543 lines
24 KiB
C++
543 lines
24 KiB
C++
/**
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* \file Ellipsoid.hpp
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* \brief Header for GeographicLib::Ellipsoid class
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*
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* Copyright (c) Charles Karney (2012-2020) <charles@karney.com> and licensed
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* under the MIT/X11 License. For more information, see
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* https://geographiclib.sourceforge.io/
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**********************************************************************/
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#if !defined(GEOGRAPHICLIB_ELLIPSOID_HPP)
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#define GEOGRAPHICLIB_ELLIPSOID_HPP 1
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#include <GeographicLib/Constants.hpp>
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#include <GeographicLib/TransverseMercator.hpp>
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#include <GeographicLib/EllipticFunction.hpp>
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#include <GeographicLib/AlbersEqualArea.hpp>
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namespace GeographicLib {
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/**
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* \brief Properties of an ellipsoid
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*
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* This class returns various properties of the ellipsoid and converts
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* between various types of latitudes. The latitude conversions are also
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* possible using the various projections supported by %GeographicLib; but
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* Ellipsoid provides more direct access (sometimes using private functions
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* of the projection classes). Ellipsoid::RectifyingLatitude,
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* Ellipsoid::InverseRectifyingLatitude, and Ellipsoid::MeridianDistance
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* provide functionality which can be provided by the Geodesic class.
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* However Geodesic uses a series approximation (valid for abs \e f < 1/150),
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* whereas Ellipsoid computes these quantities using EllipticFunction which
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* provides accurate results even when \e f is large. Use of this class
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* should be limited to −3 < \e f < 3/4 (i.e., 1/4 < b/a < 4).
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*
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* Example of use:
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* \include example-Ellipsoid.cpp
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**********************************************************************/
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class GEOGRAPHICLIB_EXPORT Ellipsoid {
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private:
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typedef Math::real real;
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static const int numit_ = 10;
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real stol_;
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real _a, _f, _f1, _f12, _e2, _es, _e12, _n, _b;
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TransverseMercator _tm;
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EllipticFunction _ell;
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AlbersEqualArea _au;
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// These are the alpha and beta coefficients in the Krueger series from
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// TransverseMercator. Thy are used by RhumbSolve to compute
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// (psi2-psi1)/(mu2-mu1).
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const Math::real* ConformalToRectifyingCoeffs() const { return _tm._alp; }
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const Math::real* RectifyingToConformalCoeffs() const { return _tm._bet; }
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friend class Rhumb; friend class RhumbLine;
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public:
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/** \name Constructor
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**********************************************************************/
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///@{
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/**
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* Constructor for a ellipsoid with
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*
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* @param[in] a equatorial radius (meters).
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* @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere.
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* Negative \e f gives a prolate ellipsoid.
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* @exception GeographicErr if \e a or (1 − \e f) \e a is not
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* positive.
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**********************************************************************/
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Ellipsoid(real a, real f);
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///@}
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/** \name %Ellipsoid dimensions.
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**********************************************************************/
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///@{
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/**
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* @return \e a the equatorial radius of the ellipsoid (meters). This is
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* the value used in the constructor.
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**********************************************************************/
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Math::real EquatorialRadius() const { return _a; }
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/**
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* @return \e b the polar semi-axis (meters).
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**********************************************************************/
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Math::real MinorRadius() const { return _b; }
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/**
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* @return \e L the distance between the equator and a pole along a
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* meridian (meters). For a sphere \e L = (π/2) \e a. The radius
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* of a sphere with the same meridian length is \e L / (π/2).
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**********************************************************************/
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Math::real QuarterMeridian() const;
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/**
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* @return \e A the total area of the ellipsoid (meters<sup>2</sup>). For
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* a sphere \e A = 4π <i>a</i><sup>2</sup>. The radius of a sphere
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* with the same area is sqrt(\e A / (4π)).
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**********************************************************************/
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Math::real Area() const;
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/**
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* @return \e V the total volume of the ellipsoid (meters<sup>3</sup>).
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* For a sphere \e V = (4π / 3) <i>a</i><sup>3</sup>. The radius of
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* a sphere with the same volume is cbrt(\e V / (4π/3)).
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**********************************************************************/
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Math::real Volume() const
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{ return (4 * Math::pi()) * Math::sq(_a) * _b / 3; }
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/**
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* \deprecated An old name for EquatorialRadius().
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**********************************************************************/
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GEOGRAPHICLIB_DEPRECATED("Use EquatorialRadius()")
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Math::real MajorRadius() const { return EquatorialRadius(); }
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///@}
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/** \name %Ellipsoid shape
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**********************************************************************/
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///@{
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/**
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* @return \e f = (\e a − \e b) / \e a, the flattening of the
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* ellipsoid. This is the value used in the constructor. This is zero,
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* positive, or negative for a sphere, oblate ellipsoid, or prolate
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* ellipsoid.
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**********************************************************************/
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Math::real Flattening() const { return _f; }
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/**
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* @return \e f ' = (\e a − \e b) / \e b, the second flattening of
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* the ellipsoid. This is zero, positive, or negative for a sphere,
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* oblate ellipsoid, or prolate ellipsoid.
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**********************************************************************/
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Math::real SecondFlattening() const { return _f / (1 - _f); }
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/**
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* @return \e n = (\e a − \e b) / (\e a + \e b), the third flattening
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* of the ellipsoid. This is zero, positive, or negative for a sphere,
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* oblate ellipsoid, or prolate ellipsoid.
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**********************************************************************/
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Math::real ThirdFlattening() const { return _n; }
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/**
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* @return <i>e</i><sup>2</sup> = (<i>a</i><sup>2</sup> −
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* <i>b</i><sup>2</sup>) / <i>a</i><sup>2</sup>, the eccentricity squared
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* of the ellipsoid. This is zero, positive, or negative for a sphere,
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* oblate ellipsoid, or prolate ellipsoid.
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**********************************************************************/
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Math::real EccentricitySq() const { return _e2; }
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/**
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* @return <i>e'</i> <sup>2</sup> = (<i>a</i><sup>2</sup> −
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* <i>b</i><sup>2</sup>) / <i>b</i><sup>2</sup>, the second eccentricity
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* squared of the ellipsoid. This is zero, positive, or negative for a
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* sphere, oblate ellipsoid, or prolate ellipsoid.
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**********************************************************************/
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Math::real SecondEccentricitySq() const { return _e12; }
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/**
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* @return <i>e''</i> <sup>2</sup> = (<i>a</i><sup>2</sup> −
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* <i>b</i><sup>2</sup>) / (<i>a</i><sup>2</sup> + <i>b</i><sup>2</sup>),
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* the third eccentricity squared of the ellipsoid. This is zero,
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* positive, or negative for a sphere, oblate ellipsoid, or prolate
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* ellipsoid.
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**********************************************************************/
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Math::real ThirdEccentricitySq() const { return _e2 / (2 - _e2); }
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///@}
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/** \name Latitude conversion.
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**********************************************************************/
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///@{
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/**
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* @param[in] phi the geographic latitude (degrees).
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* @return β the parametric latitude (degrees).
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*
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* The geographic latitude, φ, is the angle between the equatorial
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* plane and a vector normal to the surface of the ellipsoid.
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*
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* The parametric latitude (also called the reduced latitude), β,
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* allows the cartesian coordinated of a meridian to be expressed
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* conveniently in parametric form as
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* - \e R = \e a cos β
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* - \e Z = \e b sin β
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* .
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* where \e a and \e b are the equatorial radius and the polar semi-axis.
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* For a sphere β = φ.
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*
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* φ must lie in the range [−90°, 90°]; the
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* result is undefined if this condition does not hold. The returned value
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* β lies in [−90°, 90°].
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**********************************************************************/
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Math::real ParametricLatitude(real phi) const;
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/**
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* @param[in] beta the parametric latitude (degrees).
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* @return φ the geographic latitude (degrees).
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*
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* β must lie in the range [−90°, 90°]; the
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* result is undefined if this condition does not hold. The returned value
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* φ lies in [−90°, 90°].
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**********************************************************************/
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Math::real InverseParametricLatitude(real beta) const;
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/**
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* @param[in] phi the geographic latitude (degrees).
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* @return θ the geocentric latitude (degrees).
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*
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* The geocentric latitude, θ, is the angle between the equatorial
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* plane and a line between the center of the ellipsoid and a point on the
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* ellipsoid. For a sphere θ = φ.
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*
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* φ must lie in the range [−90°, 90°]; the
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* result is undefined if this condition does not hold. The returned value
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* θ lies in [−90°, 90°].
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**********************************************************************/
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Math::real GeocentricLatitude(real phi) const;
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/**
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* @param[in] theta the geocentric latitude (degrees).
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* @return φ the geographic latitude (degrees).
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*
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* θ must lie in the range [−90°, 90°]; the
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* result is undefined if this condition does not hold. The returned value
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* φ lies in [−90°, 90°].
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**********************************************************************/
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Math::real InverseGeocentricLatitude(real theta) const;
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/**
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* @param[in] phi the geographic latitude (degrees).
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* @return μ the rectifying latitude (degrees).
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*
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* The rectifying latitude, μ, has the property that the distance along
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* a meridian of the ellipsoid between two points with rectifying latitudes
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* μ<sub>1</sub> and μ<sub>2</sub> is equal to
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* (μ<sub>2</sub> - μ<sub>1</sub>) \e L / 90°,
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* where \e L = QuarterMeridian(). For a sphere μ = φ.
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*
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* φ must lie in the range [−90°, 90°]; the
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* result is undefined if this condition does not hold. The returned value
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* μ lies in [−90°, 90°].
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**********************************************************************/
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Math::real RectifyingLatitude(real phi) const;
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/**
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* @param[in] mu the rectifying latitude (degrees).
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* @return φ the geographic latitude (degrees).
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*
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* μ must lie in the range [−90°, 90°]; the
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* result is undefined if this condition does not hold. The returned value
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* φ lies in [−90°, 90°].
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**********************************************************************/
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Math::real InverseRectifyingLatitude(real mu) const;
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/**
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* @param[in] phi the geographic latitude (degrees).
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* @return ξ the authalic latitude (degrees).
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*
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* The authalic latitude, ξ, has the property that the area of the
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* ellipsoid between two circles with authalic latitudes
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* ξ<sub>1</sub> and ξ<sub>2</sub> is equal to (sin
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* ξ<sub>2</sub> - sin ξ<sub>1</sub>) \e A / 2, where \e A
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* = Area(). For a sphere ξ = φ.
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*
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* φ must lie in the range [−90°, 90°]; the
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* result is undefined if this condition does not hold. The returned value
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* ξ lies in [−90°, 90°].
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**********************************************************************/
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Math::real AuthalicLatitude(real phi) const;
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/**
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* @param[in] xi the authalic latitude (degrees).
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* @return φ the geographic latitude (degrees).
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*
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* ξ must lie in the range [−90°, 90°]; the
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* result is undefined if this condition does not hold. The returned value
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* φ lies in [−90°, 90°].
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**********************************************************************/
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Math::real InverseAuthalicLatitude(real xi) const;
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/**
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* @param[in] phi the geographic latitude (degrees).
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* @return χ the conformal latitude (degrees).
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*
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* The conformal latitude, χ, gives the mapping of the ellipsoid to a
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* sphere which which is conformal (angles are preserved) and in which the
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* equator of the ellipsoid maps to the equator of the sphere. For a
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* sphere χ = φ.
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*
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* φ must lie in the range [−90°, 90°]; the
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* result is undefined if this condition does not hold. The returned value
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* χ lies in [−90°, 90°].
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**********************************************************************/
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Math::real ConformalLatitude(real phi) const;
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/**
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* @param[in] chi the conformal latitude (degrees).
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* @return φ the geographic latitude (degrees).
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*
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* χ must lie in the range [−90°, 90°]; the
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* result is undefined if this condition does not hold. The returned value
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* φ lies in [−90°, 90°].
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**********************************************************************/
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Math::real InverseConformalLatitude(real chi) const;
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/**
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* @param[in] phi the geographic latitude (degrees).
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* @return ψ the isometric latitude (degrees).
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*
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* The isometric latitude gives the mapping of the ellipsoid to a plane
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* which which is conformal (angles are preserved) and in which the equator
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* of the ellipsoid maps to a straight line of constant scale; this mapping
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* defines the Mercator projection. For a sphere ψ =
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* sinh<sup>−1</sup> tan φ.
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*
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* φ must lie in the range [−90°, 90°]; the result is
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* undefined if this condition does not hold. The value returned for φ
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* = ±90° is some (positive or negative) large but finite value,
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* such that InverseIsometricLatitude returns the original value of φ.
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**********************************************************************/
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Math::real IsometricLatitude(real phi) const;
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/**
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* @param[in] psi the isometric latitude (degrees).
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* @return φ the geographic latitude (degrees).
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*
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* The returned value φ lies in [−90°, 90°]. For a
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* sphere φ = tan<sup>−1</sup> sinh ψ.
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**********************************************************************/
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Math::real InverseIsometricLatitude(real psi) const;
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///@}
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/** \name Other quantities.
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**********************************************************************/
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///@{
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/**
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* @param[in] phi the geographic latitude (degrees).
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* @return \e R = \e a cos β the radius of a circle of latitude
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* φ (meters). \e R (π/180°) gives meters per degree
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* longitude measured along a circle of latitude.
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*
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* φ must lie in the range [−90°, 90°]; the
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* result is undefined if this condition does not hold.
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**********************************************************************/
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Math::real CircleRadius(real phi) const;
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/**
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* @param[in] phi the geographic latitude (degrees).
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* @return \e Z = \e b sin β the distance of a circle of latitude
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* φ from the equator measured parallel to the ellipsoid axis
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* (meters).
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*
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* φ must lie in the range [−90°, 90°]; the
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* result is undefined if this condition does not hold.
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**********************************************************************/
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Math::real CircleHeight(real phi) const;
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/**
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* @param[in] phi the geographic latitude (degrees).
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* @return \e s the distance along a meridian
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* between the equator and a point of latitude φ (meters). \e s is
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* given by \e s = μ \e L / 90°, where \e L =
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* QuarterMeridian()).
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*
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* φ must lie in the range [−90°, 90°]; the
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* result is undefined if this condition does not hold.
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**********************************************************************/
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Math::real MeridianDistance(real phi) const;
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/**
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* @param[in] phi the geographic latitude (degrees).
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* @return ρ the meridional radius of curvature of the ellipsoid at
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* latitude φ (meters); this is the curvature of the meridian. \e
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* rho is given by ρ = (180°/π) d\e s / dφ,
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* where \e s = MeridianDistance(); thus ρ (π/180°)
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* gives meters per degree latitude measured along a meridian.
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*
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* φ must lie in the range [−90°, 90°]; the
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* result is undefined if this condition does not hold.
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**********************************************************************/
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Math::real MeridionalCurvatureRadius(real phi) const;
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/**
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* @param[in] phi the geographic latitude (degrees).
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* @return ν the transverse radius of curvature of the ellipsoid at
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* latitude φ (meters); this is the curvature of a curve on the
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* ellipsoid which also lies in a plane perpendicular to the ellipsoid
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* and to the meridian. ν is related to \e R = CircleRadius() by \e
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* R = ν cos φ.
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*
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* φ must lie in the range [−90°, 90°]; the
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* result is undefined if this condition does not hold.
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**********************************************************************/
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Math::real TransverseCurvatureRadius(real phi) const;
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/**
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* @param[in] phi the geographic latitude (degrees).
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* @param[in] azi the angle between the meridian and the normal section
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* (degrees).
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* @return the radius of curvature of the ellipsoid in the normal
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* section at latitude φ inclined at an angle \e azi to the
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* meridian (meters).
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*
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* φ must lie in the range [−90°, 90°]; the result is
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* undefined this condition does not hold.
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**********************************************************************/
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Math::real NormalCurvatureRadius(real phi, real azi) const;
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///@}
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/** \name Eccentricity conversions.
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**********************************************************************/
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///@{
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/**
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* @param[in] fp = \e f ' = (\e a − \e b) / \e b, the second
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* flattening.
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* @return \e f = (\e a − \e b) / \e a, the flattening.
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*
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* \e f ' should lie in (−1, ∞).
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* The returned value \e f lies in (−∞, 1).
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**********************************************************************/
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static Math::real SecondFlatteningToFlattening(real fp)
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{ return fp / (1 + fp); }
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/**
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* @param[in] f = (\e a − \e b) / \e a, the flattening.
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* @return \e f ' = (\e a − \e b) / \e b, the second flattening.
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*
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* \e f should lie in (−∞, 1).
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* The returned value \e f ' lies in (−1, ∞).
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**********************************************************************/
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static Math::real FlatteningToSecondFlattening(real f)
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{ return f / (1 - f); }
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/**
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* @param[in] n = (\e a − \e b) / (\e a + \e b), the third
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* flattening.
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* @return \e f = (\e a − \e b) / \e a, the flattening.
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*
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* \e n should lie in (−1, 1).
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* The returned value \e f lies in (−∞, 1).
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**********************************************************************/
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static Math::real ThirdFlatteningToFlattening(real n)
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{ return 2 * n / (1 + n); }
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/**
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* @param[in] f = (\e a − \e b) / \e a, the flattening.
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* @return \e n = (\e a − \e b) / (\e a + \e b), the third
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* flattening.
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*
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* \e f should lie in (−∞, 1).
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* The returned value \e n lies in (−1, 1).
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**********************************************************************/
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static Math::real FlatteningToThirdFlattening(real f)
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{ return f / (2 - f); }
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/**
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* @param[in] e2 = <i>e</i><sup>2</sup> = (<i>a</i><sup>2</sup> −
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* <i>b</i><sup>2</sup>) / <i>a</i><sup>2</sup>, the eccentricity
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* squared.
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* @return \e f = (\e a − \e b) / \e a, the flattening.
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*
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* <i>e</i><sup>2</sup> should lie in (−∞, 1).
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* The returned value \e f lies in (−∞, 1).
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|
**********************************************************************/
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static Math::real EccentricitySqToFlattening(real e2)
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{ using std::sqrt; return e2 / (sqrt(1 - e2) + 1); }
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|
|
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/**
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* @param[in] f = (\e a − \e b) / \e a, the flattening.
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* @return <i>e</i><sup>2</sup> = (<i>a</i><sup>2</sup> −
|
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* <i>b</i><sup>2</sup>) / <i>a</i><sup>2</sup>, the eccentricity
|
|
* squared.
|
|
*
|
|
* \e f should lie in (−∞, 1).
|
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* The returned value <i>e</i><sup>2</sup> lies in (−∞, 1).
|
|
**********************************************************************/
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static Math::real FlatteningToEccentricitySq(real f)
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{ return f * (2 - f); }
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|
|
|
/**
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* @param[in] ep2 = <i>e'</i> <sup>2</sup> = (<i>a</i><sup>2</sup> −
|
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* <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).
|
|
**********************************************************************/
|
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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) {
|
|
using std::sqrt;
|
|
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
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|
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#endif // GEOGRAPHICLIB_ELLIPSOID_HPP
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