GeographicLib  1.50.1
GeographicLib::Ellipsoid Class Reference

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 MinorRadius () const
 
Math::real QuarterMeridian () const
 
Math::real Area () const
 
Math::real Volume () const
 
Math::real MajorRadius () 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 EllipsoidWGS84 ()
 
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)
 

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 <iomanip>
#include <exception>
using namespace std;
using namespace GeographicLib;
int main() {
try {
// 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 39 of file Ellipsoid.hpp.

Constructor & Destructor Documentation

◆ Ellipsoid()

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

Constructor for a ellipsoid with

Parameters
[in]aequatorial radius (meters).
[in]fflattening of ellipsoid. Setting f = 0 gives a sphere. Negative f gives a prolate ellipsoid.
Exceptions
GeographicErrif a or (1 − f) a is not positive.

Definition at line 16 of file Ellipsoid.cpp.

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 80 of file Ellipsoid.hpp.

Referenced by GeographicLib::Rhumb::EquatorialRadius().

◆ MinorRadius()

Math::real GeographicLib::Ellipsoid::MinorRadius ( ) const
inline
Returns
b the polar semi-axis (meters).

Definition at line 85 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 37 of file Ellipsoid.cpp.

References GeographicLib::EllipticFunction::E().

Referenced by GeographicLib::Rhumb::GenInverse(), GeographicLib::RhumbLine::GenPosition(), and RectifyingLatitude().

◆ 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 40 of file Ellipsoid.cpp.

References GeographicLib::Math::atanh(), GeographicLib::Math::pi(), and GeographicLib::Math::sq().

Referenced by GeographicLib::Rhumb::EllipsoidArea(), and main().

◆ 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 106 of file Ellipsoid.hpp.

References GeographicLib::Math::pi(), and GeographicLib::Math::sq().

◆ MajorRadius()

Math::real GeographicLib::Ellipsoid::MajorRadius ( ) const
inline
Deprecated:
An old name for EquatorialRadius().

Definition at line 113 of file Ellipsoid.hpp.

◆ Flattening()

Math::real GeographicLib::Ellipsoid::Flattening ( ) const
inline
Returns
f = (ab) / 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 126 of file Ellipsoid.hpp.

Referenced by GeographicLib::Rhumb::Flattening().

◆ SecondFlattening()

Math::real GeographicLib::Ellipsoid::SecondFlattening ( ) const
inline
Returns
f ' = (ab) / b, the second flattening of the ellipsoid. This is zero, positive, or negative for a sphere, oblate ellipsoid, or prolate ellipsoid.

Definition at line 133 of file Ellipsoid.hpp.

◆ ThirdFlattening()

Math::real GeographicLib::Ellipsoid::ThirdFlattening ( ) const
inline
Returns
n = (ab) / (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 140 of file Ellipsoid.hpp.

◆ EccentricitySq()

Math::real GeographicLib::Ellipsoid::EccentricitySq ( ) const
inline
Returns
e2 = (a2b2) / a2, the eccentricity squared of the ellipsoid. This is zero, positive, or negative for a sphere, oblate ellipsoid, or prolate ellipsoid.

Definition at line 148 of file Ellipsoid.hpp.

◆ SecondEccentricitySq()

Math::real GeographicLib::Ellipsoid::SecondEccentricitySq ( ) const
inline
Returns
e' 2 = (a2b2) / 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 156 of file Ellipsoid.hpp.

◆ ThirdEccentricitySq()

Math::real GeographicLib::Ellipsoid::ThirdEccentricitySq ( ) const
inline
Returns
e'' 2 = (a2b2) / (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 165 of file Ellipsoid.hpp.

◆ ParametricLatitude()

Math::real GeographicLib::Ellipsoid::ParametricLatitude ( real  phi) const
Parameters
[in]phithe 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 48 of file Ellipsoid.cpp.

References GeographicLib::Math::atand(), GeographicLib::Math::LatFix(), and GeographicLib::Math::tand().

Referenced by MeridianDistance().

◆ InverseParametricLatitude()

Math::real GeographicLib::Ellipsoid::InverseParametricLatitude ( real  beta) const
Parameters
[in]betathe 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 Ellipsoid.cpp.

References GeographicLib::Math::atand(), GeographicLib::Math::LatFix(), and GeographicLib::Math::tand().

Referenced by InverseRectifyingLatitude().

◆ GeocentricLatitude()

Math::real GeographicLib::Ellipsoid::GeocentricLatitude ( real  phi) const
Parameters
[in]phithe 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 54 of file Ellipsoid.cpp.

References GeographicLib::Math::atand(), GeographicLib::Math::LatFix(), and GeographicLib::Math::tand().

◆ InverseGeocentricLatitude()

Math::real GeographicLib::Ellipsoid::InverseGeocentricLatitude ( real  theta) const
Parameters
[in]thetathe 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.

References GeographicLib::Math::atand(), GeographicLib::Math::LatFix(), and GeographicLib::Math::tand().

◆ RectifyingLatitude()

Math::real GeographicLib::Ellipsoid::RectifyingLatitude ( real  phi) const
Parameters
[in]phithe 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 Ellipsoid.cpp.

References MeridianDistance(), and QuarterMeridian().

◆ InverseRectifyingLatitude()

Math::real GeographicLib::Ellipsoid::InverseRectifyingLatitude ( real  mu) const
Parameters
[in]muthe 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 Ellipsoid.cpp.

References GeographicLib::Math::degree(), GeographicLib::EllipticFunction::E(), GeographicLib::EllipticFunction::Einv(), and InverseParametricLatitude().

Referenced by GeographicLib::RhumbLine::GenPosition().

◆ AuthalicLatitude()

Math::real GeographicLib::Ellipsoid::AuthalicLatitude ( real  phi) const
Parameters
[in]phithe 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.

References GeographicLib::Math::atand(), GeographicLib::Math::LatFix(), and GeographicLib::Math::tand().

Referenced by main().

◆ InverseAuthalicLatitude()

Math::real GeographicLib::Ellipsoid::InverseAuthalicLatitude ( real  xi) const
Parameters
[in]xithe 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 Ellipsoid.cpp.

References GeographicLib::Math::atand(), GeographicLib::Math::LatFix(), and GeographicLib::Math::tand().

◆ ConformalLatitude()

Math::real GeographicLib::Ellipsoid::ConformalLatitude ( real  phi) const
Parameters
[in]phithe 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 Ellipsoid.cpp.

References GeographicLib::Math::atand(), GeographicLib::Math::LatFix(), GeographicLib::Math::tand(), and GeographicLib::Math::taupf().

◆ InverseConformalLatitude()

Math::real GeographicLib::Ellipsoid::InverseConformalLatitude ( real  chi) const
Parameters
[in]chithe 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 Ellipsoid.cpp.

References GeographicLib::Math::atand(), GeographicLib::Math::LatFix(), GeographicLib::Math::tand(), and GeographicLib::Math::tauf().

◆ IsometricLatitude()

Math::real GeographicLib::Ellipsoid::IsometricLatitude ( real  phi) const
Parameters
[in]phithe 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 Ellipsoid.cpp.

References GeographicLib::Math::asinh(), GeographicLib::Math::degree(), GeographicLib::Math::LatFix(), GeographicLib::Math::tand(), and GeographicLib::Math::taupf().

Referenced by GeographicLib::Rhumb::GenInverse().

◆ InverseIsometricLatitude()

Math::real GeographicLib::Ellipsoid::InverseIsometricLatitude ( real  psi) const
Parameters
[in]psithe 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 Ellipsoid.cpp.

References GeographicLib::Math::atand(), GeographicLib::Math::degree(), and GeographicLib::Math::tauf().

◆ CircleRadius()

Math::real GeographicLib::Ellipsoid::CircleRadius ( real  phi) const
Parameters
[in]phithe 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 Ellipsoid.cpp.

References GeographicLib::Math::hypot(), GeographicLib::Math::LatFix(), and GeographicLib::Math::tand().

◆ CircleHeight()

Math::real GeographicLib::Ellipsoid::CircleHeight ( real  phi) const
Parameters
[in]phithe 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 Ellipsoid.cpp.

References GeographicLib::Math::hypot(), GeographicLib::Math::LatFix(), and GeographicLib::Math::tand().

◆ MeridianDistance()

Math::real GeographicLib::Ellipsoid::MeridianDistance ( real  phi) const
Parameters
[in]phithe 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 Ellipsoid.cpp.

References GeographicLib::EllipticFunction::Ed(), and ParametricLatitude().

Referenced by RectifyingLatitude().

◆ MeridionalCurvatureRadius()

Math::real GeographicLib::Ellipsoid::MeridionalCurvatureRadius ( real  phi) const
Parameters
[in]phithe 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 Ellipsoid.cpp.

References GeographicLib::Math::LatFix(), GeographicLib::Math::sind(), and GeographicLib::Math::sq().

◆ TransverseCurvatureRadius()

Math::real GeographicLib::Ellipsoid::TransverseCurvatureRadius ( real  phi) const
Parameters
[in]phithe 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 Ellipsoid.cpp.

References GeographicLib::Math::LatFix(), GeographicLib::Math::sind(), and GeographicLib::Math::sq().

◆ NormalCurvatureRadius()

Math::real GeographicLib::Ellipsoid::NormalCurvatureRadius ( real  phi,
real  azi 
) const
Parameters
[in]phithe geographic latitude (degrees).
[in]azithe 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 Ellipsoid.cpp.

References GeographicLib::Math::LatFix(), GeographicLib::Math::sincosd(), GeographicLib::Math::sind(), and GeographicLib::Math::sq().

◆ SecondFlatteningToFlattening()

static Math::real GeographicLib::Ellipsoid::SecondFlatteningToFlattening ( real  fp)
inlinestatic
Parameters
[in]fp= f ' = (ab) / b, the second flattening.
Returns
f = (ab) / a, the flattening.

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

Definition at line 422 of file Ellipsoid.hpp.

◆ FlatteningToSecondFlattening()

static Math::real GeographicLib::Ellipsoid::FlatteningToSecondFlattening ( real  f)
inlinestatic
Parameters
[in]f= (ab) / a, the flattening.
Returns
f ' = (ab) / b, the second flattening.

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

Definition at line 432 of file Ellipsoid.hpp.

◆ ThirdFlatteningToFlattening()

static Math::real GeographicLib::Ellipsoid::ThirdFlatteningToFlattening ( real  n)
inlinestatic
Parameters
[in]n= (ab) / (a + b), the third flattening.
Returns
f = (ab) / a, the flattening.

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

Definition at line 443 of file Ellipsoid.hpp.

◆ FlatteningToThirdFlattening()

static Math::real GeographicLib::Ellipsoid::FlatteningToThirdFlattening ( real  f)
inlinestatic
Parameters
[in]f= (ab) / a, the flattening.
Returns
n = (ab) / (a + b), the third flattening.

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

Definition at line 454 of file Ellipsoid.hpp.

◆ EccentricitySqToFlattening()

static Math::real GeographicLib::Ellipsoid::EccentricitySqToFlattening ( real  e2)
inlinestatic
Parameters
[in]e2= e2 = (a2b2) / a2, the eccentricity squared.
Returns
f = (ab) / a, the flattening.

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

Definition at line 466 of file Ellipsoid.hpp.

◆ FlatteningToEccentricitySq()

static Math::real GeographicLib::Ellipsoid::FlatteningToEccentricitySq ( real  f)
inlinestatic
Parameters
[in]f= (ab) / a, the flattening.
Returns
e2 = (a2b2) / a2, the eccentricity squared.

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

Definition at line 478 of file Ellipsoid.hpp.

◆ SecondEccentricitySqToFlattening()

static Math::real GeographicLib::Ellipsoid::SecondEccentricitySqToFlattening ( real  ep2)
inlinestatic
Parameters
[in]ep2= e' 2 = (a2b2) / b2, the second eccentricity squared.
Returns
f = (ab) / a, the flattening.

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

Definition at line 490 of file Ellipsoid.hpp.

◆ FlatteningToSecondEccentricitySq()

static Math::real GeographicLib::Ellipsoid::FlatteningToSecondEccentricitySq ( real  f)
inlinestatic
Parameters
[in]f= (ab) / a, the flattening.
Returns
e' 2 = (a2b2) / b2, the second eccentricity squared.

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

Definition at line 502 of file Ellipsoid.hpp.

References GeographicLib::Math::sq().

◆ ThirdEccentricitySqToFlattening()

static Math::real GeographicLib::Ellipsoid::ThirdEccentricitySqToFlattening ( real  epp2)
inlinestatic
Parameters
[in]epp2= e'' 2 = (a2b2) / (a2 + b2), the third eccentricity squared.
Returns
f = (ab) / a, the flattening.

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

Definition at line 514 of file Ellipsoid.hpp.

◆ FlatteningToThirdEccentricitySq()

static Math::real GeographicLib::Ellipsoid::FlatteningToThirdEccentricitySq ( real  f)
inlinestatic
Parameters
[in]f= (ab) / a, the flattening.
Returns
e'' 2 = (a2b2) / (a2 + b2), the third eccentricity squared.

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

Definition at line 528 of file Ellipsoid.hpp.

References GeographicLib::Math::sq().

◆ WGS84()

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

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

Definition at line 32 of file Ellipsoid.cpp.

References GeographicLib::Constants::WGS84_a(), and GeographicLib::Constants::WGS84_f().

Friends And Related Function Documentation

◆ Rhumb

friend class Rhumb
friend

Definition at line 54 of file Ellipsoid.hpp.

◆ RhumbLine

friend class RhumbLine
friend

Definition at line 54 of file Ellipsoid.hpp.


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