GeographicLib  1.35
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GeographicLib::GeodesicExact Class Reference

Exact geodesic calculations. More...

#include <GeographicLib/GeodesicExact.hpp>

Public Types

enum  mask {
  NONE, LATITUDE, LONGITUDE, AZIMUTH,
  DISTANCE, DISTANCE_IN, REDUCEDLENGTH, GEODESICSCALE,
  AREA, ALL
}
 

Public Member Functions

Constructor
 GeodesicExact (real a, real f)
 
Direct geodesic problem specified in terms of distance.
Math::real Direct (real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2, real &azi2, real &m12, real &M12, real &M21, real &S12) const throw ()
 
Math::real Direct (real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2) const throw ()
 
Math::real Direct (real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2, real &azi2) const throw ()
 
Math::real Direct (real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2, real &azi2, real &m12) const throw ()
 
Math::real Direct (real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2, real &azi2, real &M12, real &M21) const throw ()
 
Math::real Direct (real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2, real &azi2, real &m12, real &M12, real &M21) const throw ()
 
Direct geodesic problem specified in terms of arc length.
void ArcDirect (real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2, real &s12, real &m12, real &M12, real &M21, real &S12) const throw ()
 
void ArcDirect (real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2) const throw ()
 
void ArcDirect (real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2) const throw ()
 
void ArcDirect (real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2, real &s12) const throw ()
 
void ArcDirect (real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2, real &s12, real &m12) const throw ()
 
void ArcDirect (real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2, real &s12, real &M12, real &M21) const throw ()
 
void ArcDirect (real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2, real &s12, real &m12, real &M12, real &M21) const throw ()
 
General version of the direct geodesic solution.
Math::real GenDirect (real lat1, real lon1, real azi1, bool arcmode, real s12_a12, unsigned outmask, real &lat2, real &lon2, real &azi2, real &s12, real &m12, real &M12, real &M21, real &S12) const throw ()
 
Inverse geodesic problem.
Math::real Inverse (real lat1, real lon1, real lat2, real lon2, real &s12, real &azi1, real &azi2, real &m12, real &M12, real &M21, real &S12) const throw ()
 
Math::real Inverse (real lat1, real lon1, real lat2, real lon2, real &s12) const throw ()
 
Math::real Inverse (real lat1, real lon1, real lat2, real lon2, real &azi1, real &azi2) const throw ()
 
Math::real Inverse (real lat1, real lon1, real lat2, real lon2, real &s12, real &azi1, real &azi2) const throw ()
 
Math::real Inverse (real lat1, real lon1, real lat2, real lon2, real &s12, real &azi1, real &azi2, real &m12) const throw ()
 
Math::real Inverse (real lat1, real lon1, real lat2, real lon2, real &s12, real &azi1, real &azi2, real &M12, real &M21) const throw ()
 
Math::real Inverse (real lat1, real lon1, real lat2, real lon2, real &s12, real &azi1, real &azi2, real &m12, real &M12, real &M21) const throw ()
 
General version of inverse geodesic solution.
Math::real GenInverse (real lat1, real lon1, real lat2, real lon2, unsigned outmask, real &s12, real &azi1, real &azi2, real &m12, real &M12, real &M21, real &S12) const throw ()
 
Interface to GeodesicLineExact.
GeodesicLineExact Line (real lat1, real lon1, real azi1, unsigned caps=ALL) const throw ()
 
Inspector functions.
Math::real MajorRadius () const throw ()
 
Math::real Flattening () const throw ()
 
Math::real EllipsoidArea () const throw ()
 

Static Public Attributes

static const GeodesicExact WGS84
 

Friends

class GeodesicLineExact
 

Detailed Description

Exact geodesic calculations.

The equations for geodesics on an ellipsoid can be expressed in terms of incomplete elliptic integrals. The Geodesic class expands these integrals in a series in the flattening f and this provides an accurate solution for f ∈ [-0.01, 0.01]. The GeodesicExact class computes the ellitpic integrals directly and so provides a solution which is valid for all f. However, in practice, its use should be limited to about b/a [0.01, 100] or f ∈ [-99, 0.99].

For the WGS84 ellipsoid, these classes are 2–3 times slower than the series solution and 2–3 times less accurate (because it's less easy to control round-off errors with the elliptic integral formulation); i.e., the error is about 40 nm (40 nanometers) instead of 15 nm. However the error in the series solution scales as f7 while the error in the elliptic integral solution depends weakly on f. If the quarter meridian distance is 10000 km and the ratio b/a = 1 − f is varied then the approximate maximum error (expressed as a distance) is

      1 - f  error (nm)
      1/128     387
      1/64      345
      1/32      269
      1/16      210
      1/8       115
      1/4        69
      1/2        36
        1        15
        2        25
        4        96
        8       318
       16       985
       32      2352
       64      6008
      128     19024

The computation of the area in these classes is via a 30th order series. This gives accurate results for b/a [1/2, 2]; the accuracy is about 8 decimal digits for b/a [1/4, 4].

See geodellip for the formulation. See the documentation on the Geodesic class for additional information on the geodesic problems.

Example of use:

// Example of using the GeographicLib::GeodesicExact class
#include <iostream>
#include <exception>
using namespace std;
using namespace GeographicLib;
int main() {
try {
// Alternatively: const GeodesicExact& geod = GeodesicExact::WGS84;
{
// Sample direct calculation, travelling about NE from JFK
double lat1 = 40.6, lon1 = -73.8, s12 = 5.5e6, azi1 = 51;
double lat2, lon2;
geod.Direct(lat1, lon1, azi1, s12, lat2, lon2);
cout << lat2 << " " << lon2 << "\n";
}
{
// Sample inverse calculation, JFK to LHR
double
lat1 = 40.6, lon1 = -73.8, // JFK Airport
lat2 = 51.6, lon2 = -0.5; // LHR Airport
double s12;
geod.Inverse(lat1, lon1, lat2, lon2, s12);
cout << s12 << "\n";
}
}
catch (const exception& e) {
cerr << "Caught exception: " << e.what() << "\n";
return 1;
}
return 0;
}

GeodSolve is a command-line utility providing access to the functionality of GeodesicExact and GeodesicLineExact (via the -E option).

Definition at line 80 of file GeodesicExact.hpp.

Member Enumeration Documentation

Bit masks for what calculations to do. These masks do double duty. They signify to the GeodesicLineExact::GeodesicLineExact constructor and to GeodesicExact::Line what capabilities should be included in the GeodesicLineExact object. They also specify which results to return in the general routines GeodesicExact::GenDirect and GeodesicExact::GenInverse routines. GeodesicLineExact::mask is a duplication of this enum.

Enumerator
NONE 

No capabilities, no output.

LATITUDE 

Calculate latitude lat2. (It's not necessary to include this as a capability to GeodesicLineExact because this is included by default.)

LONGITUDE 

Calculate longitude lon2.

AZIMUTH 

Calculate azimuths azi1 and azi2. (It's not necessary to include this as a capability to GeodesicLineExact because this is included by default.)

DISTANCE 

Calculate distance s12.

DISTANCE_IN 

Allow distance s12 to be used as input in the direct geodesic problem.

REDUCEDLENGTH 

Calculate reduced length m12.

GEODESICSCALE 

Calculate geodesic scales M12 and M21.

AREA 

Calculate area S12.

ALL 

All capabilities, calculate everything.

Definition at line 170 of file GeodesicExact.hpp.

Constructor & Destructor Documentation

GeographicLib::GeodesicExact::GeodesicExact ( real  a,
real  f 
)

Constructor for a ellipsoid with

Parameters
[in]aequatorial radius (meters).
[in]fflattening of ellipsoid. Setting f = 0 gives a sphere. Negative f gives a prolate ellipsoid. If f > 1, set flattening to 1/f.
Exceptions
GeographicErrif a or (1 − f ) a is not positive.

Definition at line 55 of file GeodesicExact.cpp.

References GeographicLib::Math::isfinite().

Member Function Documentation

Math::real GeographicLib::GeodesicExact::Direct ( real  lat1,
real  lon1,
real  azi1,
real  s12,
real &  lat2,
real &  lon2,
real &  azi2,
real &  m12,
real &  M12,
real &  M21,
real &  S12 
) const
throw (
)
inline

Perform the direct geodesic calculation where the length of the geodesic is specified in terms of distance.

Parameters
[in]lat1latitude of point 1 (degrees).
[in]lon1longitude of point 1 (degrees).
[in]azi1azimuth at point 1 (degrees).
[in]s12distance between point 1 and point 2 (meters); it can be signed.
[out]lat2latitude of point 2 (degrees).
[out]lon2longitude of point 2 (degrees).
[out]azi2(forward) azimuth at point 2 (degrees).
[out]m12reduced length of geodesic (meters).
[out]M12geodesic scale of point 2 relative to point 1 (dimensionless).
[out]M21geodesic scale of point 1 relative to point 2 (dimensionless).
[out]S12area under the geodesic (meters2).
Returns
a12 arc length of between point 1 and point 2 (degrees).

lat1 should be in the range [−90°, 90°]; lon1 and azi1 should be in the range [−540°, 540°). The values of lon2 and azi2 returned are in the range [−180°, 180°).

If either point is at a pole, the azimuth is defined by keeping the longitude fixed, writing lat = ±(90° − ε), and taking the limit ε → 0+. An arc length greater that 180° signifies a geodesic which is not a shortest path. (For a prolate ellipsoid, an additional condition is necessary for a shortest path: the longitudinal extent must not exceed of 180°.)

The following functions are overloaded versions of GeodesicExact::Direct which omit some of the output parameters. Note, however, that the arc length is always computed and returned as the function value.

Definition at line 282 of file GeodesicExact.hpp.

Referenced by main().

Math::real GeographicLib::GeodesicExact::Direct ( real  lat1,
real  lon1,
real  azi1,
real  s12,
real &  lat2,
real &  lon2 
) const
throw (
)
inline

See the documentation for GeodesicExact::Direct.

Definition at line 296 of file GeodesicExact.hpp.

Math::real GeographicLib::GeodesicExact::Direct ( real  lat1,
real  lon1,
real  azi1,
real  s12,
real &  lat2,
real &  lon2,
real &  azi2 
) const
throw (
)
inline

See the documentation for GeodesicExact::Direct.

Definition at line 308 of file GeodesicExact.hpp.

Math::real GeographicLib::GeodesicExact::Direct ( real  lat1,
real  lon1,
real  azi1,
real  s12,
real &  lat2,
real &  lon2,
real &  azi2,
real &  m12 
) const
throw (
)
inline

See the documentation for GeodesicExact::Direct.

Definition at line 320 of file GeodesicExact.hpp.

Math::real GeographicLib::GeodesicExact::Direct ( real  lat1,
real  lon1,
real  azi1,
real  s12,
real &  lat2,
real &  lon2,
real &  azi2,
real &  M12,
real &  M21 
) const
throw (
)
inline

See the documentation for GeodesicExact::Direct.

Definition at line 332 of file GeodesicExact.hpp.

Math::real GeographicLib::GeodesicExact::Direct ( real  lat1,
real  lon1,
real  azi1,
real  s12,
real &  lat2,
real &  lon2,
real &  azi2,
real &  m12,
real &  M12,
real &  M21 
) const
throw (
)
inline

See the documentation for GeodesicExact::Direct.

Definition at line 345 of file GeodesicExact.hpp.

void GeographicLib::GeodesicExact::ArcDirect ( real  lat1,
real  lon1,
real  azi1,
real  a12,
real &  lat2,
real &  lon2,
real &  azi2,
real &  s12,
real &  m12,
real &  M12,
real &  M21,
real &  S12 
) const
throw (
)
inline

Perform the direct geodesic calculation where the length of the geodesic is specified in terms of arc length.

Parameters
[in]lat1latitude of point 1 (degrees).
[in]lon1longitude of point 1 (degrees).
[in]azi1azimuth at point 1 (degrees).
[in]a12arc length between point 1 and point 2 (degrees); it can be signed.
[out]lat2latitude of point 2 (degrees).
[out]lon2longitude of point 2 (degrees).
[out]azi2(forward) azimuth at point 2 (degrees).
[out]s12distance between point 1 and point 2 (meters).
[out]m12reduced length of geodesic (meters).
[out]M12geodesic scale of point 2 relative to point 1 (dimensionless).
[out]M21geodesic scale of point 1 relative to point 2 (dimensionless).
[out]S12area under the geodesic (meters2).

lat1 should be in the range [−90°, 90°]; lon1 and azi1 should be in the range [−540°, 540°). The values of lon2 and azi2 returned are in the range [−180°, 180°).

If either point is at a pole, the azimuth is defined by keeping the longitude fixed, writing lat = ±(90° − ε), and taking the limit ε → 0+. An arc length greater that 180° signifies a geodesic which is not a shortest path. (For a prolate ellipsoid, an additional condition is necessary for a shortest path: the longitudinal extent must not exceed of 180°.)

The following functions are overloaded versions of GeodesicExact::Direct which omit some of the output parameters.

Definition at line 395 of file GeodesicExact.hpp.

Referenced by main().

void GeographicLib::GeodesicExact::ArcDirect ( real  lat1,
real  lon1,
real  azi1,
real  a12,
real &  lat2,
real &  lon2 
) const
throw (
)
inline

See the documentation for GeodesicExact::ArcDirect.

Definition at line 408 of file GeodesicExact.hpp.

void GeographicLib::GeodesicExact::ArcDirect ( real  lat1,
real  lon1,
real  azi1,
real  a12,
real &  lat2,
real &  lon2,
real &  azi2 
) const
throw (
)
inline

See the documentation for GeodesicExact::ArcDirect.

Definition at line 419 of file GeodesicExact.hpp.

void GeographicLib::GeodesicExact::ArcDirect ( real  lat1,
real  lon1,
real  azi1,
real  a12,
real &  lat2,
real &  lon2,
real &  azi2,
real &  s12 
) const
throw (
)
inline

See the documentation for GeodesicExact::ArcDirect.

Definition at line 430 of file GeodesicExact.hpp.

void GeographicLib::GeodesicExact::ArcDirect ( real  lat1,
real  lon1,
real  azi1,
real  a12,
real &  lat2,
real &  lon2,
real &  azi2,
real &  s12,
real &  m12 
) const
throw (
)
inline

See the documentation for GeodesicExact::ArcDirect.

Definition at line 442 of file GeodesicExact.hpp.

void GeographicLib::GeodesicExact::ArcDirect ( real  lat1,
real  lon1,
real  azi1,
real  a12,
real &  lat2,
real &  lon2,
real &  azi2,
real &  s12,
real &  M12,
real &  M21 
) const
throw (
)
inline

See the documentation for GeodesicExact::ArcDirect.

Definition at line 455 of file GeodesicExact.hpp.

void GeographicLib::GeodesicExact::ArcDirect ( real  lat1,
real  lon1,
real  azi1,
real  a12,
real &  lat2,
real &  lon2,
real &  azi2,
real &  s12,
real &  m12,
real &  M12,
real &  M21 
) const
throw (
)
inline

See the documentation for GeodesicExact::ArcDirect.

Definition at line 468 of file GeodesicExact.hpp.

Math::real GeographicLib::GeodesicExact::GenDirect ( real  lat1,
real  lon1,
real  azi1,
bool  arcmode,
real  s12_a12,
unsigned  outmask,
real &  lat2,
real &  lon2,
real &  azi2,
real &  s12,
real &  m12,
real &  M12,
real &  M21,
real &  S12 
) const
throw (
)

The general direct geodesic calculation. GeodesicExact::Direct and GeodesicExact::ArcDirect are defined in terms of this function.

Parameters
[in]lat1latitude of point 1 (degrees).
[in]lon1longitude of point 1 (degrees).
[in]azi1azimuth at point 1 (degrees).
[in]arcmodeboolean flag determining the meaning of the second parameter.
[in]s12_a12if arcmode is false, this is the distance between point 1 and point 2 (meters); otherwise it is the arc length between point 1 and point 2 (degrees); it can be signed.
[in]outmaska bitor'ed combination of GeodesicExact::mask values specifying which of the following parameters should be set.
[out]lat2latitude of point 2 (degrees).
[out]lon2longitude of point 2 (degrees).
[out]azi2(forward) azimuth at point 2 (degrees).
[out]s12distance between point 1 and point 2 (meters).
[out]m12reduced length of geodesic (meters).
[out]M12geodesic scale of point 2 relative to point 1 (dimensionless).
[out]M21geodesic scale of point 1 relative to point 2 (dimensionless).
[out]S12area under the geodesic (meters2).
Returns
a12 arc length of between point 1 and point 2 (degrees).

The GeodesicExact::mask values possible for outmask are

The function value a12 is always computed and returned and this equals s12_a12 is arcmode is true. If outmask includes GeodesicExact::DISTANCE and arcmode is false, then s12 = s12_a12. It is not necessary to include GeodesicExact::DISTANCE_IN in outmask; this is automatically included is arcmode is false.

Definition at line 115 of file GeodesicExact.cpp.

Math::real GeographicLib::GeodesicExact::Inverse ( real  lat1,
real  lon1,
real  lat2,
real  lon2,
real &  s12,
real &  azi1,
real &  azi2,
real &  m12,
real &  M12,
real &  M21,
real &  S12 
) const
throw (
)
inline

Perform the inverse geodesic calculation.

Parameters
[in]lat1latitude of point 1 (degrees).
[in]lon1longitude of point 1 (degrees).
[in]lat2latitude of point 2 (degrees).
[in]lon2longitude of point 2 (degrees).
[out]s12distance between point 1 and point 2 (meters).
[out]azi1azimuth at point 1 (degrees).
[out]azi2(forward) azimuth at point 2 (degrees).
[out]m12reduced length of geodesic (meters).
[out]M12geodesic scale of point 2 relative to point 1 (dimensionless).
[out]M21geodesic scale of point 1 relative to point 2 (dimensionless).
[out]S12area under the geodesic (meters2).
Returns
a12 arc length of between point 1 and point 2 (degrees).

lat1 and lat2 should be in the range [−90°, 90°]; lon1 and lon2 should be in the range [−540°, 540°). The values of azi1 and azi2 returned are in the range [−180°, 180°).

If either point is at a pole, the azimuth is defined by keeping the longitude fixed, writing lat = ±(90° − ε), and taking the limit ε → 0+.

The following functions are overloaded versions of GeodesicExact::Inverse which omit some of the output parameters. Note, however, that the arc length is always computed and returned as the function value.

Definition at line 568 of file GeodesicExact.hpp.

Referenced by main().

Math::real GeographicLib::GeodesicExact::Inverse ( real  lat1,
real  lon1,
real  lat2,
real  lon2,
real &  s12 
) const
throw (
)
inline

See the documentation for GeodesicExact::Inverse.

Definition at line 580 of file GeodesicExact.hpp.

Math::real GeographicLib::GeodesicExact::Inverse ( real  lat1,
real  lon1,
real  lat2,
real  lon2,
real &  azi1,
real &  azi2 
) const
throw (
)
inline

See the documentation for GeodesicExact::Inverse.

Definition at line 591 of file GeodesicExact.hpp.

Math::real GeographicLib::GeodesicExact::Inverse ( real  lat1,
real  lon1,
real  lat2,
real  lon2,
real &  s12,
real &  azi1,
real &  azi2 
) const
throw (
)
inline

See the documentation for GeodesicExact::Inverse.

Definition at line 602 of file GeodesicExact.hpp.

Math::real GeographicLib::GeodesicExact::Inverse ( real  lat1,
real  lon1,
real  lat2,
real  lon2,
real &  s12,
real &  azi1,
real &  azi2,
real &  m12 
) const
throw (
)
inline

See the documentation for GeodesicExact::Inverse.

Definition at line 614 of file GeodesicExact.hpp.

Math::real GeographicLib::GeodesicExact::Inverse ( real  lat1,
real  lon1,
real  lat2,
real  lon2,
real &  s12,
real &  azi1,
real &  azi2,
real &  M12,
real &  M21 
) const
throw (
)
inline

See the documentation for GeodesicExact::Inverse.

Definition at line 626 of file GeodesicExact.hpp.

Math::real GeographicLib::GeodesicExact::Inverse ( real  lat1,
real  lon1,
real  lat2,
real  lon2,
real &  s12,
real &  azi1,
real &  azi2,
real &  m12,
real &  M12,
real &  M21 
) const
throw (
)
inline

See the documentation for GeodesicExact::Inverse.

Definition at line 638 of file GeodesicExact.hpp.

Math::real GeographicLib::GeodesicExact::GenInverse ( real  lat1,
real  lon1,
real  lat2,
real  lon2,
unsigned  outmask,
real &  s12,
real &  azi1,
real &  azi2,
real &  m12,
real &  M12,
real &  M21,
real &  S12 
) const
throw (
)

The general inverse geodesic calculation. GeodesicExact::Inverse is defined in terms of this function.

Parameters
[in]lat1latitude of point 1 (degrees).
[in]lon1longitude of point 1 (degrees).
[in]lat2latitude of point 2 (degrees).
[in]lon2longitude of point 2 (degrees).
[in]outmaska bitor'ed combination of GeodesicExact::mask values specifying which of the following parameters should be set.
[out]s12distance between point 1 and point 2 (meters).
[out]azi1azimuth at point 1 (degrees).
[out]azi2(forward) azimuth at point 2 (degrees).
[out]m12reduced length of geodesic (meters).
[out]M12geodesic scale of point 2 relative to point 1 (dimensionless).
[out]M21geodesic scale of point 1 relative to point 2 (dimensionless).
[out]S12area under the geodesic (meters2).
Returns
a12 arc length of between point 1 and point 2 (degrees).

The GeodesicExact::mask values possible for outmask are

The arc length is always computed and returned as the function value.

Definition at line 130 of file GeodesicExact.cpp.

References GeographicLib::Math::AngDiff(), GeographicLib::Math::AngNormalize(), GeographicLib::Math::hypot(), and GeographicLib::Math::sq().

GeodesicLineExact GeographicLib::GeodesicExact::Line ( real  lat1,
real  lon1,
real  azi1,
unsigned  caps = ALL 
) const
throw (
)

Set up to compute several points on a single geodesic.

Parameters
[in]lat1latitude of point 1 (degrees).
[in]lon1longitude of point 1 (degrees).
[in]azi1azimuth at point 1 (degrees).
[in]capsbitor'ed combination of GeodesicExact::mask values specifying the capabilities the GeodesicLineExact object should possess, i.e., which quantities can be returned in calls to GeodesicLineExact::Position.
Returns
a GeodesicLineExact object.

lat1 should be in the range [−90°, 90°]; lon1 and azi1 should be in the range [−540°, 540°).

The GeodesicExact::mask values are

The default value of caps is GeodesicExact::ALL which turns on all the capabilities.

If the point is at a pole, the azimuth is defined by keeping lon1 fixed, writing lat1 = ±(90 − ε), and taking the limit ε → 0+.

Definition at line 110 of file GeodesicExact.cpp.

Referenced by main().

Math::real GeographicLib::GeodesicExact::MajorRadius ( ) const
throw (
)
inline
Returns
a the equatorial radius of the ellipsoid (meters). This is the value used in the constructor.

Definition at line 747 of file GeodesicExact.hpp.

Math::real GeographicLib::GeodesicExact::Flattening ( ) const
throw (
)
inline
Returns
f the flattening of the ellipsoid. This is the value used in the constructor.

Definition at line 753 of file GeodesicExact.hpp.

Math::real GeographicLib::GeodesicExact::EllipsoidArea ( ) const
throw (
)
inline
Returns
total area of ellipsoid in meters2. The area of a polygon encircling a pole can be found by adding GeodesicExact::EllipsoidArea()/2 to the sum of S12 for each side of the polygon.

Definition at line 769 of file GeodesicExact.hpp.

Friends And Related Function Documentation

friend class GeodesicLineExact
friend

Definition at line 83 of file GeodesicExact.hpp.

Member Data Documentation

const GeodesicExact GeographicLib::GeodesicExact::WGS84
static

A global instantiation of GeodesicExact with the parameters for the WGS84 ellipsoid.

Definition at line 777 of file GeodesicExact.hpp.


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