Elliptic integrals and functions. More...

#include <GeographicLib/EllipticFunction.hpp>

Public Member Functions
Constructor
	EllipticFunction (real k2=0, real alpha2=0)

	EllipticFunction (real k2, real alpha2, real kp2, real alphap2)

void	Reset (real k2=0, real alpha2=0)

void	Reset (real k2, real alpha2, real kp2, real alphap2)

Inspector functions.
Math::real	k2 () const

Math::real	kp2 () const

Math::real	alpha2 () const

Math::real	alphap2 () const

Complete elliptic integrals.
Math::real	K () const

Math::real	E () const

Math::real	D () const

Math::real	KE () const

Math::real	Pi () const

Math::real	G () const

Math::real	H () const

Incomplete elliptic integrals.
Math::real	F (real phi) const

Math::real	E (real phi) const

Math::real	Ed (real ang) const

Math::real	Einv (real x) const

Math::real	Pi (real phi) const

Math::real	D (real phi) const

Math::real	G (real phi) const

Math::real	H (real phi) const

Incomplete integrals in terms of Jacobi elliptic functions.
Math::real	F (real sn, real cn, real dn) const

Math::real	E (real sn, real cn, real dn) const

Math::real	Pi (real sn, real cn, real dn) const

Math::real	D (real sn, real cn, real dn) const

Math::real	G (real sn, real cn, real dn) const

Math::real	H (real sn, real cn, real dn) const

Periodic versions of incomplete elliptic integrals.
Math::real	deltaF (real sn, real cn, real dn) const

Math::real	deltaE (real sn, real cn, real dn) const

Math::real	deltaEinv (real stau, real ctau) const

Math::real	deltaPi (real sn, real cn, real dn) const

Math::real	deltaD (real sn, real cn, real dn) const

Math::real	deltaG (real sn, real cn, real dn) const

Math::real	deltaH (real sn, real cn, real dn) const

Elliptic functions.
void	sncndn (real x, real &sn, real &cn, real &dn) const

Math::real	Delta (real sn, real cn) const

Static Public Member Functions
Symmetric elliptic integrals.
static real	RF (real x, real y, real z)

static real	RF (real x, real y)

static real	RC (real x, real y)

static real	RG (real x, real y, real z)

static real	RG (real x, real y)

static real	RJ (real x, real y, real z, real p)

static real	RD (real x, real y, real z)

Detailed Description

Elliptic integrals and functions.

This provides the elliptic functions and integrals needed for Ellipsoid, GeodesicExact, and TransverseMercatorExact. Two categories of function are provided:

static functions to compute symmetric elliptic integrals (https://dlmf.nist.gov/19.16.i)
member functions to compute Legrendre's elliptic integrals (https://dlmf.nist.gov/19.2.ii) and the Jacobi elliptic functions (https://dlmf.nist.gov/22.2).

In the latter case, an object is constructed giving the modulus k (and optionally the parameter α²). The modulus is always passed as its square k² which allows k to be pure imaginary (k² < 0). (Confusingly, Abramowitz and Stegun call m = k² the "parameter" and n = α² the "characteristic".)

In geodesic applications, it is convenient to separate the incomplete integrals into secular and periodic components, e.g.,

\[ E(\phi, k) = (2 E(k) / \pi) [ \phi + \delta E(\phi, k) ] \]

where δE(φ, k) is an odd periodic function with period π.

The computation of the elliptic integrals uses the algorithms given in

B. C. Carlson, Computation of real or complex elliptic integrals, Numerical Algorithms 10, 13–26 (1995); preprint.

with the additional optimizations given in https://dlmf.nist.gov/19.36.i. The computation of the Jacobi elliptic functions uses the algorithm given in

R. Bulirsch, Numerical Calculation of Elliptic Integrals and Elliptic Functions, Numericshe Mathematik 7, 78–90 (1965).

The notation follows https://dlmf.nist.gov/19 and https://dlmf.nist.gov/22

Example of use:

// Example of using the GeographicLib::EllipticFunction class
 
#include <iostream>
#include <iomanip>
#include <exception>
#include <cmath>
#include <GeographicLib/Math.hpp>
#include <GeographicLib/EllipticFunction.hpp>
 
using namespace std;
using namespace GeographicLib;
 
int main() {
  try {
    EllipticFunction ell(0.1);  // parameter m = 0.1
    // See Abramowitz and Stegun, table 17.1
    cout << ell.K() << " " << ell.E() << "\n";
    double phi = 20, sn, cn;
    Math::sincosd(phi, sn ,cn);
    // See Abramowitz and Stegun, table 17.6 with
    // alpha = asin(sqrt(m)) = 18.43 deg and phi = 20 deg
    cout << ell.E(phi * Math::degree()) << " "
         << ell.E(sn, cn, ell.Delta(sn, cn))
         << "\n";
    // See Carlson 1995, Sec 3.
    cout << fixed << setprecision(16)
         << "RF(1,2,0)      = " << EllipticFunction::RF(1,2)      << "\n"
         << "RF(2,3,4)      = " << EllipticFunction::RF(2,3,4)    << "\n"
         << "RC(0,1/4)      = " << EllipticFunction::RC(0,0.25)   << "\n"
         << "RC(9/4,2)      = " << EllipticFunction::RC(2.25,2)   << "\n"
         << "RC(1/4,-2)     = " << EllipticFunction::RC(0.25,-2)  << "\n"
         << "RJ(0,1,2,3)    = " << EllipticFunction::RJ(0,1,2,3)  << "\n"
         << "RJ(2,3,4,5)    = " << EllipticFunction::RJ(2,3,4,5)  << "\n"
         << "RD(0,2,1)      = " << EllipticFunction::RD(0,2,1)    << "\n"
         << "RD(2,3,4)      = " << EllipticFunction::RD(2,3,4)    << "\n"
         << "RG(0,16,16)    = " << EllipticFunction::RG(16,16)    << "\n"
         << "RG(2,3,4)      = " << EllipticFunction::RG(2,3,4)    << "\n"
         << "RG(0,0.0796,4) = " << EllipticFunction::RG(0.0796,4) << "\n";
  }
  catch (const exception& e) {
    cout << "Caught exception: " << e.what() << "\n";
  }
}

Definition at line 63 of file EllipticFunction.hpp.

Constructor & Destructor Documentation

◆ EllipticFunction() [1/2]

GeographicLib::EllipticFunction::EllipticFunction	(	real	k2 = `0`,
		real	alpha2 = `0`
	)

inline

Constructor specifying the modulus and parameter.

Parameters

[in]	k2	the square of the modulus k². k² must lie in (−∞, 1].
[in]	alpha2	the parameter α². α² must lie in (−∞, 1].

Exceptions

GeographicErr if k2 or alpha2 is out of its legal range.

If only elliptic integrals of the first and second kinds are needed, then set α² = 0 (the default value); in this case, we have Π(φ, 0, k) = F(φ, k), G(φ, 0, k) = E(φ, k), and H(φ, 0, k) = F(φ, k) - D(φ, k).

Definition at line 90 of file EllipticFunction.hpp.

◆ EllipticFunction() [2/2]

GeographicLib::EllipticFunction::EllipticFunction	(	real	k2,
		real	alpha2,
		real	kp2,
		real	alphap2
	)

inline

Constructor specifying the modulus and parameter and their complements.

Parameters

[in]	k2	the square of the modulus k². k² must lie in (−∞, 1].
[in]	alpha2	the parameter α². α² must lie in (−∞, 1].
[in]	kp2	the complementary modulus squared k'² = 1 − k². This must lie in [0, ∞).
[in]	alphap2	the complementary parameter α'² = 1 − α². This must lie in [0, ∞).

Exceptions

GeographicErr if k2, alpha2, kp2, or alphap2 is out of its legal range.

The arguments must satisfy k2 + kp2 = 1 and alpha2 + alphap2 = 1. (No checking is done that these conditions are met.) This constructor is provided to enable accuracy to be maintained, e.g., when k is very close to unity.

Definition at line 112 of file EllipticFunction.hpp.

Member Function Documentation

◆ Reset() [1/2]

void GeographicLib::EllipticFunction::Reset	(	real	k2 = `0`,
		real	alpha2 = `0`
	)

inline

Reset the modulus and parameter.

Parameters

[in]	k2	the new value of square of the modulus k² which must lie in (−∞, ]. done.)
[in]	alpha2	the new value of parameter α². α² must lie in (−∞, 1].

Exceptions

GeographicErr if k2 or alpha2 is out of its legal range.

Definition at line 126 of file EllipticFunction.hpp.

◆ Reset() [2/2]

void GeographicLib::EllipticFunction::Reset	(	real	k2,
		real	alpha2,
		real	kp2,
		real	alphap2
	)

Reset the modulus and parameter supplying also their complements.

Parameters

[in]	k2	the square of the modulus k². k² must lie in (−∞, 1].
[in]	alpha2	the parameter α². α² must lie in (−∞, 1].
[in]	kp2	the complementary modulus squared k'² = 1 − k². This must lie in [0, ∞).
[in]	alphap2	the complementary parameter α'² = 1 − α². This must lie in [0, ∞).

Exceptions

GeographicErr if k2, alpha2, kp2, or alphap2 is out of its legal range.

The arguments must satisfy k2 + kp2 = 1 and alpha2 + alphap2 = 1. (No checking is done that these conditions are met.) This constructor is provided to enable accuracy to be maintained, e.g., when is very small.

Definition at line 226 of file EllipticFunction.cpp.

References alpha2(), alphap2(), GeographicLib::Math::infinity(), k2(), kp2(), GeographicLib::Math::pi(), RC(), RD(), RF(), RG(), RJ(), and GeographicLib::Math::sq().

◆ k2()

Math::real GeographicLib::EllipticFunction::k2 ( ) const

inline

Returns: the square of the modulus k².

Definition at line 158 of file EllipticFunction.hpp.

Referenced by Reset().

◆ kp2()

Math::real GeographicLib::EllipticFunction::kp2 ( ) const

inline

Returns: the square of the complementary modulus k'² = 1 − k².

Definition at line 164 of file EllipticFunction.hpp.

Referenced by Reset().

◆ alpha2()

Math::real GeographicLib::EllipticFunction::alpha2 ( ) const

inline

Returns: the parameter α².

Definition at line 169 of file EllipticFunction.hpp.

Referenced by Reset().

◆ alphap2()

Math::real GeographicLib::EllipticFunction::alphap2 ( ) const

inline

Returns: the complementary parameter α'² = 1 − α².

Definition at line 175 of file EllipticFunction.hpp.

Referenced by Reset().

◆ K()

Math::real GeographicLib::EllipticFunction::K ( ) const

inline

The complete integral of the first kind.

Returns: K(k).

K(k) is defined in https://dlmf.nist.gov/19.2.E4

\[ K(k) = \int_0^{\pi/2} \frac1{\sqrt{1-k^2\sin^2\phi}}\,d\phi. \]

Definition at line 191 of file EllipticFunction.hpp.

Referenced by deltaF(), F(), GeographicLib::TransverseMercatorExact::Forward(), and GeographicLib::TransverseMercatorExact::Reverse().

◆ E() [1/3]

Math::real GeographicLib::EllipticFunction::E ( ) const

inline

The complete integral of the second kind.

Returns: E(k).

E(k) is defined in https://dlmf.nist.gov/19.2.E5

\[ E(k) = \int_0^{\pi/2} \sqrt{1-k^2\sin^2\phi}\,d\phi. \]

Definition at line 203 of file EllipticFunction.hpp.

Referenced by deltaE(), deltaEinv(), E(), Ed(), Einv(), GeographicLib::TransverseMercatorExact::Forward(), GeographicLib::Ellipsoid::InverseRectifyingLatitude(), GeographicLib::Ellipsoid::QuarterMeridian(), and GeographicLib::TransverseMercatorExact::Reverse().

◆ D() [1/3]

Math::real GeographicLib::EllipticFunction::D ( ) const

inline

Jahnke's complete integral.

Returns: D(k).

D(k) is defined in https://dlmf.nist.gov/19.2.E6

\[ D(k) = \int_0^{\pi/2} \frac{\sin^2\phi}{\sqrt{1-k^2\sin^2\phi}}\,d\phi. \]

Definition at line 216 of file EllipticFunction.hpp.

Referenced by D(), and deltaD().

◆ KE()

Math::real GeographicLib::EllipticFunction::KE ( ) const

inline

The difference between the complete integrals of the first and second kinds.

Returns: K(k) − E(k).

Definition at line 224 of file EllipticFunction.hpp.

Referenced by GeographicLib::TransverseMercatorExact::Reverse().

◆ Pi() [1/3]

Math::real GeographicLib::EllipticFunction::Pi ( ) const

inline

The complete integral of the third kind.

Returns: Π(α², k).

Π(α², k) is defined in https://dlmf.nist.gov/19.2.E7

\[ \Pi(\alpha^2, k) = \int_0^{\pi/2} \frac1{\sqrt{1-k^2\sin^2\phi}(1 - \alpha^2\sin^2\phi)}\,d\phi. \]

Definition at line 238 of file EllipticFunction.hpp.

Referenced by deltaPi(), Pi(), GeographicLib::JacobiConformal::x(), and GeographicLib::JacobiConformal::y().

◆ G() [1/3]

Math::real GeographicLib::EllipticFunction::G ( ) const

inline

Legendre's complete geodesic longitude integral.

Returns: G(α², k).

G(α², k) is given by

\[ G(\alpha^2, k) = \int_0^{\pi/2} \frac{\sqrt{1-k^2\sin^2\phi}}{1 - \alpha^2\sin^2\phi}\,d\phi. \]

Definition at line 251 of file EllipticFunction.hpp.

Referenced by deltaG(), and G().

◆ H() [1/3]

Math::real GeographicLib::EllipticFunction::H ( ) const

inline

Cayley's complete geodesic longitude difference integral.

Returns: H(α², k).

H(α², k) is given by

\[ H(\alpha^2, k) = \int_0^{\pi/2} \frac{\cos^2\phi}{(1-\alpha^2\sin^2\phi)\sqrt{1-k^2\sin^2\phi}} \,d\phi. \]

Definition at line 265 of file EllipticFunction.hpp.

Referenced by deltaH(), and H().

◆ F() [1/2]

Math::real GeographicLib::EllipticFunction::F ( real phi ) const

The incomplete integral of the first kind.

Parameters

[in] phi

Returns: F(φ, k).

F(φ, k) is defined in https://dlmf.nist.gov/19.2.E4

\[ F(\phi, k) = \int_0^\phi \frac1{\sqrt{1-k^2\sin^2\theta}}\,d\theta. \]

Definition at line 496 of file EllipticFunction.cpp.

References Delta(), deltaF(), F(), K(), and GeographicLib::Math::pi().

Referenced by deltaF(), and F().

◆ E() [2/3]

Math::real GeographicLib::EllipticFunction::E ( real phi ) const

The incomplete integral of the second kind.

Parameters

[in] phi

Returns: E(φ, k).

E(φ, k) is defined in https://dlmf.nist.gov/19.2.E5

\[ E(\phi, k) = \int_0^\phi \sqrt{1-k^2\sin^2\theta}\,d\theta. \]

Definition at line 502 of file EllipticFunction.cpp.

References Delta(), deltaE(), E(), and GeographicLib::Math::pi().

◆ Ed()

Math::real GeographicLib::EllipticFunction::Ed ( real ang ) const

The incomplete integral of the second kind with the argument given in degrees.

Parameters

[in] ang in degrees.

Returns: E(π ang/180, k).

Definition at line 508 of file EllipticFunction.cpp.

References GeographicLib::Math::AngNormalize(), Delta(), E(), GeographicLib::Math::sincosd(), and GeographicLib::Math::td.

Referenced by GeographicLib::Ellipsoid::MeridianDistance().

◆ Einv()

Math::real GeographicLib::EllipticFunction::Einv ( real x ) const

The inverse of the incomplete integral of the second kind.

Parameters

[in] x

Returns: φ = E⁻¹(x, k); i.e., the solution of such that E(φ, k) = x.

Definition at line 540 of file EllipticFunction.cpp.

References Delta(), E(), GEOGRAPHICLIB_PANIC, and GeographicLib::Math::pi().

Referenced by deltaEinv(), and GeographicLib::Ellipsoid::InverseRectifyingLatitude().

◆ Pi() [2/3]

Math::real GeographicLib::EllipticFunction::Pi ( real phi ) const

The incomplete integral of the third kind.

Parameters

[in] phi

Returns: Π(φ, α², k).

Π(φ, α², k) is defined in https://dlmf.nist.gov/19.2.E7

\[ \Pi(\phi, \alpha^2, k) = \int_0^\phi \frac1{\sqrt{1-k^2\sin^2\theta}(1 - \alpha^2\sin^2\theta)}\,d\theta. \]

Definition at line 516 of file EllipticFunction.cpp.

References Delta(), deltaPi(), GeographicLib::Math::pi(), and Pi().

◆ D() [2/3]

Math::real GeographicLib::EllipticFunction::D ( real phi ) const

Jahnke's incomplete elliptic integral.

Parameters

[in] phi

Returns: D(φ, k).

D(φ, k) is defined in https://dlmf.nist.gov/19.2.E4

\[ D(\phi, k) = \int_0^\phi \frac{\sin^2\theta}{\sqrt{1-k^2\sin^2\theta}}\,d\theta. \]

Definition at line 522 of file EllipticFunction.cpp.

References D(), Delta(), deltaD(), and GeographicLib::Math::pi().

◆ G() [2/3]

Math::real GeographicLib::EllipticFunction::G ( real phi ) const

Legendre's geodesic longitude integral.

Parameters

[in] phi

Returns: G(φ, α², k).

G(φ, α², k) is defined by

\[ \begin{align} G(\phi, \alpha^2, k) &= \frac{k^2}{\alpha^2} F(\phi, k) + \biggl(1 - \frac{k^2}{\alpha^2}\biggr) \Pi(\phi, \alpha^2, k) \\ &= \int_0^\phi \frac{\sqrt{1-k^2\sin^2\theta}}{1 - \alpha^2\sin^2\theta}\,d\theta. \end{align} \]

Legendre expresses the longitude of a point on the geodesic in terms of this combination of elliptic integrals in Exercices de Calcul Intégral, Vol. 1 (1811), p. 181, https://books.google.com/books?id=riIOAAAAQAAJ&pg=PA181.

See Geodesics in terms of elliptic integrals for the expression for the longitude in terms of this function.

Definition at line 528 of file EllipticFunction.cpp.

References Delta(), deltaG(), G(), and GeographicLib::Math::pi().

◆ H() [2/3]

Math::real GeographicLib::EllipticFunction::H ( real phi ) const

Cayley's geodesic longitude difference integral.

Parameters

[in] phi

Returns: H(φ, α², k).

H(φ, α², k) is defined by

\[ \begin{align} H(\phi, \alpha^2, k) &= \frac1{\alpha^2} F(\phi, k) + \biggl(1 - \frac1{\alpha^2}\biggr) \Pi(\phi, \alpha^2, k) \\ &= \int_0^\phi \frac{\cos^2\theta} {(1-\alpha^2\sin^2\theta)\sqrt{1-k^2\sin^2\theta}} \,d\theta. \end{align} \]

Cayley expresses the longitude difference of a point on the geodesic in terms of this combination of elliptic integrals in Phil. Mag. 40 (1870), p. 333, https://books.google.com/books?id=Zk0wAAAAIAAJ&pg=PA333.

See Geodesics in terms of elliptic integrals for the expression for the longitude in terms of this function.

Definition at line 534 of file EllipticFunction.cpp.

References Delta(), deltaH(), H(), and GeographicLib::Math::pi().

◆ F() [2/2]

Math::real GeographicLib::EllipticFunction::F	(	real	sn,
		real	cn,
		real	dn
	)		const

The incomplete integral of the first kind in terms of Jacobi elliptic functions.

Parameters

[in]	sn	= sinφ.
[in]	cn	= cosφ.
[in]	dn	= sqrt(1 − k² sin²φ).

Returns: F(φ, k) as though φ ∈ (−π, π].

Definition at line 372 of file EllipticFunction.cpp.

References K(), and RF().

◆ E() [3/3]

Math::real GeographicLib::EllipticFunction::E	(	real	sn,
		real	cn,
		real	dn
	)		const

The incomplete integral of the second kind in terms of Jacobi elliptic functions.

Parameters

[in]	sn	= sinφ.
[in]	cn	= cosφ.
[in]	dn	= sqrt(1 − k² sin²φ).

Returns: E(φ, k) as though φ ∈ (−π, π].

Definition at line 383 of file EllipticFunction.cpp.

References E(), RD(), and RF().

◆ Pi() [3/3]

Math::real GeographicLib::EllipticFunction::Pi	(	real	sn,
		real	cn,
		real	dn
	)		const

The incomplete integral of the third kind in terms of Jacobi elliptic functions.

Parameters

[in]	sn	= sinφ.
[in]	cn	= cosφ.
[in]	dn	= sqrt(1 − k² sin²φ).

Returns: Π(φ, α², k) as though φ ∈ (−π, π].

Definition at line 418 of file EllipticFunction.cpp.

References Pi(), RF(), and RJ().

◆ D() [3/3]

Math::real GeographicLib::EllipticFunction::D	(	real	sn,
		real	cn,
		real	dn
	)		const

Jahnke's incomplete elliptic integral in terms of Jacobi elliptic functions.

Parameters

[in]	sn	= sinφ.
[in]	cn	= cosφ.
[in]	dn	= sqrt(1 − k² sin²φ).

Returns: D(φ, k) as though φ ∈ (−π, π].

Definition at line 406 of file EllipticFunction.cpp.

References D(), and RD().

◆ G() [3/3]

Math::real GeographicLib::EllipticFunction::G	(	real	sn,
		real	cn,
		real	dn
	)		const

Legendre's geodesic longitude integral in terms of Jacobi elliptic functions.

Parameters

[in]	sn	= sinφ.
[in]	cn	= cosφ.
[in]	dn	= sqrt(1 − k² sin²φ).

Returns: G(φ, α², k) as though φ ∈ (−π, π].

Definition at line 433 of file EllipticFunction.cpp.

References G(), RF(), and RJ().

◆ H() [3/3]

Math::real GeographicLib::EllipticFunction::H	(	real	sn,
		real	cn,
		real	dn
	)		const

Cayley's geodesic longitude difference integral in terms of Jacobi elliptic functions.

Parameters

[in]	sn	= sinφ.
[in]	cn	= cosφ.
[in]	dn	= sqrt(1 − k² sin²φ).

Returns: H(φ, α², k) as though φ ∈ (−π, π].

Definition at line 446 of file EllipticFunction.cpp.

References H(), RF(), and RJ().

◆ deltaF()

Math::real GeographicLib::EllipticFunction::deltaF	(	real	sn,
		real	cn,
		real	dn
	)		const

The periodic incomplete integral of the first kind.

Parameters

[in]	sn	= sinφ.
[in]	cn	= cosφ.
[in]	dn	= sqrt(1 − k² sin²φ).

Returns: the periodic function π F(φ, k) / (2 K(k)) - φ.

Definition at line 460 of file EllipticFunction.cpp.

References F(), K(), and GeographicLib::Math::pi().

Referenced by F().

◆ deltaE()

Math::real GeographicLib::EllipticFunction::deltaE	(	real	sn,
		real	cn,
		real	dn
	)		const

The periodic incomplete integral of the second kind.

Parameters

[in]	sn	= sinφ.
[in]	cn	= cosφ.
[in]	dn	= sqrt(1 − k² sin²φ).

Returns: the periodic function π E(φ, k) / (2 E(k)) - φ.

Definition at line 466 of file EllipticFunction.cpp.

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

Referenced by E(), and GeographicLib::GeodesicLineExact::GenPosition().

◆ deltaEinv()

Math::real GeographicLib::EllipticFunction::deltaEinv	(	real	stau,
		real	ctau
	)		const

The periodic inverse of the incomplete integral of the second kind.

Parameters

[in]	stau	= sinτ.
[in]	ctau	= sinτ.

Returns: the periodic function E⁻¹(τ (2 E(k)/π), k) - τ.

Definition at line 565 of file EllipticFunction.cpp.

References E(), Einv(), and GeographicLib::Math::pi().

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

◆ deltaPi()

Math::real GeographicLib::EllipticFunction::deltaPi	(	real	sn,
		real	cn,
		real	dn
	)		const

The periodic incomplete integral of the third kind.

Parameters

[in]	sn	= sinφ.
[in]	cn	= cosφ.
[in]	dn	= sqrt(1 − k² sin²φ).

Returns: the periodic function π Π(φ, α², k) / (2 Π(α², k)) - φ.

Definition at line 472 of file EllipticFunction.cpp.

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

Referenced by Pi().

◆ deltaD()

Math::real GeographicLib::EllipticFunction::deltaD	(	real	sn,
		real	cn,
		real	dn
	)		const

The periodic Jahnke's incomplete elliptic integral.

Parameters

[in]	sn	= sinφ.
[in]	cn	= cosφ.
[in]	dn	= sqrt(1 − k² sin²φ).

Returns: the periodic function π D(φ, k) / (2 D(k)) - φ.

Definition at line 478 of file EllipticFunction.cpp.

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

Referenced by D(), and GeographicLib::GeodesicLineExact::GenPosition().

◆ deltaG()

Math::real GeographicLib::EllipticFunction::deltaG	(	real	sn,
		real	cn,
		real	dn
	)		const

Legendre's periodic geodesic longitude integral.

Parameters

[in]	sn	= sinφ.
[in]	cn	= cosφ.
[in]	dn	= sqrt(1 − k² sin²φ).

Returns: the periodic function π G(φ, k) / (2 G(k)) - φ.

Definition at line 484 of file EllipticFunction.cpp.

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

Referenced by G().

◆ deltaH()

Math::real GeographicLib::EllipticFunction::deltaH	(	real	sn,
		real	cn,
		real	dn
	)		const

Cayley's periodic geodesic longitude difference integral.

Parameters

[in]	sn	= sinφ.
[in]	cn	= cosφ.
[in]	dn	= sqrt(1 − k² sin²φ).

Returns: the periodic function π H(φ, k) / (2 H(k)) - φ.

Definition at line 490 of file EllipticFunction.cpp.

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

Referenced by GeographicLib::GeodesicLineExact::GenPosition(), and H().

◆ sncndn()

void GeographicLib::EllipticFunction::sncndn	(	real	x,
		real &	sn,
		real &	cn,
		real &	dn
	)		const

The Jacobi elliptic functions.

Parameters

[in]	x	the argument.
[out]	sn	sn(x, k).
[out]	cn	cn(x, k).
[out]	dn	dn(x, k).

Definition at line 317 of file EllipticFunction.cpp.

References GEOGRAPHICLIB_PANIC, and std::swap().

Referenced by GeographicLib::TransverseMercatorExact::Forward(), and GeographicLib::TransverseMercatorExact::Reverse().

◆ Delta()

Math::real GeographicLib::EllipticFunction::Delta	(	real	sn,
		real	cn
	)		const

inline

The Δ amplitude function.

Parameters

[in]	sn	sinφ.
[in]	cn	cosφ.

Returns: Δ = sqrt(1 − k² sin²φ).

Definition at line 586 of file EllipticFunction.hpp.

Referenced by D(), E(), Ed(), Einv(), F(), G(), GeographicLib::GeodesicLineExact::GenPosition(), H(), Pi(), GeographicLib::JacobiConformal::x(), and GeographicLib::JacobiConformal::y().

◆ RF() [1/2]

Math::real GeographicLib::EllipticFunction::RF	(	real	x,
		real	y,
		real	z
	)

static

Symmetric integral of the first kind R_F.

Parameters

[in]	x
[in]	y
[in]	z

Returns: R_F(x, y, z).

R_F is defined in https://dlmf.nist.gov/19.16.E1

\[ R_F(x, y, z) = \frac12 \int_0^\infty\frac1{\sqrt{(t + x) (t + y) (t + z)}}\, dt, \]

where at most one of arguments, x, y, z, can be zero and those arguments that are nonzero must be positive. 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.

Definition at line 29 of file EllipticFunction.cpp.

Referenced by E(), F(), G(), H(), Pi(), Reset(), and RG().

◆ RF() [2/2]

Math::real GeographicLib::EllipticFunction::RF	(	real	x,
		real	y
	)

static

Complete symmetric integral of the first kind, R_F with one argument zero.

Parameters

[in]	x
[in]	y

Returns: R_F(x, y, 0).

The arguments x and y must be positive.

Definition at line 66 of file EllipticFunction.cpp.

References GeographicLib::Math::pi(), and std::swap().

◆ RC()

Math::real GeographicLib::EllipticFunction::RC	(	real	x,
		real	y
	)

static

Degenerate symmetric integral of the first kind R_C.

Parameters

[in]	x
[in]	y

Returns: R_C(x, y) = R_F(x, y, y).

R_C is defined in https://dlmf.nist.gov/19.2.E17

\[ R_C(x, y) = \frac12 \int_0^\infty\frac1{\sqrt{t + x}(t + y)}\,dt, \]

where x ≥ 0 and y > 0.

Definition at line 81 of file EllipticFunction.cpp.

Referenced by Reset(), and RJ().

◆ RG() [1/2]

Math::real GeographicLib::EllipticFunction::RG	(	real	x,
		real	y,
		real	z
	)

static

Symmetric integral of the second kind R_G.

Parameters

[in]	x
[in]	y
[in]	z

Returns: R_G(x, y, z).

R_G is defined in Carlson, eq 1.5

\[ R_G(x, y, z) = \frac14 \int_0^\infty[(t + x) (t + y) (t + z)]^{-1/2} \biggl( \frac x{t + x} + \frac y{t + y} + \frac z{t + z} \biggr)t\,dt, \]

where at most one of arguments, x, y, z, can be zero and those arguments that are nonzero must be positive. See also https://dlmf.nist.gov/19.23.E6_5. 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.

Definition at line 96 of file EllipticFunction.cpp.

References RD(), RF(), and RG().

Referenced by Reset(), and RG().

◆ RG() [2/2]

Math::real GeographicLib::EllipticFunction::RG	(	real	x,
		real	y
	)

static

Complete symmetric integral of the second kind, R_G with one argument zero.

Parameters

[in]	x
[in]	y

Returns: R_G(x, y, 0).

The arguments x and y must be positive.

Definition at line 105 of file EllipticFunction.cpp.

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

◆ RJ()

Math::real GeographicLib::EllipticFunction::RJ	(	real	x,
		real	y,
		real	z,
		real	p
	)

static

Symmetric integral of the third kind R_J.

Parameters

[in]	x
[in]	y
[in]	z
[in]	p

Returns: R_J(x, y, z, p).

R_J is defined in https://dlmf.nist.gov/19.16.E2

\[ R_J(x, y, z, p) = \frac32 \int_0^\infty [(t + x) (t + y) (t + z)]^{-1/2} (t + p)^{-1}\, dt, \]

where p > 0, and x, y, z are nonnegative with at most one of them being 0.

Definition at line 128 of file EllipticFunction.cpp.

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

Referenced by G(), H(), Pi(), and Reset().

◆ RD()

Math::real GeographicLib::EllipticFunction::RD	(	real	x,
		real	y,
		real	z
	)

static

Degenerate symmetric integral of the third kind R_D.

Parameters

[in]	x
[in]	y
[in]	z

Returns: R_D(x, y, z) = R_J(x, y, z, z).

R_D is defined in https://dlmf.nist.gov/19.16.E5

\[ R_D(x, y, z) = \frac32 \int_0^\infty[(t + x) (t + y)]^{-1/2} (t + z)^{-3/2}\, dt, \]

where x, y, z are positive except that at most one of x and y can be 0.

Definition at line 182 of file EllipticFunction.cpp.

Referenced by D(), E(), Reset(), and RG().

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

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ EllipticFunction() [1/2]

◆ EllipticFunction() [2/2]

Member Function Documentation

◆ Reset() [1/2]

◆ Reset() [2/2]

◆ k2()

◆ kp2()

◆ alpha2()

◆ alphap2()

◆ K()

◆ E() [1/3]

◆ D() [1/3]

◆ KE()

◆ Pi() [1/3]

◆ G() [1/3]

◆ H() [1/3]

◆ F() [1/2]

◆ E() [2/3]

◆ Ed()

◆ Einv()

◆ Pi() [2/3]

◆ D() [2/3]

◆ G() [2/3]

◆ H() [2/3]

◆ F() [2/2]

◆ E() [3/3]

◆ Pi() [3/3]

◆ D() [3/3]

◆ G() [3/3]

◆ H() [3/3]

◆ deltaF()

◆ deltaE()

◆ deltaEinv()

◆ deltaPi()

◆ deltaD()

◆ deltaG()

◆ deltaH()

◆ sncndn()

◆ Delta()

◆ RF() [1/2]

◆ RF() [2/2]

◆ RC()

◆ RG() [1/2]

◆ RG() [2/2]

◆ RJ()

◆ RD()