Properties of an ellipsoid. More...

#include <GeographicLib/Ellipsoid.hpp>

Public Member Functions
Constructor
	Ellipsoid (real a, real f)

Ellipsoid dimensions.
Math::real	EquatorialRadius () const

Math::real	PolarRadius () 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)

Detailed Description

Properties of an ellipsoid.

This class returns various properties of the ellipsoid and converts between various types of latitudes. This is for the most part a thin wrapper on top of the AuxLatitude class which is called with exact = true so that the results are valid for arbitrary flattenings −100 < f < 99/100 (i.e., 1/100 < b/a < 100).

Example of use:

// Example of using the GeographicLib::Ellipsoid class
 
#include <iostream>
#include <iomanip>
#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";
    cout << "Table phi(deg)  beta-phi xi-phi mu-phi chi-phi theta-phi (mins)\n"
         << fixed << setprecision(2);
    for (int i = 0; i <= 90; i += 15) {
      double phi = i,
        bet = wgs84.ParametricLatitude(phi),
        xi = wgs84.AuthalicLatitude(phi),
        mu = wgs84.RectifyingLatitude(phi),
        chi = wgs84.ConformalLatitude(phi),
        theta = wgs84.GeocentricLatitude(phi);
      cout << i << " "
           << (bet-phi)*60 << " "
           << (xi-phi)*60 << " "
           << (mu-phi)*60 << " "
           << (chi-phi)*60 << " "
           << (theta-phi)*60 << "\n";
    }
  }
  catch (const exception& e) {
    cerr << "Caught exception: " << e.what() << "\n";
    return 1;
  }
}

Definition at line 31 of file Ellipsoid.hpp.

Constructor & Destructor Documentation

◆ Ellipsoid()

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

Constructor for an 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.

Member Function Documentation

◆ EquatorialRadius()

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

inline

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

Definition at line 65 of file Ellipsoid.hpp.

◆ PolarRadius()

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

inline

Returns: b the polar semi-axis (meters).

Definition at line 70 of file Ellipsoid.hpp.

◆ QuarterMeridian()

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 36 of file Ellipsoid.cpp.

◆ Area()

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 39 of file Ellipsoid.cpp.

◆ Volume()

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 91 of file Ellipsoid.hpp.

◆ Flattening()

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 105 of file Ellipsoid.hpp.

◆ SecondFlattening()

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 112 of file Ellipsoid.hpp.

◆ ThirdFlattening()

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 119 of file Ellipsoid.hpp.

◆ EccentricitySq()

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 127 of file Ellipsoid.hpp.

◆ SecondEccentricitySq()

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 135 of file Ellipsoid.hpp.

◆ ThirdEccentricitySq()

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 144 of file Ellipsoid.hpp.

◆ ParametricLatitude()

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 between 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 42 of file Ellipsoid.cpp.

◆ InverseParametricLatitude()

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 47 of file Ellipsoid.cpp.

◆ GeocentricLatitude()

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 between 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 52 of file Ellipsoid.cpp.

◆ InverseGeocentricLatitude()

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 Ellipsoid.cpp.

◆ RectifyingLatitude()

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 62 of file Ellipsoid.cpp.

◆ InverseRectifyingLatitude()

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 67 of file Ellipsoid.cpp.

◆ AuthalicLatitude()

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 Ellipsoid.cpp.

◆ InverseAuthalicLatitude()

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 77 of file Ellipsoid.cpp.

◆ ConformalLatitude()

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 82 of file Ellipsoid.cpp.

◆ InverseConformalLatitude()

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 87 of file Ellipsoid.cpp.

◆ IsometricLatitude()

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 92 of file Ellipsoid.cpp.

◆ InverseIsometricLatitude()

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 97 of file Ellipsoid.cpp.

◆ CircleRadius()

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 102 of file Ellipsoid.cpp.

References GeographicLib::AuxAngle::x().

◆ CircleHeight()

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 110 of file Ellipsoid.cpp.

References GeographicLib::AuxAngle::y().

◆ MeridianDistance()

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 118 of file Ellipsoid.cpp.

◆ MeridionalCurvatureRadius()

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 124 of file Ellipsoid.cpp.

◆ TransverseCurvatureRadius()

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 129 of file Ellipsoid.cpp.

◆ NormalCurvatureRadius()

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 134 of file Ellipsoid.cpp.

◆ SecondFlatteningToFlattening()

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 401 of file Ellipsoid.hpp.

◆ FlatteningToSecondFlattening()

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 411 of file Ellipsoid.hpp.

◆ ThirdFlatteningToFlattening()

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 422 of file Ellipsoid.hpp.

◆ FlatteningToThirdFlattening()

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 433 of file Ellipsoid.hpp.

◆ EccentricitySqToFlattening()

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 445 of file Ellipsoid.hpp.

◆ FlatteningToEccentricitySq()

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 457 of file Ellipsoid.hpp.

◆ SecondEccentricitySqToFlattening()

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 469 of file Ellipsoid.hpp.

◆ FlatteningToSecondEccentricitySq()

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 481 of file Ellipsoid.hpp.

◆ ThirdEccentricitySqToFlattening()

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 493 of file Ellipsoid.hpp.

◆ FlatteningToThirdEccentricitySq()

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 507 of file Ellipsoid.hpp.

◆ WGS84()

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

static

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

Definition at line 31 of file Ellipsoid.cpp.

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

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ Ellipsoid()

Member Function Documentation

◆ EquatorialRadius()

◆ PolarRadius()

◆ QuarterMeridian()

◆ Area()

◆ Volume()

◆ Flattening()

◆ SecondFlattening()

◆ ThirdFlattening()

◆ EccentricitySq()

◆ SecondEccentricitySq()

◆ ThirdEccentricitySq()

◆ ParametricLatitude()

◆ InverseParametricLatitude()

◆ GeocentricLatitude()

◆ InverseGeocentricLatitude()

◆ RectifyingLatitude()

◆ InverseRectifyingLatitude()

◆ AuthalicLatitude()

◆ InverseAuthalicLatitude()

◆ ConformalLatitude()

◆ InverseConformalLatitude()

◆ IsometricLatitude()

◆ InverseIsometricLatitude()

◆ CircleRadius()

◆ CircleHeight()

◆ MeridianDistance()

◆ MeridionalCurvatureRadius()

◆ TransverseCurvatureRadius()

◆ NormalCurvatureRadius()

◆ SecondFlatteningToFlattening()

◆ FlatteningToSecondFlattening()

◆ ThirdFlatteningToFlattening()

◆ FlatteningToThirdFlattening()

◆ EccentricitySqToFlattening()

◆ FlatteningToEccentricitySq()

◆ SecondEccentricitySqToFlattening()

◆ FlatteningToSecondEccentricitySq()

◆ ThirdEccentricitySqToFlattening()

◆ FlatteningToThirdEccentricitySq()

◆ WGS84()