703 lines
26 KiB
C++
703 lines
26 KiB
C++
/**
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* \file EllipticFunction.hpp
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* \brief Header for GeographicLib::EllipticFunction class
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*
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* Copyright (c) Charles Karney (2008-2021) <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_ELLIPTICFUNCTION_HPP)
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#define GEOGRAPHICLIB_ELLIPTICFUNCTION_HPP 1
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#include <GeographicLib/Constants.hpp>
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namespace GeographicLib {
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/**
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* \brief Elliptic integrals and functions
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*
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* This provides the elliptic functions and integrals needed for Ellipsoid,
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* GeodesicExact, and TransverseMercatorExact. Two categories of function
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* are provided:
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* - \e static functions to compute symmetric elliptic integrals
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* (https://dlmf.nist.gov/19.16.i)
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* - \e member functions to compute Legrendre's elliptic
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* integrals (https://dlmf.nist.gov/19.2.ii) and the
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* Jacobi elliptic functions (https://dlmf.nist.gov/22.2).
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* .
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* In the latter case, an object is constructed giving the modulus \e k (and
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* optionally the parameter α<sup>2</sup>). The modulus is always
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* passed as its square <i>k</i><sup>2</sup> which allows \e k to be pure
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* imaginary (<i>k</i><sup>2</sup> < 0). (Confusingly, Abramowitz and
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* Stegun call \e m = <i>k</i><sup>2</sup> the "parameter" and \e n =
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* α<sup>2</sup> the "characteristic".)
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*
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* In geodesic applications, it is convenient to separate the incomplete
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* integrals into secular and periodic components, e.g.,
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* \f[
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* E(\phi, k) = (2 E(k) / \pi) [ \phi + \delta E(\phi, k) ]
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* \f]
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* where δ\e E(φ, \e k) is an odd periodic function with period
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* π.
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*
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* The computation of the elliptic integrals uses the algorithms given in
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* - B. C. Carlson,
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* <a href="https://doi.org/10.1007/BF02198293"> Computation of real or
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* complex elliptic integrals</a>, Numerical Algorithms 10, 13--26 (1995)
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* .
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* with the additional optimizations given in https://dlmf.nist.gov/19.36.i.
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* The computation of the Jacobi elliptic functions uses the algorithm given
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* in
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* - R. Bulirsch,
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* <a href="https://doi.org/10.1007/BF01397975"> Numerical Calculation of
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* Elliptic Integrals and Elliptic Functions</a>, Numericshe Mathematik 7,
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* 78--90 (1965).
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* .
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* The notation follows https://dlmf.nist.gov/19 and https://dlmf.nist.gov/22
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*
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* Example of use:
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* \include example-EllipticFunction.cpp
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**********************************************************************/
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class GEOGRAPHICLIB_EXPORT EllipticFunction {
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private:
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typedef Math::real real;
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enum { num_ = 13 }; // Max depth required for sncndn; probably 5 is enough.
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real _k2, _kp2, _alpha2, _alphap2, _eps;
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real _Kc, _Ec, _Dc, _Pic, _Gc, _Hc;
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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 specifying the modulus and parameter.
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*
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* @param[in] k2 the square of the modulus <i>k</i><sup>2</sup>.
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* <i>k</i><sup>2</sup> must lie in (−∞, 1].
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* @param[in] alpha2 the parameter α<sup>2</sup>.
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* α<sup>2</sup> must lie in (−∞, 1].
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* @exception GeographicErr if \e k2 or \e alpha2 is out of its legal
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* range.
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*
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* If only elliptic integrals of the first and second kinds are needed,
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* then set α<sup>2</sup> = 0 (the default value); in this case, we
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* have Π(φ, 0, \e k) = \e F(φ, \e k), \e G(φ, 0, \e k) = \e
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* E(φ, \e k), and \e H(φ, 0, \e k) = \e F(φ, \e k) - \e
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* D(φ, \e k).
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**********************************************************************/
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EllipticFunction(real k2 = 0, real alpha2 = 0)
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{ Reset(k2, alpha2); }
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/**
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* Constructor specifying the modulus and parameter and their complements.
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*
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* @param[in] k2 the square of the modulus <i>k</i><sup>2</sup>.
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* <i>k</i><sup>2</sup> must lie in (−∞, 1].
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* @param[in] alpha2 the parameter α<sup>2</sup>.
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* α<sup>2</sup> must lie in (−∞, 1].
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* @param[in] kp2 the complementary modulus squared <i>k'</i><sup>2</sup> =
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* 1 − <i>k</i><sup>2</sup>. This must lie in [0, ∞).
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* @param[in] alphap2 the complementary parameter α'<sup>2</sup> = 1
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* − α<sup>2</sup>. This must lie in [0, ∞).
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* @exception GeographicErr if \e k2, \e alpha2, \e kp2, or \e alphap2 is
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* out of its legal range.
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*
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* The arguments must satisfy \e k2 + \e kp2 = 1 and \e alpha2 + \e alphap2
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* = 1. (No checking is done that these conditions are met.) This
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* constructor is provided to enable accuracy to be maintained, e.g., when
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* \e k is very close to unity.
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**********************************************************************/
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EllipticFunction(real k2, real alpha2, real kp2, real alphap2)
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{ Reset(k2, alpha2, kp2, alphap2); }
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/**
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* Reset the modulus and parameter.
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*
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* @param[in] k2 the new value of square of the modulus
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* <i>k</i><sup>2</sup> which must lie in (−∞, ].
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* done.)
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* @param[in] alpha2 the new value of parameter α<sup>2</sup>.
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* α<sup>2</sup> must lie in (−∞, 1].
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* @exception GeographicErr if \e k2 or \e alpha2 is out of its legal
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* range.
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**********************************************************************/
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void Reset(real k2 = 0, real alpha2 = 0)
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{ Reset(k2, alpha2, 1 - k2, 1 - alpha2); }
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/**
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* Reset the modulus and parameter supplying also their complements.
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*
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* @param[in] k2 the square of the modulus <i>k</i><sup>2</sup>.
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* <i>k</i><sup>2</sup> must lie in (−∞, 1].
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* @param[in] alpha2 the parameter α<sup>2</sup>.
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* α<sup>2</sup> must lie in (−∞, 1].
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* @param[in] kp2 the complementary modulus squared <i>k'</i><sup>2</sup> =
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* 1 − <i>k</i><sup>2</sup>. This must lie in [0, ∞).
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* @param[in] alphap2 the complementary parameter α'<sup>2</sup> = 1
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* − α<sup>2</sup>. This must lie in [0, ∞).
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* @exception GeographicErr if \e k2, \e alpha2, \e kp2, or \e alphap2 is
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* out of its legal range.
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*
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* The arguments must satisfy \e k2 + \e kp2 = 1 and \e alpha2 + \e alphap2
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* = 1. (No checking is done that these conditions are met.) This
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* constructor is provided to enable accuracy to be maintained, e.g., when
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* is very small.
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**********************************************************************/
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void Reset(real k2, real alpha2, real kp2, real alphap2);
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///@}
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/** \name Inspector functions.
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**********************************************************************/
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///@{
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/**
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* @return the square of the modulus <i>k</i><sup>2</sup>.
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**********************************************************************/
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Math::real k2() const { return _k2; }
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/**
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* @return the square of the complementary modulus <i>k'</i><sup>2</sup> =
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* 1 − <i>k</i><sup>2</sup>.
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**********************************************************************/
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Math::real kp2() const { return _kp2; }
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/**
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* @return the parameter α<sup>2</sup>.
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**********************************************************************/
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Math::real alpha2() const { return _alpha2; }
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/**
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* @return the complementary parameter α'<sup>2</sup> = 1 −
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* α<sup>2</sup>.
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**********************************************************************/
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Math::real alphap2() const { return _alphap2; }
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///@}
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/** \name Complete elliptic integrals.
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**********************************************************************/
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///@{
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/**
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* The complete integral of the first kind.
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*
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* @return \e K(\e k).
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*
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* \e K(\e k) is defined in https://dlmf.nist.gov/19.2.E4
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* \f[
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* K(k) = \int_0^{\pi/2} \frac1{\sqrt{1-k^2\sin^2\phi}}\,d\phi.
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* \f]
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**********************************************************************/
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Math::real K() const { return _Kc; }
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/**
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* The complete integral of the second kind.
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*
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* @return \e E(\e k).
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*
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* \e E(\e k) is defined in https://dlmf.nist.gov/19.2.E5
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* \f[
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* E(k) = \int_0^{\pi/2} \sqrt{1-k^2\sin^2\phi}\,d\phi.
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* \f]
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**********************************************************************/
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Math::real E() const { return _Ec; }
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/**
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* Jahnke's complete integral.
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*
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* @return \e D(\e k).
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*
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* \e D(\e k) is defined in https://dlmf.nist.gov/19.2.E6
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* \f[
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* D(k) =
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* \int_0^{\pi/2} \frac{\sin^2\phi}{\sqrt{1-k^2\sin^2\phi}}\,d\phi.
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* \f]
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**********************************************************************/
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Math::real D() const { return _Dc; }
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/**
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* The difference between the complete integrals of the first and second
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* kinds.
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*
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* @return \e K(\e k) − \e E(\e k).
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**********************************************************************/
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Math::real KE() const { return _k2 * _Dc; }
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/**
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* The complete integral of the third kind.
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*
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* @return Π(α<sup>2</sup>, \e k).
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*
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* Π(α<sup>2</sup>, \e k) is defined in
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* https://dlmf.nist.gov/19.2.E7
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* \f[
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* \Pi(\alpha^2, k) = \int_0^{\pi/2}
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* \frac1{\sqrt{1-k^2\sin^2\phi}(1 - \alpha^2\sin^2\phi)}\,d\phi.
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* \f]
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**********************************************************************/
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Math::real Pi() const { return _Pic; }
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/**
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* Legendre's complete geodesic longitude integral.
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*
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* @return \e G(α<sup>2</sup>, \e k).
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*
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* \e G(α<sup>2</sup>, \e k) is given by
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* \f[
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* G(\alpha^2, k) = \int_0^{\pi/2}
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* \frac{\sqrt{1-k^2\sin^2\phi}}{1 - \alpha^2\sin^2\phi}\,d\phi.
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* \f]
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**********************************************************************/
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Math::real G() const { return _Gc; }
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/**
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* Cayley's complete geodesic longitude difference integral.
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*
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* @return \e H(α<sup>2</sup>, \e k).
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*
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* \e H(α<sup>2</sup>, \e k) is given by
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* \f[
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* H(\alpha^2, k) = \int_0^{\pi/2}
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* \frac{\cos^2\phi}{(1-\alpha^2\sin^2\phi)\sqrt{1-k^2\sin^2\phi}}
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* \,d\phi.
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* \f]
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**********************************************************************/
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Math::real H() const { return _Hc; }
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///@}
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/** \name Incomplete elliptic integrals.
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**********************************************************************/
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///@{
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/**
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* The incomplete integral of the first kind.
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*
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* @param[in] phi
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* @return \e F(φ, \e k).
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*
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* \e F(φ, \e k) is defined in https://dlmf.nist.gov/19.2.E4
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* \f[
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* F(\phi, k) = \int_0^\phi \frac1{\sqrt{1-k^2\sin^2\theta}}\,d\theta.
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* \f]
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**********************************************************************/
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Math::real F(real phi) const;
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/**
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* The incomplete integral of the second kind.
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*
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* @param[in] phi
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* @return \e E(φ, \e k).
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*
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* \e E(φ, \e k) is defined in https://dlmf.nist.gov/19.2.E5
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* \f[
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* E(\phi, k) = \int_0^\phi \sqrt{1-k^2\sin^2\theta}\,d\theta.
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* \f]
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**********************************************************************/
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Math::real E(real phi) const;
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/**
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* The incomplete integral of the second kind with the argument given in
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* degrees.
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*
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* @param[in] ang in <i>degrees</i>.
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* @return \e E(π <i>ang</i>/180, \e k).
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**********************************************************************/
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Math::real Ed(real ang) const;
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/**
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* The inverse of the incomplete integral of the second kind.
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*
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* @param[in] x
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* @return φ = <i>E</i><sup>−1</sup>(\e x, \e k); i.e., the
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* solution of such that \e E(φ, \e k) = \e x.
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**********************************************************************/
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Math::real Einv(real x) const;
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/**
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* The incomplete integral of the third kind.
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*
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* @param[in] phi
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* @return Π(φ, α<sup>2</sup>, \e k).
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*
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* Π(φ, α<sup>2</sup>, \e k) is defined in
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* https://dlmf.nist.gov/19.2.E7
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* \f[
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* \Pi(\phi, \alpha^2, k) = \int_0^\phi
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* \frac1{\sqrt{1-k^2\sin^2\theta}(1 - \alpha^2\sin^2\theta)}\,d\theta.
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* \f]
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**********************************************************************/
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Math::real Pi(real phi) const;
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/**
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* Jahnke's incomplete elliptic integral.
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*
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* @param[in] phi
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* @return \e D(φ, \e k).
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*
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* \e D(φ, \e k) is defined in https://dlmf.nist.gov/19.2.E4
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* \f[
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* D(\phi, k) = \int_0^\phi
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* \frac{\sin^2\theta}{\sqrt{1-k^2\sin^2\theta}}\,d\theta.
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* \f]
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**********************************************************************/
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Math::real D(real phi) const;
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/**
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* Legendre's geodesic longitude integral.
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*
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* @param[in] phi
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* @return \e G(φ, α<sup>2</sup>, \e k).
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*
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* \e G(φ, α<sup>2</sup>, \e k) is defined by
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* \f[
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* \begin{align}
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* G(\phi, \alpha^2, k) &=
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* \frac{k^2}{\alpha^2} F(\phi, k) +
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* \biggl(1 - \frac{k^2}{\alpha^2}\biggr) \Pi(\phi, \alpha^2, k) \\
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* &= \int_0^\phi
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* \frac{\sqrt{1-k^2\sin^2\theta}}{1 - \alpha^2\sin^2\theta}\,d\theta.
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* \end{align}
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* \f]
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*
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* Legendre expresses the longitude of a point on the geodesic in terms of
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* this combination of elliptic integrals in Exercices de Calcul
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* Intégral, Vol. 1 (1811), p. 181,
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* https://books.google.com/books?id=riIOAAAAQAAJ&pg=PA181.
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*
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* See \ref geodellip for the expression for the longitude in terms of this
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* function.
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**********************************************************************/
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Math::real G(real phi) const;
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/**
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* Cayley's geodesic longitude difference integral.
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*
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* @param[in] phi
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* @return \e H(φ, α<sup>2</sup>, \e k).
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*
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* \e H(φ, α<sup>2</sup>, \e k) is defined by
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* \f[
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* \begin{align}
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* H(\phi, \alpha^2, k) &=
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* \frac1{\alpha^2} F(\phi, k) +
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* \biggl(1 - \frac1{\alpha^2}\biggr) \Pi(\phi, \alpha^2, k) \\
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* &= \int_0^\phi
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* \frac{\cos^2\theta}
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* {(1-\alpha^2\sin^2\theta)\sqrt{1-k^2\sin^2\theta}}
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* \,d\theta.
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* \end{align}
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* \f]
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*
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* Cayley expresses the longitude difference of a point on the geodesic in
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* terms of this combination of elliptic integrals in Phil. Mag. <b>40</b>
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* (1870), p. 333, https://books.google.com/books?id=Zk0wAAAAIAAJ&pg=PA333.
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*
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* See \ref geodellip for the expression for the longitude in terms of this
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* function.
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**********************************************************************/
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Math::real H(real phi) const;
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///@}
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/** \name Incomplete integrals in terms of Jacobi elliptic functions.
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**********************************************************************/
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///@{
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/**
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* The incomplete integral of the first kind in terms of Jacobi elliptic
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* functions.
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*
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* @param[in] sn = sinφ.
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* @param[in] cn = cosφ.
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* @param[in] dn = sqrt(1 − <i>k</i><sup>2</sup>
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* sin<sup>2</sup>φ).
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* @return \e F(φ, \e k) as though φ ∈ (−π, π].
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**********************************************************************/
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Math::real F(real sn, real cn, real dn) const;
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/**
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* The incomplete integral of the second kind in terms of Jacobi elliptic
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* functions.
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*
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* @param[in] sn = sinφ.
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* @param[in] cn = cosφ.
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* @param[in] dn = sqrt(1 − <i>k</i><sup>2</sup>
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* sin<sup>2</sup>φ).
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* @return \e E(φ, \e k) as though φ ∈ (−π, π].
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**********************************************************************/
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Math::real E(real sn, real cn, real dn) const;
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/**
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* The incomplete integral of the third kind in terms of Jacobi elliptic
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* functions.
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*
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* @param[in] sn = sinφ.
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* @param[in] cn = cosφ.
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* @param[in] dn = sqrt(1 − <i>k</i><sup>2</sup>
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* sin<sup>2</sup>φ).
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* @return Π(φ, α<sup>2</sup>, \e k) as though φ ∈
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* (−π, π].
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**********************************************************************/
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Math::real Pi(real sn, real cn, real dn) const;
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/**
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* Jahnke's incomplete elliptic integral in terms of Jacobi elliptic
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* functions.
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*
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* @param[in] sn = sinφ.
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* @param[in] cn = cosφ.
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* @param[in] dn = sqrt(1 − <i>k</i><sup>2</sup>
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* sin<sup>2</sup>φ).
|
|
* @return \e D(φ, \e k) as though φ ∈ (−π, π].
|
|
**********************************************************************/
|
|
Math::real D(real sn, real cn, real dn) const;
|
|
|
|
/**
|
|
* Legendre's geodesic longitude integral in terms of Jacobi elliptic
|
|
* functions.
|
|
*
|
|
* @param[in] sn = sinφ.
|
|
* @param[in] cn = cosφ.
|
|
* @param[in] dn = sqrt(1 − <i>k</i><sup>2</sup>
|
|
* sin<sup>2</sup>φ).
|
|
* @return \e G(φ, α<sup>2</sup>, \e k) as though φ ∈
|
|
* (−π, π].
|
|
**********************************************************************/
|
|
Math::real G(real sn, real cn, real dn) const;
|
|
|
|
/**
|
|
* Cayley's geodesic longitude difference integral in terms of Jacobi
|
|
* elliptic functions.
|
|
*
|
|
* @param[in] sn = sinφ.
|
|
* @param[in] cn = cosφ.
|
|
* @param[in] dn = sqrt(1 − <i>k</i><sup>2</sup>
|
|
* sin<sup>2</sup>φ).
|
|
* @return \e H(φ, α<sup>2</sup>, \e k) as though φ ∈
|
|
* (−π, π].
|
|
**********************************************************************/
|
|
Math::real H(real sn, real cn, real dn) const;
|
|
///@}
|
|
|
|
/** \name Periodic versions of incomplete elliptic integrals.
|
|
**********************************************************************/
|
|
///@{
|
|
/**
|
|
* The periodic incomplete integral of the first kind.
|
|
*
|
|
* @param[in] sn = sinφ.
|
|
* @param[in] cn = cosφ.
|
|
* @param[in] dn = sqrt(1 − <i>k</i><sup>2</sup>
|
|
* sin<sup>2</sup>φ).
|
|
* @return the periodic function π \e F(φ, \e k) / (2 \e K(\e k)) -
|
|
* φ.
|
|
**********************************************************************/
|
|
Math::real deltaF(real sn, real cn, real dn) const;
|
|
|
|
/**
|
|
* The periodic incomplete integral of the second kind.
|
|
*
|
|
* @param[in] sn = sinφ.
|
|
* @param[in] cn = cosφ.
|
|
* @param[in] dn = sqrt(1 − <i>k</i><sup>2</sup>
|
|
* sin<sup>2</sup>φ).
|
|
* @return the periodic function π \e E(φ, \e k) / (2 \e E(\e k)) -
|
|
* φ.
|
|
**********************************************************************/
|
|
Math::real deltaE(real sn, real cn, real dn) const;
|
|
|
|
/**
|
|
* The periodic inverse of the incomplete integral of the second kind.
|
|
*
|
|
* @param[in] stau = sinτ.
|
|
* @param[in] ctau = sinτ.
|
|
* @return the periodic function <i>E</i><sup>−1</sup>(τ (2 \e
|
|
* E(\e k)/π), \e k) - τ.
|
|
**********************************************************************/
|
|
Math::real deltaEinv(real stau, real ctau) const;
|
|
|
|
/**
|
|
* The periodic incomplete integral of the third kind.
|
|
*
|
|
* @param[in] sn = sinφ.
|
|
* @param[in] cn = cosφ.
|
|
* @param[in] dn = sqrt(1 − <i>k</i><sup>2</sup>
|
|
* sin<sup>2</sup>φ).
|
|
* @return the periodic function π Π(φ, α<sup>2</sup>,
|
|
* \e k) / (2 Π(α<sup>2</sup>, \e k)) - φ.
|
|
**********************************************************************/
|
|
Math::real deltaPi(real sn, real cn, real dn) const;
|
|
|
|
/**
|
|
* The periodic Jahnke's incomplete elliptic integral.
|
|
*
|
|
* @param[in] sn = sinφ.
|
|
* @param[in] cn = cosφ.
|
|
* @param[in] dn = sqrt(1 − <i>k</i><sup>2</sup>
|
|
* sin<sup>2</sup>φ).
|
|
* @return the periodic function π \e D(φ, \e k) / (2 \e D(\e k)) -
|
|
* φ.
|
|
**********************************************************************/
|
|
Math::real deltaD(real sn, real cn, real dn) const;
|
|
|
|
/**
|
|
* Legendre's periodic geodesic longitude integral.
|
|
*
|
|
* @param[in] sn = sinφ.
|
|
* @param[in] cn = cosφ.
|
|
* @param[in] dn = sqrt(1 − <i>k</i><sup>2</sup>
|
|
* sin<sup>2</sup>φ).
|
|
* @return the periodic function π \e G(φ, \e k) / (2 \e G(\e k)) -
|
|
* φ.
|
|
**********************************************************************/
|
|
Math::real deltaG(real sn, real cn, real dn) const;
|
|
|
|
/**
|
|
* Cayley's periodic geodesic longitude difference integral.
|
|
*
|
|
* @param[in] sn = sinφ.
|
|
* @param[in] cn = cosφ.
|
|
* @param[in] dn = sqrt(1 − <i>k</i><sup>2</sup>
|
|
* sin<sup>2</sup>φ).
|
|
* @return the periodic function π \e H(φ, \e k) / (2 \e H(\e k)) -
|
|
* φ.
|
|
**********************************************************************/
|
|
Math::real deltaH(real sn, real cn, real dn) const;
|
|
///@}
|
|
|
|
/** \name Elliptic functions.
|
|
**********************************************************************/
|
|
///@{
|
|
/**
|
|
* The Jacobi elliptic functions.
|
|
*
|
|
* @param[in] x the argument.
|
|
* @param[out] sn sn(\e x, \e k).
|
|
* @param[out] cn cn(\e x, \e k).
|
|
* @param[out] dn dn(\e x, \e k).
|
|
**********************************************************************/
|
|
void sncndn(real x, real& sn, real& cn, real& dn) const;
|
|
|
|
/**
|
|
* The Δ amplitude function.
|
|
*
|
|
* @param[in] sn sinφ.
|
|
* @param[in] cn cosφ.
|
|
* @return Δ = sqrt(1 − <i>k</i><sup>2</sup>
|
|
* sin<sup>2</sup>φ).
|
|
**********************************************************************/
|
|
Math::real Delta(real sn, real cn) const {
|
|
using std::sqrt;
|
|
return sqrt(_k2 < 0 ? 1 - _k2 * sn*sn : _kp2 + _k2 * cn*cn);
|
|
}
|
|
///@}
|
|
|
|
/** \name Symmetric elliptic integrals.
|
|
**********************************************************************/
|
|
///@{
|
|
/**
|
|
* Symmetric integral of the first kind <i>R</i><sub><i>F</i></sub>.
|
|
*
|
|
* @param[in] x
|
|
* @param[in] y
|
|
* @param[in] z
|
|
* @return <i>R</i><sub><i>F</i></sub>(\e x, \e y, \e z).
|
|
*
|
|
* <i>R</i><sub><i>F</i></sub> is defined in https://dlmf.nist.gov/19.16.E1
|
|
* \f[ R_F(x, y, z) = \frac12
|
|
* \int_0^\infty\frac1{\sqrt{(t + x) (t + y) (t + z)}}\, dt \f]
|
|
* If one of the arguments is zero, it is more efficient to call the
|
|
* two-argument version of this function with the non-zero arguments.
|
|
**********************************************************************/
|
|
static real RF(real x, real y, real z);
|
|
|
|
/**
|
|
* Complete symmetric integral of the first kind,
|
|
* <i>R</i><sub><i>F</i></sub> with one argument zero.
|
|
*
|
|
* @param[in] x
|
|
* @param[in] y
|
|
* @return <i>R</i><sub><i>F</i></sub>(\e x, \e y, 0).
|
|
**********************************************************************/
|
|
static real RF(real x, real y);
|
|
|
|
/**
|
|
* Degenerate symmetric integral of the first kind
|
|
* <i>R</i><sub><i>C</i></sub>.
|
|
*
|
|
* @param[in] x
|
|
* @param[in] y
|
|
* @return <i>R</i><sub><i>C</i></sub>(\e x, \e y) =
|
|
* <i>R</i><sub><i>F</i></sub>(\e x, \e y, \e y).
|
|
*
|
|
* <i>R</i><sub><i>C</i></sub> is defined in https://dlmf.nist.gov/19.2.E17
|
|
* \f[ R_C(x, y) = \frac12
|
|
* \int_0^\infty\frac1{\sqrt{t + x}(t + y)}\,dt \f]
|
|
**********************************************************************/
|
|
static real RC(real x, real y);
|
|
|
|
/**
|
|
* Symmetric integral of the second kind <i>R</i><sub><i>G</i></sub>.
|
|
*
|
|
* @param[in] x
|
|
* @param[in] y
|
|
* @param[in] z
|
|
* @return <i>R</i><sub><i>G</i></sub>(\e x, \e y, \e z).
|
|
*
|
|
* <i>R</i><sub><i>G</i></sub> is defined in Carlson, eq 1.5
|
|
* \f[ R_G(x, y, z) = \frac14
|
|
* \int_0^\infty[(t + x) (t + y) (t + z)]^{-1/2}
|
|
* \biggl(
|
|
* \frac x{t + x} + \frac y{t + y} + \frac z{t + z}
|
|
* \biggr)t\,dt \f]
|
|
* See also https://dlmf.nist.gov/19.16.E3.
|
|
* If one of the arguments is zero, it is more efficient to call the
|
|
* two-argument version of this function with the non-zero arguments.
|
|
**********************************************************************/
|
|
static real RG(real x, real y, real z);
|
|
|
|
/**
|
|
* Complete symmetric integral of the second kind,
|
|
* <i>R</i><sub><i>G</i></sub> with one argument zero.
|
|
*
|
|
* @param[in] x
|
|
* @param[in] y
|
|
* @return <i>R</i><sub><i>G</i></sub>(\e x, \e y, 0).
|
|
**********************************************************************/
|
|
static real RG(real x, real y);
|
|
|
|
/**
|
|
* Symmetric integral of the third kind <i>R</i><sub><i>J</i></sub>.
|
|
*
|
|
* @param[in] x
|
|
* @param[in] y
|
|
* @param[in] z
|
|
* @param[in] p
|
|
* @return <i>R</i><sub><i>J</i></sub>(\e x, \e y, \e z, \e p).
|
|
*
|
|
* <i>R</i><sub><i>J</i></sub> is defined in https://dlmf.nist.gov/19.16.E2
|
|
* \f[ R_J(x, y, z, p) = \frac32
|
|
* \int_0^\infty
|
|
* [(t + x) (t + y) (t + z)]^{-1/2} (t + p)^{-1}\, dt \f]
|
|
**********************************************************************/
|
|
static real RJ(real x, real y, real z, real p);
|
|
|
|
/**
|
|
* Degenerate symmetric integral of the third kind
|
|
* <i>R</i><sub><i>D</i></sub>.
|
|
*
|
|
* @param[in] x
|
|
* @param[in] y
|
|
* @param[in] z
|
|
* @return <i>R</i><sub><i>D</i></sub>(\e x, \e y, \e z) =
|
|
* <i>R</i><sub><i>J</i></sub>(\e x, \e y, \e z, \e z).
|
|
*
|
|
* <i>R</i><sub><i>D</i></sub> is defined in https://dlmf.nist.gov/19.16.E5
|
|
* \f[ R_D(x, y, z) = \frac32
|
|
* \int_0^\infty[(t + x) (t + y)]^{-1/2} (t + z)^{-3/2}\, dt \f]
|
|
**********************************************************************/
|
|
static real RD(real x, real y, real z);
|
|
///@}
|
|
|
|
};
|
|
|
|
} // namespace GeographicLib
|
|
|
|
#endif // GEOGRAPHICLIB_ELLIPTICFUNCTION_HPP
|