GeographicLib  1.35
Static Public Attributes | List of all members
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 MajorRadius () const throw ()
 
Math::real MinorRadius () const throw ()
 
Math::real QuarterMeridian () const throw ()
 
Math::real Area () const throw ()
 
Math::real Volume () const throw ()
 
Ellipsoid shape
Math::real Flattening () const throw ()
 
Math::real SecondFlattening () const throw ()
 
Math::real ThirdFlattening () const throw ()
 
Math::real EccentricitySq () const throw ()
 
Math::real SecondEccentricitySq () const throw ()
 
Math::real ThirdEccentricitySq () const throw ()
 
Latitude conversion.
Math::real ParametricLatitude (real phi) const throw ()
 
Math::real InverseParametricLatitude (real beta) const throw ()
 
Math::real GeocentricLatitude (real phi) const throw ()
 
Math::real InverseGeocentricLatitude (real theta) const throw ()
 
Math::real RectifyingLatitude (real phi) const throw ()
 
Math::real InverseRectifyingLatitude (real mu) const throw ()
 
Math::real AuthalicLatitude (real phi) const throw ()
 
Math::real InverseAuthalicLatitude (real xi) const throw ()
 
Math::real ConformalLatitude (real phi) const throw ()
 
Math::real InverseConformalLatitude (real chi) const throw ()
 
Math::real IsometricLatitude (real phi) const throw ()
 
Math::real InverseIsometricLatitude (real psi) const throw ()
 
Other quantities.
Math::real CircleRadius (real phi) const throw ()
 
Math::real CircleHeight (real phi) const throw ()
 
Math::real MeridianDistance (real phi) const throw ()
 
Math::real MeridionalCurvatureRadius (real phi) const throw ()
 
Math::real TransverseCurvatureRadius (real phi) const throw ()
 
Math::real NormalCurvatureRadius (real phi, real azi) const throw ()
 

Static Public Member Functions

Eccentricity conversions.
static Math::real SecondFlatteningToFlattening (real fp) throw ()
 
static Math::real FlatteningToSecondFlattening (real f) throw ()
 
static Math::real ThirdFlatteningToFlattening (real n) throw ()
 
static Math::real FlatteningToThirdFlattening (real f) throw ()
 
static Math::real EccentricitySqToFlattening (real e2) throw ()
 
static Math::real FlatteningToEccentricitySq (real f) throw ()
 
static Math::real SecondEccentricitySqToFlattening (real ep2) throw ()
 
static Math::real FlatteningToSecondEccentricitySq (real f) throw ()
 
static Math::real ThirdEccentricitySqToFlattening (real epp2) throw ()
 
static Math::real FlatteningToThirdEccentricitySq (real f) throw ()
 

Static Public Attributes

static const Ellipsoid WGS84
 

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>
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";
}
catch (const exception& e) {
cerr << "Caught exception: " << e.what() << "\n";
return 1;
}
return 0;
}

Definition at line 39 of file Ellipsoid.hpp.

Constructor & Destructor Documentation

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. If f > 1, set flattening to 1/f.
Exceptions
GeographicErrif a or (1 − f ) a is not positive.

Definition at line 19 of file Ellipsoid.cpp.

Member Function Documentation

Math::real GeographicLib::Ellipsoid::MajorRadius ( ) const
throw (
)
inline
Returns
a the equatorial radius of the ellipsoid (meters). This is the value used in the constructor.

Definition at line 84 of file Ellipsoid.hpp.

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

Definition at line 89 of file Ellipsoid.hpp.

Math::real GeographicLib::Ellipsoid::QuarterMeridian ( ) const
throw (
)
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.

References GeographicLib::EllipticFunction::E().

Math::real GeographicLib::Ellipsoid::Area ( ) const
throw (
)
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.

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

Math::real GeographicLib::Ellipsoid::Volume ( ) const
throw (
)
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 110 of file Ellipsoid.hpp.

References GeographicLib::Math::sq().

Math::real GeographicLib::Ellipsoid::Flattening ( ) const
throw (
)
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 124 of file Ellipsoid.hpp.

Math::real GeographicLib::Ellipsoid::SecondFlattening ( ) const
throw (
)
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 131 of file Ellipsoid.hpp.

Math::real GeographicLib::Ellipsoid::ThirdFlattening ( ) const
throw (
)
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 138 of file Ellipsoid.hpp.

Math::real GeographicLib::Ellipsoid::EccentricitySq ( ) const
throw (
)
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 146 of file Ellipsoid.hpp.

Math::real GeographicLib::Ellipsoid::SecondEccentricitySq ( ) const
throw (
)
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 154 of file Ellipsoid.hpp.

Math::real GeographicLib::Ellipsoid::ThirdEccentricitySq ( ) const
throw (
)
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 163 of file Ellipsoid.hpp.

Math::real GeographicLib::Ellipsoid::ParametricLatitude ( real  phi) const
throw (
)
Parameters
[in]phithe 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 47 of file Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::InverseParametricLatitude ( real  beta) const
throw (
)
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 50 of file Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::GeocentricLatitude ( real  phi) const
throw (
)
Parameters
[in]phithe 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 53 of file Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::InverseGeocentricLatitude ( real  theta) const
throw (
)
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 56 of file Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::RectifyingLatitude ( real  phi) const
throw (
)
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 59 of file Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::InverseRectifyingLatitude ( real  mu) const
throw (
)
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 64 of file Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::AuthalicLatitude ( real  phi) const
throw (
)
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 71 of file Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::InverseAuthalicLatitude ( real  xi) const
throw (
)
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 74 of file Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::ConformalLatitude ( real  phi) const
throw (
)
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 77 of file Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::InverseConformalLatitude ( real  chi) const
throw (
)
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 80 of file Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::IsometricLatitude ( real  phi) const
throw (
)
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.

Definition at line 83 of file Ellipsoid.cpp.

References GeographicLib::Math::asinh().

Math::real GeographicLib::Ellipsoid::InverseIsometricLatitude ( real  psi) const
throw (
)
Parameters
[in]psithe isometric latitude (degrees).
Returns
φ the geographic latitude (degrees).

The returned value φ lies in [−90°, 90°].

Definition at line 86 of file Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::CircleRadius ( real  phi) const
throw (
)
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 89 of file Ellipsoid.cpp.

References GeographicLib::Math::hypot().

Math::real GeographicLib::Ellipsoid::CircleHeight ( real  phi) const
throw (
)
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 95 of file Ellipsoid.cpp.

References GeographicLib::Math::hypot().

Math::real GeographicLib::Ellipsoid::MeridianDistance ( real  phi) const
throw (
)
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 101 of file Ellipsoid.cpp.

Math::real GeographicLib::Ellipsoid::MeridionalCurvatureRadius ( real  phi) const
throw (
)
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 104 of file Ellipsoid.cpp.

References GeographicLib::Math::sq().

Math::real GeographicLib::Ellipsoid::TransverseCurvatureRadius ( real  phi) const
throw (
)
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 109 of file Ellipsoid.cpp.

References GeographicLib::Math::sq().

Math::real GeographicLib::Ellipsoid::NormalCurvatureRadius ( real  phi,
real  azi 
) const
throw (
)
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°] and azi must lie in the range [−540°, 540°); the result is undefined if either of conditions does not hold.

Definition at line 114 of file Ellipsoid.cpp.

References GeographicLib::Math::sq().

static Math::real GeographicLib::Ellipsoid::SecondFlatteningToFlattening ( real  fp)
throw (
)
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 418 of file Ellipsoid.hpp.

static Math::real GeographicLib::Ellipsoid::FlatteningToSecondFlattening ( real  f)
throw (
)
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 428 of file Ellipsoid.hpp.

static Math::real GeographicLib::Ellipsoid::ThirdFlatteningToFlattening ( real  n)
throw (
)
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 439 of file Ellipsoid.hpp.

static Math::real GeographicLib::Ellipsoid::FlatteningToThirdFlattening ( real  f)
throw (
)
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 450 of file Ellipsoid.hpp.

static Math::real GeographicLib::Ellipsoid::EccentricitySqToFlattening ( real  e2)
throw (
)
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 462 of file Ellipsoid.hpp.

static Math::real GeographicLib::Ellipsoid::FlatteningToEccentricitySq ( real  f)
throw (
)
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 474 of file Ellipsoid.hpp.

static Math::real GeographicLib::Ellipsoid::SecondEccentricitySqToFlattening ( real  ep2)
throw (
)
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 486 of file Ellipsoid.hpp.

static Math::real GeographicLib::Ellipsoid::FlatteningToSecondEccentricitySq ( real  f)
throw (
)
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 498 of file Ellipsoid.hpp.

References GeographicLib::Math::sq().

static Math::real GeographicLib::Ellipsoid::ThirdEccentricitySqToFlattening ( real  epp2)
throw (
)
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 510 of file Ellipsoid.hpp.

static Math::real GeographicLib::Ellipsoid::FlatteningToThirdEccentricitySq ( real  f)
throw (
)
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 522 of file Ellipsoid.hpp.

References GeographicLib::Math::sq().

Member Data Documentation

const Ellipsoid GeographicLib::Ellipsoid::WGS84
static

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

Definition at line 531 of file Ellipsoid.hpp.


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