialRadius() const
Definition TransverseMercator.hpp:187
GeographicLib::TransverseMercator::Forward
void Forward(real lon0, real lat, real lon, real &x, real &y, real &gamma, real &k) const
Definition TransverseMercator.cpp:355
GeographicLib
Namespace for GeographicLib.
Definition Accumulator.cpp:12