An exact implementation of the transverse Mercator projection. More...

#include <GeographicLib/TransverseMercatorExact.hpp>

Public Member Functions
	TransverseMercatorExact (real a, real f, real k0, bool extendp=false)

void	Forward (real lon0, real lat, real lon, real &x, real &y, real &gamma, real &k) const

void	Reverse (real lon0, real x, real y, real &lat, real &lon, real &gamma, real &k) const

void	Forward (real lon0, real lat, real lon, real &x, real &y) const

void	Reverse (real lon0, real x, real y, real &lat, real &lon) const

Inspector functions
Math::real	EquatorialRadius () const

Math::real	Flattening () const

Math::real	CentralScale () const

Static Public Member Functions
static const TransverseMercatorExact &	UTM ()

Friends
class	TransverseMercator

Detailed Description

An exact implementation of the transverse Mercator projection.

Implementation of the Transverse Mercator Projection given in

L. P. Lee, Conformal Projections Based On Jacobian Elliptic Functions, Part V of Conformal Projections Based on Elliptic Functions, (B. V. Gutsell, Toronto, 1976), 128pp., ISBN: 0919870163 (also appeared as: Monograph 16, Suppl. No. 1 to Canadian Cartographer, Vol 13); borrow from archive.org.
C. F. F. Karney, Transverse Mercator with an accuracy of a few nanometers, J. Geodesy 85(8), 475–485 (Aug. 2011); preprint arXiv:1002.1417.

Lee gives the correct results for forward and reverse transformations subject to the branch cut rules (see the description of the extendp argument to the constructor). The maximum error is about 8 nm (8 nanometers), ground distance, for the forward and reverse transformations. The error in the convergence is 2 × 10⁻¹⁵", the relative error in the scale is 7 × 10⁻¹²%%. See Sec. 3 of arXiv:1002.1417 for details. The method is "exact" in the sense that the errors are close to the round-off limit and that no changes are needed in the algorithms for them to be used with reals of a higher precision. Thus the errors using long double (with a 64-bit fraction) are about 2000 times smaller than using double (with a 53-bit fraction).

This algorithm is about 4.5 times slower than the 6th-order Krüger method, TransverseMercator, taking about 11 us for a combined forward and reverse projection on a 2.66 GHz Intel machine (g++, version 4.3.0, -O3).

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 lon0 argument of the TransverseMercatorExact::Forward and TransverseMercatorExact::Reverse functions. The latitude of origin is taken to be the equator. See the documentation on TransverseMercator for how to include a false easting, false northing, or a latitude of origin.

See tm-grid.kmz, for an illustration of the transverse Mercator grid in Google Earth.

This class also returns the meridian convergence gamma and scale k. The meridian convergence is the bearing of grid north (the y axis) measured clockwise from true north.

See TransverseMercatorExact.cpp for more information on the implementation.

See Transverse Mercator projection for a discussion of this projection.

Example of use:

// Example of using the GeographicLib::TransverseMercatorExact class
 
#include <iostream>
#include <iomanip>
#include <exception>
#include <GeographicLib/TransverseMercatorExact.hpp>
 
using namespace std;
using namespace GeographicLib;
 
int main() {
  try {
    TransverseMercatorExact proj(Constants::WGS84_a(), Constants::WGS84_f(),
                                 Constants::UTM_k0());
    // Alternatively:
    // const TransverseMercatorExact& proj = TransverseMercatorExact::UTM();
    double lon0 = -75;          // Central meridian for UTM zone 18
    {
      // Sample forward calculation
      double lat = 40.3, lon = -74.7; // Princeton, NJ
      double x, y;
      proj.Forward(lon0, lat, lon, x, y);
      cout << x << " " << y << "\n";
    }
    {
      // Sample reverse calculation
      double x = 25e3, y = 4461e3;
      double lat, lon;
      proj.Reverse(lon0, x, y, lat, lon);
      cout << lat << " " << lon << "\n";
    }
  }
  catch (const exception& e) {
    cerr << "Caught exception: " << e.what() << "\n";
    return 1;
  }
}

TransverseMercatorProj is a command-line utility providing access to the functionality of TransverseMercator and TransverseMercatorExact.

Definition at line 85 of file TransverseMercatorExact.hpp.

Constructor & Destructor Documentation

◆ TransverseMercatorExact()

GeographicLib::TransverseMercatorExact::TransverseMercatorExact	(	real	a,
		real	f,
		real	k0,
		bool	extendp = `false`
	)

Constructor for an ellipsoid with

Parameters

[in]	a	equatorial radius (meters).
[in]	f	flattening of ellipsoid.
[in]	k0	central scale factor.
[in]	extendp	if true, use extended domain (default false).

Exceptions

GeographicErr if a, f, or k0 is not positive.

The transverse Mercator projection has a branch point singularity at lat = 0 and lon − lon0 = 90 (1 − e) or (for TransverseMercatorExact::UTM) x = 18381 km, y = 0m. The extendp argument governs where the branch cut is placed. With extendp = false, the "standard" convention is followed, namely the cut is placed along x > 18381 km, y = 0m. Forward can be called with any lat and lon then produces the transformation shown in Lee, Fig 46. Reverse analytically continues this in the ± x direction. As a consequence, Reverse may map multiple points to the same geographic location; for example, for TransverseMercatorExact::UTM, x = 22051449.037349 m, y = −7131237.022729 m and x = 29735142.378357 m, y = 4235043.607933 m both map to lat = −2°, lon = 88°.

With extendp = true, the branch cut is moved to the lower left quadrant. The various symmetries of the transverse Mercator projection can be used to explore the projection on any sheet. In this mode the domains of lat, lon, x, and y are restricted to

the union of
- lat in [0, 90] and lon − lon0 in [0, 90]
- lat in (-90, 0] and lon − lon0 in [90 (1 − e), 90]
the union of
- x/(k0 a) in [0, ∞) and y/(k0 a) in [0, E(e²)]
- x/(k0 a) in [K(1 − e²) − E(1 − e²), ∞) and y/(k0 a) in (−∞, 0]

See Sec. 5 of arXiv:1002.1417 for a full discussion of the treatment of the branch cut.

The method will work for all ellipsoids used in terrestrial geodesy. The method cannot be applied directly to the case of a sphere (f = 0) because some the constants characterizing this method diverge in that limit, and in practice, f should be larger than about numeric_limits<real>::epsilon(). However, TransverseMercator treats the sphere exactly.

Definition at line 50 of file TransverseMercatorExact.cpp.

Member Function Documentation

◆ Forward() [1/2]

void GeographicLib::TransverseMercatorExact::Forward	(	real	lon0,
		real	lat,
		real	lon,
		real &	x,
		real &	y,
		real &	gamma,
		real &	k
	)		const

Forward projection, from geographic to transverse Mercator.

Parameters

[in]	lon0	central meridian of the projection (degrees).
[in]	lat	latitude of point (degrees).
[in]	lon	longitude of point (degrees).
[out]	x	easting of point (meters).
[out]	y	northing of point (meters).
[out]	gamma	meridian convergence at point (degrees).
[out]	k	scale of projection at point.

No false easting or northing is added. lat should be in the range [−90°, 90°].

Definition at line 353 of file TransverseMercatorExact.cpp.

References GeographicLib::EllipticFunction::am(), GeographicLib::Math::AngDiff(), GeographicLib::Math::degree(), GeographicLib::EllipticFunction::E(), GeographicLib::Math::hd, GeographicLib::EllipticFunction::K(), GeographicLib::Math::LatFix(), GeographicLib::Math::qd, GeographicLib::Math::tand(), GeographicLib::Math::tauf(), and GeographicLib::Math::taupf().

Referenced by GeographicLib::TransverseMercator::Forward().

◆ Reverse() [1/2]

void GeographicLib::TransverseMercatorExact::Reverse	(	real	lon0,
		real	x,
		real	y,
		real &	lat,
		real &	lon,
		real &	gamma,
		real &	k
	)		const

Reverse projection, from transverse Mercator to geographic.

Parameters

[in]	lon0	central meridian of the projection (degrees).
[in]	x	easting of point (meters).
[in]	y	northing of point (meters).
[out]	lat	latitude of point (degrees).
[out]	lon	longitude of point (degrees).
[out]	gamma	meridian convergence at point (degrees).
[out]	k	scale of projection at point.

No false easting or northing is added. The value of lon returned is in the range [−180°, 180°].

Definition at line 413 of file TransverseMercatorExact.cpp.

References GeographicLib::EllipticFunction::am(), GeographicLib::Math::AngNormalize(), GeographicLib::Math::degree(), GeographicLib::EllipticFunction::E(), GeographicLib::Math::hd, GeographicLib::EllipticFunction::K(), GeographicLib::EllipticFunction::KE(), GeographicLib::Math::qd, and GeographicLib::Math::tauf().

Referenced by GeographicLib::TransverseMercator::Reverse().

◆ Forward() [2/2]

void GeographicLib::TransverseMercatorExact::Forward	(	real	lon0,
		real	lat,
		real	lon,
		real &	x,
		real &	y
	)		const

inline

TransverseMercatorExact::Forward without returning the convergence and scale.

Definition at line 214 of file TransverseMercatorExact.hpp.

◆ Reverse() [2/2]

void GeographicLib::TransverseMercatorExact::Reverse	(	real	lon0,
		real	x,
		real	y,
		real &	lat,
		real &	lon
	)		const

inline

TransverseMercatorExact::Reverse without returning the convergence and scale.

Definition at line 224 of file TransverseMercatorExact.hpp.

◆ EquatorialRadius()

Math::real GeographicLib::TransverseMercatorExact::EquatorialRadius ( ) const

inline

Returns: a the equatorial radius of the ellipsoid (meters). This is the value used in the constructor.

Definition at line 237 of file TransverseMercatorExact.hpp.

◆ Flattening()

Math::real GeographicLib::TransverseMercatorExact::Flattening ( ) const

inline

Returns: f the flattening of the ellipsoid. This is the value used in the constructor.

Definition at line 243 of file TransverseMercatorExact.hpp.

◆ CentralScale()

Math::real GeographicLib::TransverseMercatorExact::CentralScale ( ) const

inline

Returns: k0 central scale for the projection. This is the value of k0 used in the constructor and is the scale on the central meridian.

Definition at line 249 of file TransverseMercatorExact.hpp.

◆ UTM()

const TransverseMercatorExact & GeographicLib::TransverseMercatorExact::UTM ( )

static

A global instantiation of TransverseMercatorExact with the WGS84 ellipsoid and the UTM scale factor. However, unlike UTM, no false easting or northing is added.

Definition at line 75 of file TransverseMercatorExact.cpp.

References GeographicLib::Constants::UTM_k0(), GeographicLib::Constants::WGS84_a(), and GeographicLib::Constants::WGS84_f().