/usr/include/GeographicLib/Ellipsoid.hpp is in libgeographic-dev 1.45-2.
This file is owned by root:root, with mode 0o644.
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* \file Ellipsoid.hpp
* \brief Header for GeographicLib::Ellipsoid class
*
* Copyright (c) Charles Karney (2012-2015) <charles@karney.com> and licensed
* under the MIT/X11 License. For more information, see
* http://geographiclib.sourceforge.net/
**********************************************************************/
#if !defined(GEOGRAPHICLIB_ELLIPSOID_HPP)
#define GEOGRAPHICLIB_ELLIPSOID_HPP 1
#include <GeographicLib/Constants.hpp>
#include <GeographicLib/TransverseMercator.hpp>
#include <GeographicLib/EllipticFunction.hpp>
#include <GeographicLib/AlbersEqualArea.hpp>
namespace GeographicLib {
/**
* \brief 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 \e f < 1/150),
* whereas Ellipsoid computes these quantities using EllipticFunction which
* provides accurate results even when \e f is large. Use of this class
* should be limited to −3 < \e f < 3/4 (i.e., 1/4 < b/a < 4).
*
* Example of use:
* \include example-Ellipsoid.cpp
**********************************************************************/
class GEOGRAPHICLIB_EXPORT Ellipsoid {
private:
typedef Math::real real;
static const int numit_ = 10;
real stol_;
real _a, _f, _f1, _f12, _e2, _es, _e12, _n, _b;
TransverseMercator _tm;
EllipticFunction _ell;
AlbersEqualArea _au;
// These are the alpha and beta coefficients in the Krueger series from
// TransverseMercator. Thy are used by RhumbSolve to compute
// (psi2-psi1)/(mu2-mu1).
const Math::real* ConformalToRectifyingCoeffs() const { return _tm._alp; }
const Math::real* RectifyingToConformalCoeffs() const { return _tm._bet; }
friend class Rhumb; friend class RhumbLine;
public:
/** \name Constructor
**********************************************************************/
///@{
/**
* Constructor for a ellipsoid with
*
* @param[in] a equatorial radius (meters).
* @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere.
* Negative \e f gives a prolate ellipsoid.
* @exception GeographicErr if \e a or (1 − \e f) \e a is not
* positive.
**********************************************************************/
Ellipsoid(real a, real f);
///@}
/** \name %Ellipsoid dimensions.
**********************************************************************/
///@{
/**
* @return \e a the equatorial radius of the ellipsoid (meters). This is
* the value used in the constructor.
**********************************************************************/
Math::real MajorRadius() const { return _a; }
/**
* @return \e b the polar semi-axis (meters).
**********************************************************************/
Math::real MinorRadius() const { return _b; }
/**
* @return \e L the distance between the equator and a pole along a
* meridian (meters). For a sphere \e L = (π/2) \e a. The radius
* of a sphere with the same meridian length is \e L / (π/2).
**********************************************************************/
Math::real QuarterMeridian() const;
/**
* @return \e A the total area of the ellipsoid (meters<sup>2</sup>). For
* a sphere \e A = 4π <i>a</i><sup>2</sup>. The radius of a sphere
* with the same area is sqrt(\e A / (4π)).
**********************************************************************/
Math::real Area() const;
/**
* @return \e V the total volume of the ellipsoid (meters<sup>3</sup>).
* For a sphere \e V = (4π / 3) <i>a</i><sup>3</sup>. The radius of
* a sphere with the same volume is cbrt(\e V / (4π/3)).
**********************************************************************/
Math::real Volume() const
{ return (4 * Math::pi()) * Math::sq(_a) * _b / 3; }
///@}
/** \name %Ellipsoid shape
**********************************************************************/
///@{
/**
* @return \e f = (\e a − \e b) / \e 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.
**********************************************************************/
Math::real Flattening() const { return _f; }
/**
* @return \e f ' = (\e a − \e b) / \e b, the second flattening of
* the ellipsoid. This is zero, positive, or negative for a sphere,
* oblate ellipsoid, or prolate ellipsoid.
**********************************************************************/
Math::real SecondFlattening() const { return _f / (1 - _f); }
/**
* @return \e n = (\e a − \e b) / (\e a + \e b), the third flattening
* of the ellipsoid. This is zero, positive, or negative for a sphere,
* oblate ellipsoid, or prolate ellipsoid.
**********************************************************************/
Math::real ThirdFlattening() const { return _n; }
/**
* @return <i>e</i><sup>2</sup> = (<i>a</i><sup>2</sup> −
* <i>b</i><sup>2</sup>) / <i>a</i><sup>2</sup>, the eccentricity squared
* of the ellipsoid. This is zero, positive, or negative for a sphere,
* oblate ellipsoid, or prolate ellipsoid.
**********************************************************************/
Math::real EccentricitySq() const { return _e2; }
/**
* @return <i>e'</i> <sup>2</sup> = (<i>a</i><sup>2</sup> −
* <i>b</i><sup>2</sup>) / <i>b</i><sup>2</sup>, the second eccentricity
* squared of the ellipsoid. This is zero, positive, or negative for a
* sphere, oblate ellipsoid, or prolate ellipsoid.
**********************************************************************/
Math::real SecondEccentricitySq() const { return _e12; }
/**
* @return <i>e''</i> <sup>2</sup> = (<i>a</i><sup>2</sup> −
* <i>b</i><sup>2</sup>) / (<i>a</i><sup>2</sup> + <i>b</i><sup>2</sup>),
* the third eccentricity squared of the ellipsoid. This is zero,
* positive, or negative for a sphere, oblate ellipsoid, or prolate
* ellipsoid.
**********************************************************************/
Math::real ThirdEccentricitySq() const { return _e2 / (2 - _e2); }
///@}
/** \name Latitude conversion.
**********************************************************************/
///@{
/**
* @param[in] phi the geographic latitude (degrees).
* @return 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
* - \e R = \e a cos β
* - \e Z = \e b sin β
* .
* where \e a and \e 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°].
**********************************************************************/
Math::real ParametricLatitude(real phi) const;
/**
* @param[in] beta the parametric latitude (degrees).
* @return 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°].
**********************************************************************/
Math::real InverseParametricLatitude(real beta) const;
/**
* @param[in] phi the geographic latitude (degrees).
* @return 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°].
**********************************************************************/
Math::real GeocentricLatitude(real phi) const;
/**
* @param[in] theta the geocentric latitude (degrees).
* @return 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°].
**********************************************************************/
Math::real InverseGeocentricLatitude(real theta) const;
/**
* @param[in] phi the geographic latitude (degrees).
* @return 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
* μ<sub>1</sub> and μ<sub>2</sub> is equal to
* (μ<sub>2</sub> - μ<sub>1</sub>) \e L / 90°,
* where \e 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°].
**********************************************************************/
Math::real RectifyingLatitude(real phi) const;
/**
* @param[in] mu the rectifying latitude (degrees).
* @return 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°].
**********************************************************************/
Math::real InverseRectifyingLatitude(real mu) const;
/**
* @param[in] phi the geographic latitude (degrees).
* @return ξ the authalic latitude (degrees).
*
* The authalic latitude, ξ, has the property that the area of the
* ellipsoid between two circles with authalic latitudes
* ξ<sub>1</sub> and ξ<sub>2</sub> is equal to (sin
* ξ<sub>2</sub> - sin ξ<sub>1</sub>) \e A / 2, where \e 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°].
**********************************************************************/
Math::real AuthalicLatitude(real phi) const;
/**
* @param[in] xi the authalic latitude (degrees).
* @return 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°].
**********************************************************************/
Math::real InverseAuthalicLatitude(real xi) const;
/**
* @param[in] phi the geographic latitude (degrees).
* @return 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°].
**********************************************************************/
Math::real ConformalLatitude(real phi) const;
/**
* @param[in] chi the conformal latitude (degrees).
* @return 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°].
**********************************************************************/
Math::real InverseConformalLatitude(real chi) const;
/**
* @param[in] phi the geographic latitude (degrees).
* @return 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<sup>−1</sup> 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.
**********************************************************************/
Math::real IsometricLatitude(real phi) const;
/**
* @param[in] psi the isometric latitude (degrees).
* @return φ the geographic latitude (degrees).
*
* The returned value φ lies in [−90°, 90°]. For a
* sphere φ = tan<sup>−1</sup> sinh ψ.
**********************************************************************/
Math::real InverseIsometricLatitude(real psi) const;
///@}
/** \name Other quantities.
**********************************************************************/
///@{
/**
* @param[in] phi the geographic latitude (degrees).
* @return \e R = \e a cos β the radius of a circle of latitude
* φ (meters). \e 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.
**********************************************************************/
Math::real CircleRadius(real phi) const;
/**
* @param[in] phi the geographic latitude (degrees).
* @return \e Z = \e 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.
**********************************************************************/
Math::real CircleHeight(real phi) const;
/**
* @param[in] phi the geographic latitude (degrees).
* @return \e s the distance along a meridian
* between the equator and a point of latitude φ (meters). \e s is
* given by \e s = μ \e L / 90°, where \e L =
* QuarterMeridian()).
*
* φ must lie in the range [−90°, 90°]; the
* result is undefined if this condition does not hold.
**********************************************************************/
Math::real MeridianDistance(real phi) const;
/**
* @param[in] phi the geographic latitude (degrees).
* @return ρ the meridional radius of curvature of the ellipsoid at
* latitude φ (meters); this is the curvature of the meridian. \e
* rho is given by ρ = (180°/π) d\e s / dφ,
* where \e 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.
**********************************************************************/
Math::real MeridionalCurvatureRadius(real phi) const;
/**
* @param[in] phi the geographic latitude (degrees).
* @return 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 \e R = CircleRadius() by \e
* R = ν cos φ.
*
* φ must lie in the range [−90°, 90°]; the
* result is undefined if this condition does not hold.
**********************************************************************/
Math::real TransverseCurvatureRadius(real phi) const;
/**
* @param[in] phi the geographic latitude (degrees).
* @param[in] azi the angle between the meridian and the normal section
* (degrees).
* @return the radius of curvature of the ellipsoid in the normal
* section at latitude φ inclined at an angle \e azi to the
* meridian (meters).
*
* φ must lie in the range [−90°, 90°]; the result is
* undefined this condition does not hold.
**********************************************************************/
Math::real NormalCurvatureRadius(real phi, real azi) const;
///@}
/** \name Eccentricity conversions.
**********************************************************************/
///@{
/**
* @param[in] fp = \e f ' = (\e a − \e b) / \e b, the second
* flattening.
* @return \e f = (\e a − \e b) / \e a, the flattening.
*
* \e f ' should lie in (−1, ∞).
* The returned value \e f lies in (−∞, 1).
**********************************************************************/
static Math::real SecondFlatteningToFlattening(real fp)
{ return fp / (1 + fp); }
/**
* @param[in] f = (\e a − \e b) / \e a, the flattening.
* @return \e f ' = (\e a − \e b) / \e b, the second flattening.
*
* \e f should lie in (−∞, 1).
* The returned value \e f ' lies in (−1, ∞).
**********************************************************************/
static Math::real FlatteningToSecondFlattening(real f)
{ return f / (1 - f); }
/**
* @param[in] n = (\e a − \e b) / (\e a + \e b), the third
* flattening.
* @return \e f = (\e a − \e b) / \e a, the flattening.
*
* \e n should lie in (−1, 1).
* The returned value \e f lies in (−∞, 1).
**********************************************************************/
static Math::real ThirdFlatteningToFlattening(real n)
{ return 2 * n / (1 + n); }
/**
* @param[in] f = (\e a − \e b) / \e a, the flattening.
* @return \e n = (\e a − \e b) / (\e a + \e b), the third
* flattening.
*
* \e f should lie in (−∞, 1).
* The returned value \e n lies in (−1, 1).
**********************************************************************/
static Math::real FlatteningToThirdFlattening(real f)
{ return f / (2 - f); }
/**
* @param[in] e2 = <i>e</i><sup>2</sup> = (<i>a</i><sup>2</sup> −
* <i>b</i><sup>2</sup>) / <i>a</i><sup>2</sup>, the eccentricity
* squared.
* @return \e f = (\e a − \e b) / \e a, the flattening.
*
* <i>e</i><sup>2</sup> should lie in (−∞, 1).
* The returned value \e f lies in (−∞, 1).
**********************************************************************/
static Math::real EccentricitySqToFlattening(real e2)
{ using std::sqrt; return e2 / (sqrt(1 - e2) + 1); }
/**
* @param[in] f = (\e a − \e b) / \e a, the flattening.
* @return <i>e</i><sup>2</sup> = (<i>a</i><sup>2</sup> −
* <i>b</i><sup>2</sup>) / <i>a</i><sup>2</sup>, the eccentricity
* squared.
*
* \e f should lie in (−∞, 1).
* The returned value <i>e</i><sup>2</sup> lies in (−∞, 1).
**********************************************************************/
static Math::real FlatteningToEccentricitySq(real f)
{ return f * (2 - f); }
/**
* @param[in] ep2 = <i>e'</i> <sup>2</sup> = (<i>a</i><sup>2</sup> −
* <i>b</i><sup>2</sup>) / <i>b</i><sup>2</sup>, the second eccentricity
* squared.
* @return \e f = (\e a − \e b) / \e a, the flattening.
*
* <i>e'</i> <sup>2</sup> should lie in (−1, ∞).
* The returned value \e f lies in (−∞, 1).
**********************************************************************/
static Math::real SecondEccentricitySqToFlattening(real ep2)
{ using std::sqrt; return ep2 / (sqrt(1 + ep2) + 1 + ep2); }
/**
* @param[in] f = (\e a − \e b) / \e a, the flattening.
* @return <i>e'</i> <sup>2</sup> = (<i>a</i><sup>2</sup> −
* <i>b</i><sup>2</sup>) / <i>b</i><sup>2</sup>, the second eccentricity
* squared.
*
* \e f should lie in (−∞, 1).
* The returned value <i>e'</i> <sup>2</sup> lies in (−1, ∞).
**********************************************************************/
static Math::real FlatteningToSecondEccentricitySq(real f)
{ return f * (2 - f) / Math::sq(1 - f); }
/**
* @param[in] epp2 = <i>e''</i> <sup>2</sup> = (<i>a</i><sup>2</sup>
* − <i>b</i><sup>2</sup>) / (<i>a</i><sup>2</sup> +
* <i>b</i><sup>2</sup>), the third eccentricity squared.
* @return \e f = (\e a − \e b) / \e a, the flattening.
*
* <i>e''</i> <sup>2</sup> should lie in (−1, 1).
* The returned value \e f lies in (−∞, 1).
**********************************************************************/
static Math::real ThirdEccentricitySqToFlattening(real epp2)
{ return 2 * epp2 / (sqrt((1 - epp2) * (1 + epp2)) + 1 + epp2); }
/**
* @param[in] f = (\e a − \e b) / \e a, the flattening.
* @return <i>e''</i> <sup>2</sup> = (<i>a</i><sup>2</sup> −
* <i>b</i><sup>2</sup>) / (<i>a</i><sup>2</sup> + <i>b</i><sup>2</sup>),
* the third eccentricity squared.
*
* \e f should lie in (−∞, 1).
* The returned value <i>e''</i> <sup>2</sup> lies in (−1, 1).
**********************************************************************/
static Math::real FlatteningToThirdEccentricitySq(real f)
{ return f * (2 - f) / (1 + Math::sq(1 - f)); }
///@}
/**
* A global instantiation of Ellipsoid with the parameters for the WGS84
* ellipsoid.
**********************************************************************/
static const Ellipsoid& WGS84();
};
} // namespace GeographicLib
#endif // GEOGRAPHICLIB_ELLIPSOID_HPP
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