Geodesic calculations More...

#include <Geodesic.hpp>

Public Types
enum	mask { NONE, LATITUDE, LONGITUDE, AZIMUTH, DISTANCE, DISTANCE_IN, REDUCEDLENGTH, GEODESICSCALE, AREA, LONG_UNROLL, 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

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 GeodesicLine.
GeodesicLine	Line (real lat1, real lon1, real azi1, unsigned caps=ALL) const

GeodesicLine	InverseLine (real lat1, real lon1, real lat2, real lon2, unsigned caps=ALL) const

GeodesicLine	DirectLine (real lat1, real lon1, real azi1, real s12, unsigned caps=ALL) const

GeodesicLine	ArcDirectLine (real lat1, real lon1, real azi1, real a12, unsigned caps=ALL) const

GeodesicLine	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 Geodesic &	WGS84 ()

Private Types
enum	captype { CAP_NONE = 0U, CAP_C1 = 1U<<0, CAP_C1p = 1U<<1, CAP_C2 = 1U<<2, CAP_C3 = 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	A3coeff ()

real	A3f (real eps) const

void	C3coeff ()

void	C3f (real eps, real c[]) const

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 (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, real Ca[]) 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, real &eps, real &domg12, bool diffp, real &dlam12, real Ca[]) const

void	Lengths (real eps, 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, real Ca[]) const

Static Private Member Functions
static real	A1m1f (real eps)

static real	A2m1f (real eps)

static real	Astroid (real x, real y)

static void	C1f (real eps, real c[])

static void	C1pf (real eps, real c[])

static void	C2f (real eps, real c[])

static real	SinCosSeries (bool sinp, real sinx, real cosx, const real c[], int n)

Private Attributes
real	_a

real	_A3x [nA3x_]

real	_b

real	_c2

real	_C3x [nC3x_]

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	nA1_ = GEOGRAPHICLIB_GEODESIC_ORDER

static const int	nA2_ = GEOGRAPHICLIB_GEODESIC_ORDER

static const int	nA3_ = GEOGRAPHICLIB_GEODESIC_ORDER

static const int	nA3x_ = nA3_

static const int	nC1_ = GEOGRAPHICLIB_GEODESIC_ORDER

static const int	nC1p_ = GEOGRAPHICLIB_GEODESIC_ORDER

static const int	nC2_ = GEOGRAPHICLIB_GEODESIC_ORDER

static const int	nC3_ = GEOGRAPHICLIB_GEODESIC_ORDER

static const int	nC3x_ = (nC3_ * (nC3_ - 1)) / 2

static const int	nC4_ = GEOGRAPHICLIB_GEODESIC_ORDER

static const int	nC4x_ = (nC4_ * (nC4_ + 1)) / 2

static const int	nC_ = GEOGRAPHICLIB_GEODESIC_ORDER + 1

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.

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:

reduced length. If we fix the first point and increase azi1 by dazi1 (radians), the second point is displaced m12 dazi1 in the direction azi2 + 90°. The quantity m12 is called the "reduced length" and is symmetric under interchange of the two points. On a curved surface the reduced length obeys a symmetry relation, m12 + m21 = 0. On a flat surface, we have m12 = s12. The ratio s12/m12 gives the azimuthal scale for an azimuthal equidistant projection.
geodesic scale. Consider a reference geodesic and a second geodesic parallel to this one at point 1 and separated by a small distance dt. The separation of the two geodesics at point 2 is M12 dt where M12 is called the "geodesic scale". M21 is defined similarly (with the geodesics being parallel at point 2). On a flat surface, we have M12 = M21 = 1. The quantity 1/M12 gives the scale of the Cassini-Soldner projection.
area. The area between the geodesic from point 1 to point 2 and the equation is represented by S12; it is the area, measured counter-clockwise, of the geodesic quadrilateral with corners (lat1,lon1), (0,lon1), (0,lon2), and (lat2,lon2). It can be used to compute the area of any simple geodesic polygon.

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:

s13 = s12 + s23
a13 = a12 + a23
S13 = S12 + S23
m13 = m12 M23 + m23 M21
M13 = M12 M23 − (1 − M12 M21) m23 / m12
M31 = M32 M21 − (1 − M23 M32) m12 / m23

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:

lat1 = −lat2 (with neither point at a pole). If azi1 = azi2, the geodesic is unique. Otherwise there are two geodesics and the second one is obtained by setting [azi1, azi2] → [azi2, azi1], [M12, M21] → [M21, M12], S12 → −S12. (This occurs when the longitude difference is near ±180° for oblate ellipsoids.)
lon2 = lon1 ± 180° (with neither point at a pole). If azi1 = 0° or ±180°, the geodesic is unique. Otherwise there are two geodesics and the second one is obtained by setting [azi1, azi2] → [−azi1, −azi2], S12 → −S12. (This occurs when lat2 is near −lat1 for prolate ellipsoids.)
Points 1 and 2 at opposite poles. There are infinitely many geodesics which can be generated by setting [azi1, azi2] → [azi1, azi2] + [d, −d], for arbitrary d. (For spheres, this prescription applies when points 1 and 2 are antipodal.)
s12 = 0 (coincident points). There are infinitely many geodesics which can be generated by setting [azi1, azi2] → [azi1, azi2] + [d, d], for arbitrary d.

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 equatorial 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

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

For more information on geodesics see geodesic.

Example of use:

// Example of using the GeographicLib::Geodesic class
#include <iostream>
#include <exception>
#include <GeographicLib/Geodesic.hpp>
#include <GeographicLib/Constants.hpp>
using namespace std;
using namespace GeographicLib;
int main() {
  try {
    Geodesic geod(Constants::WGS84_a(), Constants::WGS84_f());
    // 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;
  }
}

GeodSolve is a command-line utility providing access to the functionality of Geodesic and GeodesicLine.

Definition at line 172 of file Geodesic.hpp.

Member Typedef Documentation

◆ real

typedef Math::real GeographicLib::Geodesic::real

private

Definition at line 174 of file Geodesic.hpp.

Member Enumeration Documentation

◆ captype

enum GeographicLib::Geodesic::captype

private

Enumerator
CAP_NONE
CAP_C1
CAP_C1p
CAP_C2
CAP_C3
CAP_C4
CAP_ALL
CAP_MASK
OUT_ALL
OUT_MASK

Definition at line 194 of file Geodesic.hpp.

◆ mask

enum GeographicLib::Geodesic::mask

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

Definition at line 263 of file Geodesic.hpp.

Constructor & Destructor Documentation

◆ Geodesic()

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

Member Function Documentation

◆ A1m1f()

Math::real GeographicLib::Geodesic::A1m1f ( real eps )

staticprivate

Definition at line 956 of file src/Geodesic.cpp.

◆ A2m1f()

Math::real GeographicLib::Geodesic::A2m1f ( real eps )

staticprivate

Definition at line 1201 of file src/Geodesic.cpp.

◆ A3coeff()

void GeographicLib::Geodesic::A3coeff ( )

private

Definition at line 1340 of file src/Geodesic.cpp.

◆ A3f()

Math::real GeographicLib::Geodesic::A3f ( real eps ) const

private

Definition at line 899 of file src/Geodesic.cpp.

◆ ArcDirect() [1/7]

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

inline

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

Definition at line 491 of file Geodesic.hpp.

◆ ArcDirect() [2/7]

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

inline

See the documentation for Geodesic::ArcDirect.

Definition at line 504 of file Geodesic.hpp.

◆ ArcDirect() [3/7]

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

inline

See the documentation for Geodesic::ArcDirect.

Definition at line 515 of file Geodesic.hpp.

◆ ArcDirect() [4/7]

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

inline

See the documentation for Geodesic::ArcDirect.

Definition at line 526 of file Geodesic.hpp.

◆ ArcDirect() [5/7]

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

Definition at line 538 of file Geodesic.hpp.

◆ ArcDirect() [6/7]

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

inline

See the documentation for Geodesic::ArcDirect.

Definition at line 551 of file Geodesic.hpp.

◆ ArcDirect() [7/7]

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

inline

See the documentation for Geodesic::ArcDirect.

Definition at line 564 of file Geodesic.hpp.

◆ ArcDirectLine()

GeodesicLine GeographicLib::Geodesic::ArcDirectLine	(	real	lat1,
		real	lon1,
		real	azi1,
		real	a12,
		unsigned	caps = `ALL`
	)		const

Define a GeodesicLine 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 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.

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

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

Definition at line 154 of file src/Geodesic.cpp.

◆ Astroid()

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

staticprivate

Definition at line 587 of file src/Geodesic.cpp.

◆ C1f()

void GeographicLib::Geodesic::C1f	(	real	eps,
		real	c[]
	)

staticprivate

Definition at line 989 of file src/Geodesic.cpp.

◆ C1pf()

void GeographicLib::Geodesic::C1pf	(	real	eps,
		real	c[]
	)

staticprivate

Definition at line 1095 of file src/Geodesic.cpp.

◆ C2f()

void GeographicLib::Geodesic::C2f	(	real	eps,
		real	c[]
	)

staticprivate

Definition at line 1234 of file src/Geodesic.cpp.

◆ C3coeff()

void GeographicLib::Geodesic::C3coeff ( )

private

Definition at line 1442 of file src/Geodesic.cpp.

◆ C3f()

void GeographicLib::Geodesic::C3f	(	real	eps,
		real	c[]
	)		const

private

Definition at line 904 of file src/Geodesic.cpp.

◆ C4coeff()

void GeographicLib::Geodesic::C4coeff ( )

private

Definition at line 1647 of file src/Geodesic.cpp.

◆ C4f()

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

private

Definition at line 918 of file src/Geodesic.cpp.

◆ Direct() [1/6]

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

inline

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

◆ Direct() [2/6]

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

inline

See the documentation for Geodesic::Direct.

Definition at line 393 of file Geodesic.hpp.

◆ Direct() [3/6]

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

inline

See the documentation for Geodesic::Direct.

Definition at line 405 of file Geodesic.hpp.

◆ Direct() [4/6]

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

inline

See the documentation for Geodesic::Direct.

Definition at line 417 of file Geodesic.hpp.

◆ Direct() [5/6]

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

Definition at line 429 of file Geodesic.hpp.

◆ Direct() [6/6]

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

inline

See the documentation for Geodesic::Direct.

Definition at line 442 of file Geodesic.hpp.

◆ DirectLine()

GeodesicLine GeographicLib::Geodesic::DirectLine	(	real	lat1,
		real	lon1,
		real	azi1,
		real	s12,
		unsigned	caps = `ALL`
	)		const

Define a GeodesicLine 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 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.

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

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

Definition at line 149 of file src/Geodesic.cpp.

◆ EllipsoidArea()

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

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 957 of file Geodesic.hpp.

◆ Flattening()

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

inline

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

Definition at line 949 of file Geodesic.hpp.

◆ GenDirect()

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

The general direct geodesic problem. Geodesic::Direct and Geodesic::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 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]	outmask	a bitor'ed combination of Geodesic::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 Geodesic::mask values possible for outmask are

outmask |= Geodesic::LATITUDE for the latitude lat2;
outmask |= Geodesic::LONGITUDE for the latitude lon2;
outmask |= Geodesic::AZIMUTH for the latitude azi2;
outmask |= Geodesic::DISTANCE for the distance s12;
outmask |= Geodesic::REDUCEDLENGTH for the reduced length m12;
outmask |= Geodesic::GEODESICSCALE for the geodesic scales M12 and M21;
outmask |= Geodesic::AREA for the area S12;
outmask |= Geodesic::ALL for all of the above;
outmask |= Geodesic::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 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.

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

Definition at line 123 of file src/Geodesic.cpp.

◆ GenDirectLine()

GeodesicLine GeographicLib::Geodesic::GenDirectLine	(	real	lat1,
		real	lon1,
		real	azi1,
		bool	arcmode,
		real	s12_a12,
		unsigned	caps = `ALL`
	)		const

Define a GeodesicLine 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 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.

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

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

Definition at line 136 of file src/Geodesic.cpp.

◆ GenInverse() [1/2]

Math::real GeographicLib::Geodesic::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 159 of file src/Geodesic.cpp.

◆ GenInverse() [2/2]

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

The general inverse geodesic calculation. Geodesic::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 Geodesic::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 Geodesic::mask values possible for outmask are

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

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

Definition at line 493 of file src/Geodesic.cpp.

◆ Inverse() [1/7]

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

inline

Solve 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).
[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 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 674 of file Geodesic.hpp.

◆ Inverse() [2/7]

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

inline

See the documentation for Geodesic::Inverse.

Definition at line 686 of file Geodesic.hpp.

◆ Inverse() [3/7]

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

inline

See the documentation for Geodesic::Inverse.

Definition at line 697 of file Geodesic.hpp.

◆ Inverse() [4/7]

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

inline

See the documentation for Geodesic::Inverse.

Definition at line 708 of file Geodesic.hpp.

◆ Inverse() [5/7]

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

inline

See the documentation for Geodesic::Inverse.

Definition at line 720 of file Geodesic.hpp.

◆ Inverse() [6/7]

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

Definition at line 732 of file Geodesic.hpp.

◆ Inverse() [7/7]

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

inline

See the documentation for Geodesic::Inverse.

Definition at line 744 of file Geodesic.hpp.

◆ InverseLine()

GeodesicLine GeographicLib::Geodesic::InverseLine	(	real	lat1,
		real	lon1,
		real	lat2,
		real	lon2,
		unsigned	caps = `ALL`
	)		const

Define a GeodesicLine 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 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.

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

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

Definition at line 510 of file src/Geodesic.cpp.

◆ InverseStart()

Math::real GeographicLib::Geodesic::InverseStart	(	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,
		real	Ca[]
	)		const

private

Definition at line 639 of file src/Geodesic.cpp.

◆ Lambda12()

Math::real GeographicLib::Geodesic::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,
		real &	eps,
		real &	domg12,
		bool	diffp,
		real &	dlam12,
		real	Ca[]
	)		const

private

Definition at line 812 of file src/Geodesic.cpp.

◆ Lengths()

void GeographicLib::Geodesic::Lengths	(	real	eps,
		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,
		real	Ca[]
	)		const

private

Definition at line 525 of file src/Geodesic.cpp.

◆ Line()

GeodesicLine GeographicLib::Geodesic::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 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°].

The Geodesic::mask values are

caps |= Geodesic::LATITUDE for the latitude lat2; this is added automatically;
caps |= Geodesic::LONGITUDE for the latitude lon2;
caps |= Geodesic::AZIMUTH for the azimuth azi2; this is added automatically;
caps |= Geodesic::DISTANCE for the distance s12;
caps |= Geodesic::REDUCEDLENGTH for the reduced length m12;
caps |= Geodesic::GEODESICSCALE for the geodesic scales M12 and M21;
caps |= Geodesic::AREA for the area S12;
caps |= Geodesic::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 |= Geodesic::ALL for all of the above.

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 118 of file src/Geodesic.cpp.

◆ MajorRadius()

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

inline

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

Definition at line 943 of file Geodesic.hpp.

◆ SinCosSeries()

Math::real GeographicLib::Geodesic::SinCosSeries	(	bool	sinp,
		real	sinx,
		real	cosx,
		const real	c[],
		int	n
	)

staticprivate

Definition at line 94 of file src/Geodesic.cpp.

◆ WGS84()

const Geodesic & GeographicLib::Geodesic::WGS84 ( )

static

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

Definition at line 89 of file src/Geodesic.cpp.

Friends And Related Function Documentation

◆ GeodesicLine

friend class GeodesicLine

real GeographicLib::Geodesic::tol2_

private

Definition at line 192 of file Geodesic.hpp.

◆ tolb_

real GeographicLib::Geodesic::tolb_

private

Definition at line 192 of file Geodesic.hpp.

◆ xthresh_

real GeographicLib::Geodesic::xthresh_

private

Definition at line 192 of file Geodesic.hpp.

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

Public Types

Public Member Functions

Static Public Member Functions

Private Types

Private Member Functions

Static Private Member Functions

Private Attributes

Static Private Attributes

Friends

Detailed Description

Member Typedef Documentation

◆ real

Member Enumeration Documentation

◆ captype

◆ mask

Constructor & Destructor Documentation

◆ Geodesic()

Member Function Documentation

◆ A1m1f()

◆ A2m1f()

◆ A3coeff()

◆ A3f()

◆ ArcDirect() [1/7]

◆ ArcDirect() [2/7]

◆ ArcDirect() [3/7]

◆ ArcDirect() [4/7]

◆ ArcDirect() [5/7]

◆ ArcDirect() [6/7]

◆ ArcDirect() [7/7]

◆ ArcDirectLine()

◆ Astroid()

◆ C1f()

◆ C1pf()

◆ C2f()

◆ C3coeff()

◆ C3f()

◆ C4coeff()

◆ C4f()

◆ Direct() [1/6]

◆ Direct() [2/6]

◆ Direct() [3/6]

◆ Direct() [4/6]

◆ Direct() [5/6]

◆ Direct() [6/6]

◆ DirectLine()

◆ EllipsoidArea()

◆ Flattening()

◆ GenDirect()

◆ GenDirectLine()

◆ GenInverse() [1/2]

◆ GenInverse() [2/2]

◆ Inverse() [1/7]

◆ Inverse() [2/7]

◆ Inverse() [3/7]

◆ Inverse() [4/7]

◆ Inverse() [5/7]

◆ Inverse() [6/7]

◆ Inverse() [7/7]

◆ InverseLine()

◆ InverseStart()

◆ Lambda12()

◆ Lengths()

◆ Line()

◆ MajorRadius()

◆ SinCosSeries()

◆ WGS84()

Friends And Related Function Documentation

◆ GeodesicLine

Member Data Documentation

◆ _a

◆ _A3x

◆ _b

◆ _c2

◆ _C3x

◆ _C4x

◆ _e2

◆ _ep2

◆ _etol2

◆ _f

◆ _f1