GeographicLib::Ellipsoid Class Reference

Properties of an ellipsoid. More...

#include <Ellipsoid.hpp>

Public Member Functions
Constructor
	Ellipsoid (real a, real f)
Ellipsoid dimensions.
Math::real	MajorRadius () const
Math::real	MinorRadius () const
Math::real	QuarterMeridian () const
Math::real	Area () const
Math::real	Volume () const
Ellipsoid shape
Math::real	Flattening () const
Math::real	SecondFlattening () const
Math::real	ThirdFlattening () const
Math::real	EccentricitySq () const
Math::real	SecondEccentricitySq () const
Math::real	ThirdEccentricitySq () const
Latitude conversion.
Math::real	ParametricLatitude (real phi) const
Math::real	InverseParametricLatitude (real beta) const
Math::real	GeocentricLatitude (real phi) const
Math::real	InverseGeocentricLatitude (real theta) const
Math::real	RectifyingLatitude (real phi) const
Math::real	InverseRectifyingLatitude (real mu) const
Math::real	AuthalicLatitude (real phi) const
Math::real	InverseAuthalicLatitude (real xi) const
Math::real	ConformalLatitude (real phi) const
Math::real	InverseConformalLatitude (real chi) const
Math::real	IsometricLatitude (real phi) const
Math::real	InverseIsometricLatitude (real psi) const
Other quantities.
Math::real	CircleRadius (real phi) const
Math::real	CircleHeight (real phi) const
Math::real	MeridianDistance (real phi) const
Math::real	MeridionalCurvatureRadius (real phi) const
Math::real	TransverseCurvatureRadius (real phi) const
Math::real	NormalCurvatureRadius (real phi, real azi) const

Static Public Member Functions
static const Ellipsoid &	WGS84 ()
Eccentricity conversions.
static Math::real	SecondFlatteningToFlattening (real fp)
static Math::real	FlatteningToSecondFlattening (real f)
static Math::real	ThirdFlatteningToFlattening (real n)
static Math::real	FlatteningToThirdFlattening (real f)
static Math::real	EccentricitySqToFlattening (real e2)
static Math::real	FlatteningToEccentricitySq (real f)
static Math::real	SecondEccentricitySqToFlattening (real ep2)
static Math::real	FlatteningToSecondEccentricitySq (real f)
static Math::real	ThirdEccentricitySqToFlattening (real epp2)
static Math::real	FlatteningToThirdEccentricitySq (real f)

Private Types
typedef Math::real	real

Private Member Functions
const Math::real *	ConformalToRectifyingCoeffs () const
const Math::real *	RectifyingToConformalCoeffs () const

Private Attributes
real	_a
AlbersEqualArea	_au
real	_b
real	_e12
real	_e2
EllipticFunction	_ell
real	_es
real	_f
real	_f1
real	_f12
real	_n
TransverseMercator	_tm
real	stol_

Static Private Attributes
static const int	numit_ = 10

Friends
class	Rhumb
class	RhumbLine

Detailed Description

Properties of an ellipsoid.

This class returns various properties of the ellipsoid and converts between various types of latitudes. The latitude conversions are also possible using the various projections supported by GeographicLib; but Ellipsoid provides more direct access (sometimes using private functions of the projection classes). Ellipsoid::RectifyingLatitude, Ellipsoid::InverseRectifyingLatitude, and Ellipsoid::MeridianDistance provide functionality which can be provided by the Geodesic class. However Geodesic uses a series approximation (valid for abs f < 1/150), whereas Ellipsoid computes these quantities using EllipticFunction which provides accurate results even when f is large. Use of this class should be limited to −3 < f < 3/4 (i.e., 1/4 < b/a < 4).

Example of use:

// Example of using the GeographicLib::Ellipsoid class

#include <iostream>
#include <exception>
#include <GeographicLib/Ellipsoid.hpp>

using namespace std;
using namespace GeographicLib;

int main() {
  try {
    Ellipsoid wgs84(Constants::WGS84_a(), Constants::WGS84_f());
    // Alternatively: const Ellipsoid& wgs84 = Ellipsoid::WGS84();
    cout << "The latitude half way between the equator and the pole is "
         << wgs84.InverseRectifyingLatitude(45) << "\n";
    cout << "Half the area of the ellipsoid lies between latitudes +/- "
         << wgs84.InverseAuthalicLatitude(30) << "\n";
    cout << "The northernmost edge of a square Mercator map is at latitude "
         << wgs84.InverseIsometricLatitude(180) << "\n";
  }
  catch (const exception& e) {
    cerr << "Caught exception: " << e.what() << "\n";
    return 1;
  }
}

Definition at line 39 of file Ellipsoid.hpp.

Member Typedef Documentation

typedef Math::real GeographicLib::Ellipsoid::real

private

Definition at line 41 of file Ellipsoid.hpp.

Constructor & Destructor Documentation

GeographicLib::Ellipsoid::Ellipsoid	(	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 16 of file src/Ellipsoid.cpp.

Member Function Documentation

Math::real GeographicLib::Ellipsoid::Area

(

)

const

Returns

A the total area of the ellipsoid (meters²). For a sphere A = 4π a². The radius of a sphere with the same area is sqrt(A / (4π)).

Definition at line 40 of file src/Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::AuthalicLatitude

(

real

phi

)

const

Parameters

[in]

phi

the geographic latitude (degrees).

Returns

ξ the authalic latitude (degrees).

The authalic latitude, ξ, has the property that the area of the ellipsoid between two circles with authalic latitudes ξ₁ and ξ₂ is equal to (sin ξ₂ - sin ξ₁) A / 2, where A = Area(). For a sphere ξ = φ.

φ must lie in the range [−90°, 90°]; the result is undefined if this condition does not hold. The returned value ξ lies in [−90°, 90°].

Definition at line 72 of file src/Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::CircleHeight

(

real

phi

)

const

Parameters

[in]

phi

the geographic latitude (degrees).

Returns

Z = b sin β the distance of a circle of latitude φ from the equator measured parallel to the ellipsoid axis (meters).

φ must lie in the range [−90°, 90°]; the result is undefined if this condition does not hold.

Definition at line 97 of file src/Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::CircleRadius

(

real

phi

)

const

Parameters

[in]

phi

the geographic latitude (degrees).

Returns

R = a cos β the radius of a circle of latitude φ (meters). R (π/180°) gives meters per degree longitude measured along a circle of latitude.

φ must lie in the range [−90°, 90°]; the result is undefined if this condition does not hold.

Definition at line 91 of file src/Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::ConformalLatitude

(

real

phi

)

const

Parameters

[in]

phi

the geographic latitude (degrees).

Returns

χ the conformal latitude (degrees).

The conformal latitude, χ, gives the mapping of the ellipsoid to a sphere which which is conformal (angles are preserved) and in which the equator of the ellipsoid maps to the equator of the sphere. For a sphere χ = φ.

φ must lie in the range [−90°, 90°]; the result is undefined if this condition does not hold. The returned value χ lies in [−90°, 90°].

Definition at line 78 of file src/Ellipsoid.cpp.

const Math::real* GeographicLib::Ellipsoid::ConformalToRectifyingCoeffs ( ) const

inlineprivate

Definition at line 52 of file Ellipsoid.hpp.

Math::real GeographicLib::Ellipsoid::EccentricitySq ( ) const

inline

Returns

e² = (a² − b²) / a², the eccentricity squared of the ellipsoid. This is zero, positive, or negative for a sphere, oblate ellipsoid, or prolate ellipsoid.

Definition at line 142 of file Ellipsoid.hpp.

static Math::real GeographicLib::Ellipsoid::EccentricitySqToFlattening ( real  e2 )

inlinestatic

Parameters

[in]

e2

= e2 = (a2 − b2) / a2, the eccentricity squared.

Returns

f = (a − b) / a, the flattening.

e² should lie in (−∞, 1). The returned value f lies in (−∞, 1).

Definition at line 460 of file Ellipsoid.hpp.

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

inline

Returns

f = (a − b) / a, the flattening of the ellipsoid. This is the value used in the constructor. This is zero, positive, or negative for a sphere, oblate ellipsoid, or prolate ellipsoid.

Definition at line 120 of file Ellipsoid.hpp.

static Math::real GeographicLib::Ellipsoid::FlatteningToEccentricitySq ( real  f )

inlinestatic

Parameters

[in]

f

= (a − b) / a, the flattening.

Returns

e² = (a² − b²) / a², the eccentricity squared.

f should lie in (−∞, 1). The returned value e² lies in (−∞, 1).

Definition at line 472 of file Ellipsoid.hpp.

static Math::real GeographicLib::Ellipsoid::FlatteningToSecondEccentricitySq ( real  f )

inlinestatic

Parameters

[in]

f

= (a − b) / a, the flattening.

Returns

e' ² = (a² − b²) / b², the second eccentricity squared.

f should lie in (−∞, 1). The returned value e' ² lies in (−1, ∞).

Definition at line 496 of file Ellipsoid.hpp.

static Math::real GeographicLib::Ellipsoid::FlatteningToSecondFlattening ( real  f )

inlinestatic

Parameters

[in]

f

= (a − b) / a, the flattening.

Returns

f ' = (a − b) / b, the second flattening.

f should lie in (−∞, 1). The returned value f ' lies in (−1, ∞).

Definition at line 426 of file Ellipsoid.hpp.

static Math::real GeographicLib::Ellipsoid::FlatteningToThirdEccentricitySq ( real  f )

inlinestatic

Parameters

[in]

f

= (a − b) / a, the flattening.

Returns

e'' ² = (a² − b²) / (a² + b²), the third eccentricity squared.

f should lie in (−∞, 1). The returned value e'' ² lies in (−1, 1).

Definition at line 522 of file Ellipsoid.hpp.

static Math::real GeographicLib::Ellipsoid::FlatteningToThirdFlattening ( real  f )

inlinestatic

Parameters

[in]

f

= (a − b) / a, the flattening.

Returns

n = (a − b) / (a + b), the third flattening.

f should lie in (−∞, 1). The returned value n lies in (−1, 1).

Definition at line 448 of file Ellipsoid.hpp.

Math::real GeographicLib::Ellipsoid::GeocentricLatitude

(

real

phi

)

const

Parameters

[in]

phi

the geographic latitude (degrees).

Returns

θ the geocentric latitude (degrees).

The geocentric latitude, θ, is the angle beween the equatorial plane and a line between the center of the ellipsoid and a point on the ellipsoid. For a sphere θ = φ.

φ must lie in the range [−90°, 90°]; the result is undefined if this condition does not hold. The returned value θ lies in [−90°, 90°].

Definition at line 54 of file src/Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::InverseAuthalicLatitude

(

real

xi

)

const

Parameters

[in]

xi

the authalic latitude (degrees).

Returns

φ the geographic latitude (degrees).

ξ must lie in the range [−90°, 90°]; the result is undefined if this condition does not hold. The returned value φ lies in [−90°, 90°].

Definition at line 75 of file src/Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::InverseConformalLatitude

(

real

chi

)

const

Parameters

[in]

chi

the conformal latitude (degrees).

Returns

φ the geographic latitude (degrees).

χ must lie in the range [−90°, 90°]; the result is undefined if this condition does not hold. The returned value φ lies in [−90°, 90°].

Definition at line 81 of file src/Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::InverseGeocentricLatitude

(

real

theta

)

const

Parameters

[in]

theta

the geocentric latitude (degrees).

Returns

φ the geographic latitude (degrees).

θ must lie in the range [−90°, 90°]; the result is undefined if this condition does not hold. The returned value φ lies in [−90°, 90°].

Definition at line 57 of file src/Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::InverseIsometricLatitude

(

real

psi

)

const

Parameters

[in]

psi

the isometric latitude (degrees).

Returns

φ the geographic latitude (degrees).

The returned value φ lies in [−90°, 90°]. For a sphere φ = tan⁻¹ sinh ψ.

Definition at line 88 of file src/Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::InverseParametricLatitude

(

real

beta

)

const

Parameters

[in]

beta

the parametric latitude (degrees).

Returns

φ the geographic latitude (degrees).

β must lie in the range [−90°, 90°]; the result is undefined if this condition does not hold. The returned value φ lies in [−90°, 90°].

Definition at line 51 of file src/Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::InverseRectifyingLatitude

(

real

mu

)

const

Parameters

[in]

mu

the rectifying latitude (degrees).

Returns

φ the geographic latitude (degrees).

μ must lie in the range [−90°, 90°]; the result is undefined if this condition does not hold. The returned value φ lies in [−90°, 90°].

Definition at line 65 of file src/Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::IsometricLatitude

(

real

phi

)

const

Parameters

[in]

phi

the geographic latitude (degrees).

Returns

ψ the isometric latitude (degrees).

The isometric latitude gives the mapping of the ellipsoid to a plane which which is conformal (angles are preserved) and in which the equator of the ellipsoid maps to a straight line of constant scale; this mapping defines the Mercator projection. For a sphere ψ = sinh⁻¹ tan φ.

φ must lie in the range [−90°, 90°]; the result is undefined if this condition does not hold. The value returned for φ = ±90° is some (positive or negative) large but finite value, such that InverseIsometricLatitude returns the original value of φ.

Definition at line 84 of file src/Ellipsoid.cpp.

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

inline

Returns

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

Definition at line 80 of file Ellipsoid.hpp.

Math::real GeographicLib::Ellipsoid::MeridianDistance

(

real

phi

)

const

Parameters

[in]

phi

the geographic latitude (degrees).

Returns

s the distance along a meridian between the equator and a point of latitude φ (meters). s is given by s = μ L / 90°, where L = QuarterMeridian()).

φ must lie in the range [−90°, 90°]; the result is undefined if this condition does not hold.

Definition at line 104 of file src/Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::MeridionalCurvatureRadius

(

real

phi

)

const

Parameters

[in]

phi

the geographic latitude (degrees).

Returns

ρ the meridional radius of curvature of the ellipsoid at latitude φ (meters); this is the curvature of the meridian. rho is given by ρ = (180°/π) ds / dφ, where s = MeridianDistance(); thus ρ (π/180°) gives meters per degree latitude measured along a meridian.

φ must lie in the range [−90°, 90°]; the result is undefined if this condition does not hold.

Definition at line 107 of file src/Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::MinorRadius ( ) const

inline

Returns

b the polar semi-axis (meters).

Definition at line 85 of file Ellipsoid.hpp.

Math::real GeographicLib::Ellipsoid::NormalCurvatureRadius	(	real	phi,
		real	azi
	)		const

Parameters

[in]	phi	the geographic latitude (degrees).
[in]	azi	the angle between the meridian and the normal section (degrees).

Returns

the radius of curvature of the ellipsoid in the normal section at latitude φ inclined at an angle azi to the meridian (meters).

φ must lie in the range [−90°, 90°]; the result is undefined this condition does not hold.

Definition at line 117 of file src/Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::ParametricLatitude

(

real

phi

)

const

Parameters

[in]

phi

the geographic latitude (degrees).

Returns

β the parametric latitude (degrees).

The geographic latitude, φ, is the angle beween the equatorial plane and a vector normal to the surface of the ellipsoid.

The parametric latitude (also called the reduced latitude), β, allows the cartesian coordinated of a meridian to be expressed conveniently in parametric form as

• R = a cos β
• Z = b sin β

where a and b are the equatorial radius and the polar semi-axis. For a sphere β = φ.

φ must lie in the range [−90°, 90°]; the result is undefined if this condition does not hold. The returned value β lies in [−90°, 90°].

Definition at line 48 of file src/Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::QuarterMeridian

(

)

const

Returns

L the distance between the equator and a pole along a meridian (meters). For a sphere L = (π/2) a. The radius of a sphere with the same meridian length is L / (π/2).

Definition at line 37 of file src/Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::RectifyingLatitude

(

real

phi

)

const

Parameters

[in]

phi

the geographic latitude (degrees).

Returns

μ the rectifying latitude (degrees).

The rectifying latitude, μ, has the property that the distance along a meridian of the ellipsoid between two points with rectifying latitudes μ₁ and μ₂ is equal to (μ₂ - μ₁) L / 90°, where L = QuarterMeridian(). For a sphere μ = φ.

φ must lie in the range [−90°, 90°]; the result is undefined if this condition does not hold. The returned value μ lies in [−90°, 90°].

Definition at line 60 of file src/Ellipsoid.cpp.

const Math::real* GeographicLib::Ellipsoid::RectifyingToConformalCoeffs ( ) const

inlineprivate

Definition at line 53 of file Ellipsoid.hpp.

Math::real GeographicLib::Ellipsoid::SecondEccentricitySq ( ) const

inline

Returns

e' ² = (a² − b²) / b², the second eccentricity squared of the ellipsoid. This is zero, positive, or negative for a sphere, oblate ellipsoid, or prolate ellipsoid.

Definition at line 150 of file Ellipsoid.hpp.

static Math::real GeographicLib::Ellipsoid::SecondEccentricitySqToFlattening ( real  ep2 )

inlinestatic

Parameters

[in]

ep2

= e' 2 = (a2 − b2) / b2, the second eccentricity squared.

Returns

f = (a − b) / a, the flattening.

e' ² should lie in (−1, ∞). The returned value f lies in (−∞, 1).

Definition at line 484 of file Ellipsoid.hpp.

Math::real GeographicLib::Ellipsoid::SecondFlattening ( ) const

inline

Returns

f ' = (a − b) / b, the second flattening of the ellipsoid. This is zero, positive, or negative for a sphere, oblate ellipsoid, or prolate ellipsoid.

Definition at line 127 of file Ellipsoid.hpp.

static Math::real GeographicLib::Ellipsoid::SecondFlatteningToFlattening ( real  fp )

inlinestatic

Parameters

[in]

fp

= f ' = (a − b) / b, the second flattening.

Returns

f = (a − b) / a, the flattening.

f ' should lie in (−1, ∞). The returned value f lies in (−∞, 1).

Definition at line 416 of file Ellipsoid.hpp.

Math::real GeographicLib::Ellipsoid::ThirdEccentricitySq ( ) const

inline

Returns

e'' ² = (a² − b²) / (a² + b²), the third eccentricity squared of the ellipsoid. This is zero, positive, or negative for a sphere, oblate ellipsoid, or prolate ellipsoid.

Definition at line 159 of file Ellipsoid.hpp.

static Math::real GeographicLib::Ellipsoid::ThirdEccentricitySqToFlattening ( real  epp2 )

inlinestatic

Parameters

[in]

epp2

= e'' 2 = (a2 − b2) / (a2 + b2), the third eccentricity squared.

Returns

f = (a − b) / a, the flattening.

e'' ² should lie in (−1, 1). The returned value f lies in (−∞, 1).

Definition at line 508 of file Ellipsoid.hpp.

Math::real GeographicLib::Ellipsoid::ThirdFlattening ( ) const

inline

Returns

n = (a − b) / (a + b), the third flattening of the ellipsoid. This is zero, positive, or negative for a sphere, oblate ellipsoid, or prolate ellipsoid.

Definition at line 134 of file Ellipsoid.hpp.

static Math::real GeographicLib::Ellipsoid::ThirdFlatteningToFlattening ( real  n )

inlinestatic

Parameters

[in]

n

= (a − b) / (a + b), the third flattening.

Returns

f = (a − b) / a, the flattening.

n should lie in (−1, 1). The returned value f lies in (−∞, 1).

Definition at line 437 of file Ellipsoid.hpp.

Math::real GeographicLib::Ellipsoid::TransverseCurvatureRadius

(

real

phi

)

const

Parameters

[in]

phi

the geographic latitude (degrees).

Returns

ν the transverse radius of curvature of the ellipsoid at latitude φ (meters); this is the curvature of a curve on the ellipsoid which also lies in a plane perpendicular to the ellipsoid and to the meridian. ν is related to R = CircleRadius() by R = ν cos φ.

φ must lie in the range [−90°, 90°]; the result is undefined if this condition does not hold.

Definition at line 112 of file src/Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::Volume ( ) const

inline

Returns

V the total volume of the ellipsoid (meters³). For a sphere V = (4π / 3) a³. The radius of a sphere with the same volume is cbrt(V / (4π/3)).

Definition at line 106 of file Ellipsoid.hpp.

const Ellipsoid & GeographicLib::Ellipsoid::WGS84 ( )

static

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

Definition at line 32 of file src/Ellipsoid.cpp.

Friends And Related Function Documentation

friend class Rhumb

friend

Definition at line 54 of file Ellipsoid.hpp.

friend class RhumbLine

friend

Definition at line 54 of file Ellipsoid.hpp.

Member Data Documentation

real GeographicLib::Ellipsoid::_a

private

Definition at line 44 of file Ellipsoid.hpp.

AlbersEqualArea GeographicLib::Ellipsoid::_au

private

Definition at line 47 of file Ellipsoid.hpp.

real GeographicLib::Ellipsoid::_b

private

Definition at line 44 of file Ellipsoid.hpp.

real GeographicLib::Ellipsoid::_e12

private

Definition at line 44 of file Ellipsoid.hpp.

real GeographicLib::Ellipsoid::_e2

private

Definition at line 44 of file Ellipsoid.hpp.

EllipticFunction GeographicLib::Ellipsoid::_ell

private

Definition at line 46 of file Ellipsoid.hpp.

real GeographicLib::Ellipsoid::_es

private

Definition at line 44 of file Ellipsoid.hpp.

real GeographicLib::Ellipsoid::_f

private

Definition at line 44 of file Ellipsoid.hpp.

real GeographicLib::Ellipsoid::_f1

private

Definition at line 44 of file Ellipsoid.hpp.

real GeographicLib::Ellipsoid::_f12

private

Definition at line 44 of file Ellipsoid.hpp.

real GeographicLib::Ellipsoid::_n

private

Definition at line 44 of file Ellipsoid.hpp.

TransverseMercator GeographicLib::Ellipsoid::_tm

private

Definition at line 45 of file Ellipsoid.hpp.

const int GeographicLib::Ellipsoid::numit_ = 10

staticprivate

Definition at line 42 of file Ellipsoid.hpp.

real GeographicLib::Ellipsoid::stol_

private

Definition at line 43 of file Ellipsoid.hpp.

---

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

• Ellipsoid.hpp
• src/Ellipsoid.cpp