Gnomonic projection More...

#include <Gnomonic.hpp>

Public Member Functions
void	Forward (real lat0, real lon0, real lat, real lon, real &x, real &y, real &azi, real &rk) const

void	Forward (real lat0, real lon0, real lat, real lon, real &x, real &y) const

	Gnomonic (const Geodesic &earth=Geodesic::WGS84())

void	Reverse (real lat0, real lon0, real x, real y, real &lat, real &lon, real &azi, real &rk) const

void	Reverse (real lat0, real lon0, real x, real y, real &lat, real &lon) const

Inspector functions
Math::real	MajorRadius () const

Math::real	Flattening () const

Private Types
typedef Math::real	real

Private Attributes
real	_a

Geodesic	_earth

real	_f

real	eps0_

real	eps_

Static Private Attributes
static const int	numit_ = 10

Detailed Description

Gnomonic projection

Gnomonic projection centered at an arbitrary position C on the ellipsoid. This projection is derived in Section 8 of

C. F. F. Karney, Algorithms for geodesics, J. Geodesy 87, 43–55 (2013); DOI: 10.1007/s00190-012-0578-z; addenda: geod-addenda.html.

The projection of P is defined as follows: compute the geodesic line from C to P; compute the reduced length m12, geodesic scale M12, and ρ = m12/M12; finally x = ρ sin azi1; y = ρ cos azi1, where azi1 is the azimuth of the geodesic at C. The Gnomonic::Forward and Gnomonic::Reverse methods also return the azimuth azi of the geodesic at P and reciprocal scale rk in the azimuthal direction. The scale in the radial direction if 1/rk².

For a sphere, ρ is reduces to a tan(s12/a), where s12 is the length of the geodesic from C to P, and the gnomonic projection has the property that all geodesics appear as straight lines. For an ellipsoid, this property holds only for geodesics interesting the centers. However geodesic segments close to the center are approximately straight.

Consider a geodesic segment of length l. Let T be the point on the geodesic (extended if necessary) closest to C the center of the projection and t be the distance CT. To lowest order, the maximum deviation (as a true distance) of the corresponding gnomonic line segment (i.e., with the same end points) from the geodesic is

(K(T) - K(C)) l² t / 32.

where K is the Gaussian curvature.

This result applies for any surface. For an ellipsoid of revolution, consider all geodesics whose end points are within a distance r of C. For a given r, the deviation is maximum when the latitude of C is 45°, when endpoints are a distance r away, and when their azimuths from the center are ± 45° or ± 135°. To lowest order in r and the flattening f, the deviation is f (r/2a)³ r.

The conversions all take place using a Geodesic object (by default Geodesic::WGS84()). For more information on geodesics see geodesic.

Warning

The definition of this projection for a sphere is standard. However, there is no standard for how it should be extended to an ellipsoid. The choices are:

Declare that the projection is undefined for an ellipsoid.
Project to a tangent plane from the center of the ellipsoid. This causes great ellipses to appear as straight lines in the projection; i.e., it generalizes the spherical great circle to a great ellipse. This was proposed by independently by Bowring and Williams in 1997.
Project to the conformal sphere with the constant of integration chosen so that the values of the latitude match for the center point and perform a central projection onto the plane tangent to the conformal sphere at the center point. This causes normal sections through the center point to appear as straight lines in the projection; i.e., it generalizes the spherical great circle to a normal section. This was proposed by I. G. Letoval'tsev, Generalization of the gnomonic projection for a spheroid and the principal geodetic problems involved in the alignment of surface routes, Geodesy and Aerophotography (5), 271–274 (1963).
The projection given here. This causes geodesics close to the center point to appear as straight lines in the projection; i.e., it generalizes the spherical great circle to a geodesic.

Example of use:

// Example of using the GeographicLib::Gnomonic class
#include <iostream>
#include <exception>
#include <GeographicLib/Geodesic.hpp>
#include <GeographicLib/Gnomonic.hpp>
using namespace std;
using namespace GeographicLib;
int main() {
  try {
    Geodesic geod(Constants::WGS84_a(), Constants::WGS84_f());
    // Alternatively: const Geodesic& geod = Geodesic::WGS84();
    const double lat0 = 48 + 50/60.0, lon0 = 2 + 20/60.0; // Paris
    Gnomonic proj(geod);
    {
      // Sample forward calculation
      double lat = 50.9, lon = 1.8; // Calais
      double x, y;
      proj.Forward(lat0, lon0, lat, lon, x, y);
      cout << x << " " << y << "\n";
    }
    {
      // Sample reverse calculation
      double x = -38e3, y = 230e3;
      double lat, lon;
      proj.Reverse(lat0, lon0, x, y, lat, lon);
      cout << lat << " " << lon << "\n";
    }
  }
  catch (const exception& e) {
    cerr << "Caught exception: " << e.what() << "\n";
    return 1;
  }
}

GeodesicProj is a command-line utility providing access to the functionality of AzimuthalEquidistant, Gnomonic, and CassiniSoldner.

Definition at line 102 of file Gnomonic.hpp.

Member Typedef Documentation

typedef Math::real GeographicLib::Gnomonic::real

private

Definition at line 104 of file Gnomonic.hpp.

Constructor & Destructor Documentation

GeographicLib::Gnomonic::Gnomonic ( const Geodesic & earth = Geodesic::WGS84() )

explicit

Constructor for Gnomonic.

Parameters

[in] earth the Geodesic object to use for geodesic calculations. By default this uses the WGS84 ellipsoid.

Definition at line 22 of file src/Gnomonic.cpp.

Member Function Documentation

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

inline

Returns: f the flattening of the ellipsoid. This is the value inherited from the Geodesic object used in the constructor.

Definition at line 200 of file Gnomonic.hpp.

void GeographicLib::Gnomonic::Forward	(	real	lat0,
		real	lon0,
		real	lat,
		real	lon,
		real &	x,
		real &	y,
		real &	azi,
		real &	rk
	)		const

Forward projection, from geographic to gnomonic.

Parameters

[in]	lat0	latitude of center point of projection (degrees).
[in]	lon0	longitude of center point of 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]	azi	azimuth of geodesic at point (degrees).
[out]	rk	reciprocal of azimuthal scale at point.

lat0 and lat should be in the range [−90°, 90°]. The scale of the projection is 1/rk² in the "radial" direction, azi clockwise from true north, and is 1/rk in the direction perpendicular to this. If the point lies "over the horizon", i.e., if rk ≤ 0, then NaNs are returned for x and y (the correct values are returned for azi and rk). A call to Forward followed by a call to Reverse will return the original (lat, lon) (to within roundoff) provided the point in not over the horizon.

Definition at line 30 of file src/Gnomonic.cpp.

void GeographicLib::Gnomonic::Forward	(	real	lat0,
		real	lon0,
		real	lat,
		real	lon,
		real &	x,
		real &	y
	)		const

inline

Gnomonic::Forward without returning the azimuth and scale.

Definition at line 172 of file Gnomonic.hpp.

Math::real GeographicLib::Gnomonic::MajorRadius ( ) const

inline

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

Definition at line 194 of file Gnomonic.hpp.

void GeographicLib::Gnomonic::Reverse	(	real	lat0,
		real	lon0,
		real	x,
		real	y,
		real &	lat,
		real &	lon,
		real &	azi,
		real &	rk
	)		const

Reverse projection, from gnomonic to geographic.

Parameters

[in]	lat0	latitude of center point of projection (degrees).
[in]	lon0	longitude of center point of 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]	azi	azimuth of geodesic at point (degrees).
[out]	rk	reciprocal of azimuthal scale at point.

lat0 should be in the range [−90°, 90°]. lat will be in the range [−90°, 90°] and lon will be in the range [−180°, 180°]. The scale of the projection is 1/rk² in the "radial" direction, azi clockwise from true north, and is 1/rk in the direction perpendicular to this. Even though all inputs should return a valid lat and lon, it's possible that the procedure fails to converge for very large x or y; in this case NaNs are returned for all the output arguments. A call to Reverse followed by a call to Forward will return the original (x, y) (to roundoff).

Definition at line 47 of file src/Gnomonic.cpp.

void GeographicLib::Gnomonic::Reverse	(	real	lat0,
		real	lon0,
		real	x,
		real	y,
		real &	lat,
		real &	lon
	)		const

inline

Gnomonic::Reverse without returning the azimuth and scale.

Definition at line 181 of file Gnomonic.hpp.

Member Data Documentation

real GeographicLib::Gnomonic::_a

private

Definition at line 107 of file Gnomonic.hpp.

Geodesic GeographicLib::Gnomonic::_earth

private

Definition at line 106 of file Gnomonic.hpp.

real GeographicLib::Gnomonic::_f

private

Definition at line 107 of file Gnomonic.hpp.

real GeographicLib::Gnomonic::eps0_

private

Definition at line 105 of file Gnomonic.hpp.

real GeographicLib::Gnomonic::eps_

private

Definition at line 105 of file Gnomonic.hpp.

const int GeographicLib::Gnomonic::numit_ = 10

staticprivate

Definition at line 108 of file Gnomonic.hpp.

The documentation for this class was generated from the following files:

Public Member Functions

Private Types

Private Attributes

Static Private Attributes

Detailed Description

Member Typedef Documentation

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation