GeographicLib::GeodesicExact Class Reference

Exact geodesic calculations. More...

#include <GeodesicExact.hpp>

Public Types
enum	mask {   NONE, LATITUDE, LONGITUDE, AZIMUTH,   DISTANCE, DISTANCE_IN, REDUCEDLENGTH, GEODESICSCALE,   AREA, LONG_UNROLL, 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
Math::real	Direct (real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2) const
Math::real	Direct (real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2, real &azi2) const
Math::real	Direct (real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2, real &azi2, real &m12) const
Math::real	Direct (real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2, real &azi2, real &M12, real &M21) const
Math::real	Direct (real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2, real &azi2, real &m12, real &M12, real &M21) const
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
void	ArcDirect (real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2) const
void	ArcDirect (real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2) const
void	ArcDirect (real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2, real &s12) const
void	ArcDirect (real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2, real &s12, real &m12) const
void	ArcDirect (real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2, real &s12, real &M12, real &M21) const
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
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
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
Math::real	Inverse (real lat1, real lon1, real lat2, real lon2, real &s12) const
Math::real	Inverse (real lat1, real lon1, real lat2, real lon2, real &azi1, real &azi2) const
Math::real	Inverse (real lat1, real lon1, real lat2, real lon2, real &s12, real &azi1, real &azi2) const
Math::real	Inverse (real lat1, real lon1, real lat2, real lon2, real &s12, real &azi1, real &azi2, real &m12) const
Math::real	Inverse (real lat1, real lon1, real lat2, real lon2, real &s12, real &azi1, real &azi2, real &M12, real &M21) const
Math::real	Inverse (real lat1, real lon1, real lat2, real lon2, real &s12, real &azi1, real &azi2, real &m12, real &M12, real &M21) const
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
Interface to GeodesicLineExact.
GeodesicLineExact	Line (real lat1, real lon1, real azi1, unsigned caps=ALL) const
GeodesicLineExact	InverseLine (real lat1, real lon1, real lat2, real lon2, unsigned caps=ALL) const
GeodesicLineExact	DirectLine (real lat1, real lon1, real azi1, real s12, unsigned caps=ALL) const
GeodesicLineExact	ArcDirectLine (real lat1, real lon1, real azi1, real a12, unsigned caps=ALL) const
GeodesicLineExact	GenDirectLine (real lat1, real lon1, real azi1, bool arcmode, real s12_a12, unsigned caps=ALL) const
Inspector functions.
Math::real	MajorRadius () const
Math::real	Flattening () const
Math::real	EllipsoidArea () const

Static Public Member Functions
static const GeodesicExact &	WGS84 ()

Private Types
enum	captype {   CAP_NONE = 0U, CAP_E = 1U<<0, CAP_D = 1U<<2, CAP_H = 1U<<3,   CAP_C4 = 1U<<4, CAP_ALL = 0x1FU, CAP_MASK = CAP_ALL, OUT_ALL = 0x7F80U,   OUT_MASK = 0xFF80U }
typedef Math::real	real

Private Member Functions
void	C4coeff ()
void	C4f (real k2, real c[]) const
real	GenInverse (real lat1, real lon1, real lat2, real lon2, unsigned outmask, real &s12, real &salp1, real &calp1, real &salp2, real &calp2, real &m12, real &M12, real &M21, real &S12) const
real	InverseStart (EllipticFunction &E, real sbet1, real cbet1, real dn1, real sbet2, real cbet2, real dn2, real lam12, real slam12, real clam12, real &salp1, real &calp1, real &salp2, real &calp2, real &dnm) const
real	Lambda12 (real sbet1, real cbet1, real dn1, real sbet2, real cbet2, real dn2, real salp1, real calp1, real slam120, real clam120, real &salp2, real &calp2, real &sig12, real &ssig1, real &csig1, real &ssig2, real &csig2, EllipticFunction &E, real &domg12, bool diffp, real &dlam12) const
void	Lengths (const EllipticFunction &E, real sig12, real ssig1, real csig1, real dn1, real ssig2, real csig2, real dn2, real cbet1, real cbet2, unsigned outmask, real &s12s, real &m12a, real &m0, real &M12, real &M21) const

Static Private Member Functions
static real	Astroid (real x, real y)
static real	CosSeries (real sinx, real cosx, const real c[], int n)
static Math::real	reale (long long hi, long long lo)

Private Attributes
real	_a
real	_b
real	_c2
real	_C4x [nC4x_]
real	_e2
real	_ep2
real	_etol2
real	_f
real	_f1
real	_n
unsigned	maxit2_
real	tiny_
real	tol0_
real	tol1_
real	tol2_
real	tolb_
real	xthresh_

Static Private Attributes
static const unsigned	maxit1_ = 20
static const int	nC4_ = GEOGRAPHICLIB_GEODESICEXACT_ORDER
static const int	nC4x_ = (nC4_ * (nC4_ + 1)) / 2

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>
#include <GeographicLib/GeodesicExact.hpp>
#include <GeographicLib/Constants.hpp>

using namespace std;
using namespace GeographicLib;

int main() {
  try {
    GeodesicExact geod(Constants::WGS84_a(), Constants::WGS84_f());
    // 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;
  }
}

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 Typedef Documentation

typedef Math::real GeographicLib::GeodesicExact::real

private

Definition at line 82 of file GeodesicExact.hpp.

Member Enumeration Documentation

enum GeographicLib::GeodesicExact::captype

private

Enumerator
CAP_NONE
CAP_E
CAP_D
CAP_H
CAP_C4
CAP_ALL
CAP_MASK
OUT_ALL
OUT_MASK

Definition at line 90 of file GeodesicExact.hpp.

enum GeographicLib::GeodesicExact::mask

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.
LONG_UNROLL	Unroll lon2 in the direct calculation.
ALL	All capabilities, calculate everything. (LONG_UNROLL is not included in this mask.)

Definition at line 158 of file GeodesicExact.hpp.

Constructor & Destructor Documentation

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

Constructor for a ellipsoid with

Parameters

[in]	a	equatorial radius (meters).
[in]	f	flattening of ellipsoid. Setting f = 0 gives a sphere. Negative f gives a prolate ellipsoid.

Exceptions

GeographicErr

if a or (1 − f) a is not positive.

Definition at line 42 of file src/GeodesicExact.cpp.

Member Function Documentation

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

inline

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

Parameters

[in]	lat1	latitude of point 1 (degrees).
[in]	lon1	longitude of point 1 (degrees).
[in]	azi1	azimuth at point 1 (degrees).
[in]	a12	arc length between point 1 and point 2 (degrees); it can be signed.
[out]	lat2	latitude of point 2 (degrees).
[out]	lon2	longitude of point 2 (degrees).
[out]	azi2	(forward) azimuth at point 2 (degrees).
[out]	s12	distance between point 1 and point 2 (meters).
[out]	m12	reduced length of geodesic (meters).
[out]	M12	geodesic scale of point 2 relative to point 1 (dimensionless).
[out]	M21	geodesic scale of point 1 relative to point 2 (dimensionless).
[out]	S12	area under the geodesic (meters2).

lat1 should be in the range [−90°, 90°]. 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 386 of file GeodesicExact.hpp.

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

inline

See the documentation for GeodesicExact::ArcDirect.

Definition at line 399 of file GeodesicExact.hpp.

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

inline

See the documentation for GeodesicExact::ArcDirect.

Definition at line 410 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

inline

See the documentation for GeodesicExact::ArcDirect.

Definition at line 421 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

inline

See the documentation for GeodesicExact::ArcDirect.

Definition at line 433 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

inline

See the documentation for GeodesicExact::ArcDirect.

Definition at line 446 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

inline

See the documentation for GeodesicExact::ArcDirect.

Definition at line 459 of file GeodesicExact.hpp.

GeodesicLineExact GeographicLib::GeodesicExact::ArcDirectLine	(	real	lat1,
		real	lon1,
		real	azi1,
		real	a12,
		unsigned	caps = ALL
	)		const

Define a GeodesicLineExact in terms of the direct geodesic problem specified in terms of arc length.

Parameters

[in]	lat1	latitude of point 1 (degrees).
[in]	lon1	longitude of point 1 (degrees).
[in]	azi1	azimuth at point 1 (degrees).
[in]	a12	arc length between point 1 and point 2 (degrees); it can be negative.
[in]	caps	bitor'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.

This function sets point 3 of the GeodesicLineExact to correspond to point 2 of the direct geodesic problem.

lat1 should be in the range [−90°, 90°].

Definition at line 159 of file src/GeodesicExact.cpp.

Math::real GeographicLib::GeodesicExact::Astroid ( real  x, real  y  )

staticprivate

Definition at line 597 of file src/GeodesicExact.cpp.

void GeographicLib::GeodesicExact::C4coeff ( )

private

Definition at line 36 of file GeodesicExactC4.cpp.

void GeographicLib::GeodesicExact::C4f ( real  k2, real  c[]  ) const

private

Definition at line 915 of file src/GeodesicExact.cpp.

Math::real GeographicLib::GeodesicExact::CosSeries ( real  sinx, real  cosx, const real  c[], int  n  )

staticprivate

Definition at line 99 of file src/GeodesicExact.cpp.

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

inline

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

Parameters

[in]	lat1	latitude of point 1 (degrees).
[in]	lon1	longitude of point 1 (degrees).
[in]	azi1	azimuth at point 1 (degrees).
[in]	s12	distance between point 1 and point 2 (meters); it can be signed.
[out]	lat2	latitude of point 2 (degrees).
[out]	lon2	longitude of point 2 (degrees).
[out]	azi2	(forward) azimuth at point 2 (degrees).
[out]	m12	reduced length of geodesic (meters).
[out]	M12	geodesic scale of point 2 relative to point 1 (dimensionless).
[out]	M21	geodesic scale of point 1 relative to point 2 (dimensionless).
[out]	S12	area under the geodesic (meters2).

Returns

a12 arc length of between point 1 and point 2 (degrees).

lat1 should be in the range [−90°, 90°]. 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 274 of file GeodesicExact.hpp.

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

inline

See the documentation for GeodesicExact::Direct.

Definition at line 288 of file GeodesicExact.hpp.

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

inline

See the documentation for GeodesicExact::Direct.

Definition at line 300 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

inline

See the documentation for GeodesicExact::Direct.

Definition at line 312 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

inline

See the documentation for GeodesicExact::Direct.

Definition at line 324 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

inline

See the documentation for GeodesicExact::Direct.

Definition at line 337 of file GeodesicExact.hpp.

GeodesicLineExact GeographicLib::GeodesicExact::DirectLine	(	real	lat1,
		real	lon1,
		real	azi1,
		real	s12,
		unsigned	caps = ALL
	)		const

Define a GeodesicLineExact in terms of the direct geodesic problem specified in terms of distance.

Parameters

[in]	lat1	latitude of point 1 (degrees).
[in]	lon1	longitude of point 1 (degrees).
[in]	azi1	azimuth at point 1 (degrees).
[in]	s12	distance between point 1 and point 2 (meters); it can be negative.
[in]	caps	bitor'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.

This function sets point 3 of the GeodesicLineExact to correspond to point 2 of the direct geodesic problem.

lat1 should be in the range [−90°, 90°].

Definition at line 153 of file src/GeodesicExact.cpp.

Math::real GeographicLib::GeodesicExact::EllipsoidArea ( ) const

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 849 of file GeodesicExact.hpp.

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

inline

Returns

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

Definition at line 841 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

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

Parameters

[in]	lat1	latitude of point 1 (degrees).
[in]	lon1	longitude of point 1 (degrees).
[in]	azi1	azimuth at point 1 (degrees).
[in]	arcmode	boolean flag determining the meaning of the second parameter.
[in]	s12_a12	if 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]	outmask	a bitor'ed combination of GeodesicExact::mask values specifying which of the following parameters should be set.
[out]	lat2	latitude of point 2 (degrees).
[out]	lon2	longitude of point 2 (degrees).
[out]	azi2	(forward) azimuth at point 2 (degrees).
[out]	s12	distance between point 1 and point 2 (meters).
[out]	m12	reduced length of geodesic (meters).
[out]	M12	geodesic scale of point 2 relative to point 1 (dimensionless).
[out]	M21	geodesic scale of point 1 relative to point 2 (dimensionless).
[out]	S12	area under the geodesic (meters2).

Returns

a12 arc length of between point 1 and point 2 (degrees).

The GeodesicExact::mask values possible for outmask are

• outmask |= GeodesicExact::LATITUDE for the latitude lat2;
• outmask |= GeodesicExact::LONGITUDE for the latitude lon2;
• outmask |= GeodesicExact::AZIMUTH for the latitude azi2;
• outmask |= GeodesicExact::DISTANCE for the distance s12;
• outmask |= GeodesicExact::REDUCEDLENGTH for the reduced length m12;
• outmask |= GeodesicExact::GEODESICSCALE for the geodesic scales M12 and M21;
• outmask |= GeodesicExact::AREA for the area S12;
• outmask |= GeodesicExact::ALL for all of the above;
• outmask |= GeodesicExact::LONG_UNROLL to unroll lon2 instead of wrapping it into the range [−180°, 180°].

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.

With the GeodesicExact::LONG_UNROLL bit set, the quantity lon2 − lon1 indicates how many times and in what sense the geodesic encircles the ellipsoid.

Definition at line 124 of file src/GeodesicExact.cpp.

GeodesicLineExact GeographicLib::GeodesicExact::GenDirectLine	(	real	lat1,
		real	lon1,
		real	azi1,
		bool	arcmode,
		real	s12_a12,
		unsigned	caps = ALL
	)		const

Define a GeodesicLineExact in terms of the direct geodesic problem specified in terms of either distance or arc length.

Parameters

[in]	lat1	latitude of point 1 (degrees).
[in]	lon1	longitude of point 1 (degrees).
[in]	azi1	azimuth at point 1 (degrees).
[in]	arcmode	boolean flag determining the meaning of the s12_a12.
[in]	s12_a12	if 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]	caps	bitor'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.

This function sets point 3 of the GeodesicLineExact to correspond to point 2 of the direct geodesic problem.

lat1 should be in the range [−90°, 90°].

Definition at line 139 of file src/GeodesicExact.cpp.

Math::real GeographicLib::GeodesicExact::GenInverse ( real  lat1, real  lon1, real  lat2, real  lon2, unsigned  outmask, real &  s12, real &  salp1, real &  calp1, real &  salp2, real &  calp2, real &  m12, real &  M12, real &  M21, real &  S12  ) const

private

Definition at line 165 of file src/GeodesicExact.cpp.

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

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

Parameters

[in]	lat1	latitude of point 1 (degrees).
[in]	lon1	longitude of point 1 (degrees).
[in]	lat2	latitude of point 2 (degrees).
[in]	lon2	longitude of point 2 (degrees).
[in]	outmask	a bitor'ed combination of GeodesicExact::mask values specifying which of the following parameters should be set.
[out]	s12	distance between point 1 and point 2 (meters).
[out]	azi1	azimuth at point 1 (degrees).
[out]	azi2	(forward) azimuth at point 2 (degrees).
[out]	m12	reduced length of geodesic (meters).
[out]	M12	geodesic scale of point 2 relative to point 1 (dimensionless).
[out]	M21	geodesic scale of point 1 relative to point 2 (dimensionless).
[out]	S12	area under the geodesic (meters2).

Returns

a12 arc length of between point 1 and point 2 (degrees).

The GeodesicExact::mask values possible for outmask are

• outmask |= GeodesicExact::DISTANCE for the distance s12;
• outmask |= GeodesicExact::AZIMUTH for the latitude azi2;
• outmask |= GeodesicExact::REDUCEDLENGTH for the reduced length m12;
• outmask |= GeodesicExact::GEODESICSCALE for the geodesic scales M12 and M21;
• outmask |= GeodesicExact::AREA for the area S12;
• outmask |= GeodesicExact::ALL for all of the above.

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

Definition at line 520 of file src/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

inline

Perform the inverse geodesic calculation.

Parameters

[in]	lat1	latitude of point 1 (degrees).
[in]	lon1	longitude of point 1 (degrees).
[in]	lat2	latitude of point 2 (degrees).
[in]	lon2	longitude of point 2 (degrees).
[out]	s12	distance between point 1 and point 2 (meters).
[out]	azi1	azimuth at point 1 (degrees).
[out]	azi2	(forward) azimuth at point 2 (degrees).
[out]	m12	reduced length of geodesic (meters).
[out]	M12	geodesic scale of point 2 relative to point 1 (dimensionless).
[out]	M21	geodesic scale of point 1 relative to point 2 (dimensionless).
[out]	S12	area 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°]. 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 565 of file GeodesicExact.hpp.

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

inline

See the documentation for GeodesicExact::Inverse.

Definition at line 577 of file GeodesicExact.hpp.

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

inline

See the documentation for GeodesicExact::Inverse.

Definition at line 588 of file GeodesicExact.hpp.

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

inline

See the documentation for GeodesicExact::Inverse.

Definition at line 599 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

inline

See the documentation for GeodesicExact::Inverse.

Definition at line 611 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

inline

See the documentation for GeodesicExact::Inverse.

Definition at line 623 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

inline

See the documentation for GeodesicExact::Inverse.

Definition at line 635 of file GeodesicExact.hpp.

GeodesicLineExact GeographicLib::GeodesicExact::InverseLine	(	real	lat1,
		real	lon1,
		real	lat2,
		real	lon2,
		unsigned	caps = ALL
	)		const

Define a GeodesicLineExact in terms of the inverse geodesic problem.

Parameters

[in]	lat1	latitude of point 1 (degrees).
[in]	lon1	longitude of point 1 (degrees).
[in]	lat2	latitude of point 2 (degrees).
[in]	lon2	longitude of point 2 (degrees).
[in]	caps	bitor'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.

This function sets point 3 of the GeodesicLineExact to correspond to point 2 of the inverse geodesic problem.

lat1 and lat2 should be in the range [−90°, 90°].

Definition at line 538 of file src/GeodesicExact.cpp.

Math::real GeographicLib::GeodesicExact::InverseStart ( EllipticFunction &  E, real  sbet1, real  cbet1, real  dn1, real  sbet2, real  cbet2, real  dn2, real  lam12, real  slam12, real  clam12, real &  salp1, real &  calp1, real &  salp2, real &  calp2, real &  dnm  ) const

private

Definition at line 649 of file src/GeodesicExact.cpp.

Math::real GeographicLib::GeodesicExact::Lambda12 ( real  sbet1, real  cbet1, real  dn1, real  sbet2, real  cbet2, real  dn2, real  salp1, real  calp1, real  slam120, real  clam120, real &  salp2, real &  calp2, real &  sig12, real &  ssig1, real &  csig1, real &  ssig2, real &  csig2, EllipticFunction &  E, real &  domg12, bool  diffp, real &  dlam12  ) const

private

Definition at line 819 of file src/GeodesicExact.cpp.

void GeographicLib::GeodesicExact::Lengths ( const EllipticFunction &  E, real  sig12, real  ssig1, real  csig1, real  dn1, real  ssig2, real  csig2, real  dn2, real  cbet1, real  cbet2, unsigned  outmask, real &  s12s, real &  m12a, real &  m0, real &  M12, real &  M21  ) const

private

Definition at line 553 of file src/GeodesicExact.cpp.

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

Set up to compute several points on a single geodesic.

Parameters

[in]	lat1	latitude of point 1 (degrees).
[in]	lon1	longitude of point 1 (degrees).
[in]	azi1	azimuth at point 1 (degrees).
[in]	caps	bitor'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°].

The GeodesicExact::mask values are

• caps |= GeodesicExact::LATITUDE for the latitude lat2; this is added automatically;
• caps |= GeodesicExact::LONGITUDE for the latitude lon2;
• caps |= GeodesicExact::AZIMUTH for the azimuth azi2; this is added automatically;
• caps |= GeodesicExact::DISTANCE for the distance s12;
• caps |= GeodesicExact::REDUCEDLENGTH for the reduced length m12;
• caps |= GeodesicExact::GEODESICSCALE for the geodesic scales M12 and M21;
• caps |= GeodesicExact::AREA for the area S12;
• caps |= GeodesicExact::DISTANCE_IN permits the length of the geodesic to be given in terms of s12; without this capability the length can only be specified in terms of arc length;
• caps |= GeodesicExact::ALL for all of the above.

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 119 of file src/GeodesicExact.cpp.

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

inline

Returns

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

Definition at line 835 of file GeodesicExact.hpp.

static Math::real GeographicLib::GeodesicExact::reale ( long long  hi, long long  lo  )

inlinestaticprivate

Definition at line 142 of file GeodesicExact.hpp.

const GeodesicExact & GeographicLib::GeodesicExact::WGS84 ( )

static

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

Definition at line 93 of file src/GeodesicExact.cpp.

Friends And Related Function Documentation

friend class GeodesicLineExact

friend

Definition at line 83 of file GeodesicExact.hpp.

Member Data Documentation

real GeographicLib::GeodesicExact::_a

private

Definition at line 106 of file GeodesicExact.hpp.

real GeographicLib::GeodesicExact::_b

private

Definition at line 106 of file GeodesicExact.hpp.

real GeographicLib::GeodesicExact::_c2

private

Definition at line 106 of file GeodesicExact.hpp.

real GeographicLib::GeodesicExact::_C4x[nC4x_]

private

Definition at line 107 of file GeodesicExact.hpp.

real GeographicLib::GeodesicExact::_e2

private

Definition at line 106 of file GeodesicExact.hpp.

real GeographicLib::GeodesicExact::_ep2

private

Definition at line 106 of file GeodesicExact.hpp.

real GeographicLib::GeodesicExact::_etol2

private

Definition at line 106 of file GeodesicExact.hpp.

real GeographicLib::GeodesicExact::_f

private

Definition at line 106 of file GeodesicExact.hpp.

real GeographicLib::GeodesicExact::_f1

private

Definition at line 106 of file GeodesicExact.hpp.

real GeographicLib::GeodesicExact::_n

private

Definition at line 106 of file GeodesicExact.hpp.

const unsigned GeographicLib::GeodesicExact::maxit1_ = 20

staticprivate

Definition at line 86 of file GeodesicExact.hpp.

unsigned GeographicLib::GeodesicExact::maxit2_

private

Definition at line 87 of file GeodesicExact.hpp.

const int GeographicLib::GeodesicExact::nC4_ = GEOGRAPHICLIB_GEODESICEXACT_ORDER

staticprivate

Definition at line 84 of file GeodesicExact.hpp.

const int GeographicLib::GeodesicExact::nC4x_ = (nC4_ * (nC4_ + 1)) / 2

staticprivate

Definition at line 85 of file GeodesicExact.hpp.

real GeographicLib::GeodesicExact::tiny_

private

Definition at line 88 of file GeodesicExact.hpp.

real GeographicLib::GeodesicExact::tol0_

private

Definition at line 88 of file GeodesicExact.hpp.

real GeographicLib::GeodesicExact::tol1_

private

Definition at line 88 of file GeodesicExact.hpp.

real GeographicLib::GeodesicExact::tol2_

private

Definition at line 88 of file GeodesicExact.hpp.

real GeographicLib::GeodesicExact::tolb_

private

Definition at line 88 of file GeodesicExact.hpp.

real GeographicLib::GeodesicExact::xthresh_

private

Definition at line 88 of file GeodesicExact.hpp.

---

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

• GeodesicExact.hpp
• src/GeodesicExact.cpp
• GeodesicExactC4.cpp