GeographicLib  1.35
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GeographicLib::Geodesic Class Reference

Geodesic calculations More...

#include <GeographicLib/Geodesic.hpp>

Public Types

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

Public Member Functions

Constructor
 Geodesic (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 GeodesicLine.
GeodesicLine 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 Geodesic WGS84
 

Friends

class GeodesicLine
 

Detailed Description

Geodesic calculations

The shortest path between two points on a ellipsoid at (lat1, lon1) and (lat2, lon2) is called the geodesic. Its length is s12 and the geodesic from point 1 to point 2 has azimuths azi1 and azi2 at the two end points. (The azimuth is the heading measured clockwise from north. azi2 is the "forward" azimuth, i.e., the heading that takes you beyond point 2 not back to point 1.) In the figure below, latitude if labeled φ, longitude λ (with λ12 = λ2 − λ1), and azimuth α.

spheroidal triangle

Given lat1, lon1, azi1, and s12, we can determine lat2, lon2, and azi2. This is the direct geodesic problem and its solution is given by the function Geodesic::Direct. (If s12 is sufficiently large that the geodesic wraps more than halfway around the earth, there will be another geodesic between the points with a smaller s12.)

Given lat1, lon1, lat2, and lon2, we can determine azi1, azi2, and s12. This is the inverse geodesic problem, whose solution is given by Geodesic::Inverse. Usually, the solution to the inverse problem is unique. In cases where there are multiple solutions (all with the same s12, of course), all the solutions can be easily generated once a particular solution is provided.

The standard way of specifying the direct problem is the specify the distance s12 to the second point. However it is sometimes useful instead to specify the arc length a12 (in degrees) on the auxiliary sphere. This is a mathematical construct used in solving the geodesic problems. The solution of the direct problem in this form is provided by Geodesic::ArcDirect. An arc length in excess of 180° indicates that the geodesic is not a shortest path. In addition, the arc length between an equatorial crossing and the next extremum of latitude for a geodesic is 90°.

This class can also calculate several other quantities related to geodesics. These are:

Overloaded versions of Geodesic::Direct, Geodesic::ArcDirect, and Geodesic::Inverse allow these quantities to be returned. In addition there are general functions Geodesic::GenDirect, and Geodesic::GenInverse which allow an arbitrary set of results to be computed. The quantities m12, M12, M21 which all specify the behavior of nearby geodesics obey addition rules. If points 1, 2, and 3 all lie on a single geodesic, then the following rules hold:

Additional functionality is provided by the GeodesicLine class, which allows a sequence of points along a geodesic to be computed.

The shortest distance returned by the solution of the inverse problem is (obviously) uniquely defined. However, in a few special cases there are multiple azimuths which yield the same shortest distance. Here is a catalog of those cases:

The calculations are accurate to better than 15 nm (15 nanometers) for the WGS84 ellipsoid. See Sec. 9 of arXiv:1102.1215v1 for details. The algorithms used by this class are based on series expansions using the flattening f as a small parameter. These are only accurate for |f| < 0.02; however reasonably accurate results will be obtained for |f| < 0.2. Here is a table of the approximate maximum error (expressed as a distance) for an ellipsoid with the same major radius as the WGS84 ellipsoid and different values of the flattening.

    |f|      error
    0.01     25 nm
    0.02     30 nm
    0.05     10 um
    0.1     1.5 mm
    0.2     300 mm

For very eccentric ellipsoids, use GeodesicExact instead.

The algorithms are described in

For more information on geodesics see geodesic.

Example of use:

// Example of using the GeographicLib::Geodesic class
#include <iostream>
#include <exception>
using namespace std;
using namespace GeographicLib;
int main() {
try {
// Alternatively: const Geodesic& geod = Geodesic::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 Geodesic and GeodesicLine.

Definition at line 169 of file Geodesic.hpp.

Member Enumeration Documentation

Bit masks for what calculations to do. These masks do double duty. They signify to the GeodesicLine::GeodesicLine constructor and to Geodesic::Line what capabilities should be included in the GeodesicLine object. They also specify which results to return in the general routines Geodesic::GenDirect and Geodesic::GenInverse routines. GeodesicLine::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 GeodesicLine 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 GeodesicLine 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 278 of file Geodesic.hpp.

Constructor & Destructor Documentation

GeographicLib::Geodesic::Geodesic ( 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 Geodesic.cpp.

References GeographicLib::Math::isfinite().

Member Function Documentation

Math::real GeographicLib::Geodesic::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

Solve the direct geodesic problem 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 negative.
[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 Geodesic::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 390 of file Geodesic.hpp.

Referenced by main().

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

See the documentation for Geodesic::Direct.

Definition at line 404 of file Geodesic.hpp.

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

See the documentation for Geodesic::Direct.

Definition at line 416 of file Geodesic.hpp.

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

See the documentation for Geodesic::Direct.

Definition at line 428 of file Geodesic.hpp.

Math::real GeographicLib::Geodesic::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 Geodesic::Direct.

Definition at line 440 of file Geodesic.hpp.

Math::real GeographicLib::Geodesic::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 Geodesic::Direct.

Definition at line 453 of file Geodesic.hpp.

void GeographicLib::Geodesic::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

Solve the direct geodesic problem 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 negative.
[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 Geodesic::Direct which omit some of the output parameters.

Definition at line 503 of file Geodesic.hpp.

Referenced by main().

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

See the documentation for Geodesic::ArcDirect.

Definition at line 516 of file Geodesic.hpp.

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

See the documentation for Geodesic::ArcDirect.

Definition at line 527 of file Geodesic.hpp.

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

See the documentation for Geodesic::ArcDirect.

Definition at line 538 of file Geodesic.hpp.

void GeographicLib::Geodesic::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 Geodesic::ArcDirect.

Definition at line 550 of file Geodesic.hpp.

void GeographicLib::Geodesic::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 Geodesic::ArcDirect.

Definition at line 563 of file Geodesic.hpp.

void GeographicLib::Geodesic::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 Geodesic::ArcDirect.

Definition at line 576 of file Geodesic.hpp.

Math::real GeographicLib::Geodesic::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 problem. Geodesic::Direct and Geodesic::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 s12_a12.
[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 negative.
[in]outmaska bitor'ed combination of Geodesic::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 Geodesic::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 Geodesic::DISTANCE and arcmode is false, then s12 = s12_a12. It is not necessary to include Geodesic::DISTANCE_IN in outmask; this is automatically included is arcmode is false.

Definition at line 121 of file Geodesic.cpp.

Math::real GeographicLib::Geodesic::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

Solve the inverse geodesic problem.

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 solution to the inverse problem is found using Newton's method. If this fails to converge (this is very unlikely in geodetic applications but does occur for very eccentric ellipsoids), then the bisection method is used to refine the solution.

The following functions are overloaded versions of Geodesic::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 681 of file Geodesic.hpp.

Referenced by main().

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

See the documentation for Geodesic::Inverse.

Definition at line 693 of file Geodesic.hpp.

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

See the documentation for Geodesic::Inverse.

Definition at line 704 of file Geodesic.hpp.

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

See the documentation for Geodesic::Inverse.

Definition at line 715 of file Geodesic.hpp.

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

See the documentation for Geodesic::Inverse.

Definition at line 727 of file Geodesic.hpp.

Math::real GeographicLib::Geodesic::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 Geodesic::Inverse.

Definition at line 739 of file Geodesic.hpp.

Math::real GeographicLib::Geodesic::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 Geodesic::Inverse.

Definition at line 751 of file Geodesic.hpp.

Math::real GeographicLib::Geodesic::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. Geodesic::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 Geodesic::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 Geodesic::mask values possible for outmask are

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

Definition at line 134 of file Geodesic.cpp.

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

GeodesicLine GeographicLib::Geodesic::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 Geodesic::mask values specifying the capabilities the GeodesicLine object should possess, i.e., which quantities can be returned in calls to GeodesicLine::Position.
Returns
a GeodesicLine object.

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

The Geodesic::mask values are

The default value of caps is Geodesic::ALL.

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 116 of file Geodesic.cpp.

Referenced by main().

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

Definition at line 859 of file Geodesic.hpp.

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

Definition at line 865 of file Geodesic.hpp.

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

Definition at line 881 of file Geodesic.hpp.

Friends And Related Function Documentation

friend class GeodesicLine
friend

Definition at line 172 of file Geodesic.hpp.

Member Data Documentation

const Geodesic GeographicLib::Geodesic::WGS84
static

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

Definition at line 889 of file Geodesic.hpp.


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