.NET wrapper for GeographicLib::EllipticFunction. More...

#include <EllipticFunction.h>

Public Member Functions
Constructor
	EllipticFunction (double k2, double alpha2)

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

	~EllipticFunction ()

void	Reset (double k2, double alpha2)

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

Complete elliptic integrals.
double	K ()

double	E ()

double	D ()

double	KE ()

double	Pi ()

double	G ()

double	H ()

Incomplete elliptic integrals.
double	F (double phi)

double	E (double phi)

double	Ed (double ang)

double	Einv (double x)

double	Pi (double phi)

double	D (double phi)

double	G (double phi)

double	H (double phi)

Incomplete integrals in terms of Jacobi elliptic functions.
double	F (double sn, double cn, double dn)

double	E (double sn, double cn, double dn)

double	Pi (double sn, double cn, double dn)

double	D (double sn, double cn, double dn)

double	G (double sn, double cn, double dn)

double	H (double sn, double cn, double dn)

Periodic versions of incomplete elliptic integrals.
double	deltaF (double sn, double cn, double dn)

double	deltaE (double sn, double cn, double dn)

double	deltaEinv (double stau, double ctau)

double	deltaPi (double sn, double cn, double dn)

double	deltaD (double sn, double cn, double dn)

double	deltaG (double sn, double cn, double dn)

double	deltaH (double sn, double cn, double dn)

Elliptic functions.
void	sncndn (double x, [System::Runtime::InteropServices::Out] double%sn, [System::Runtime::InteropServices::Out] double%cn, [System::Runtime::InteropServices::Out] double%dn)

double	Delta (double sn, double cn)

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

static double	RF (double x, double y)

static double	RC (double x, double y)

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

static double	RG (double x, double y)

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

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

Public Attributes
Inspector functions.
property double	k2 { double get()

property double	kp2 { double get()

property double	alpha2 { double get()

property double	alphap2 { double get()

Private Member Functions
	!EllipticFunction ()

Private Attributes
GeographicLib::EllipticFunction *	m_pEllipticFunction

Detailed Description

.NET wrapper for GeographicLib::EllipticFunction.

This class allows .NET applications to access GeographicLib::EllipticFunction.

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

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

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)

with the additional optimizations given in http://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 http://dlmf.nist.gov/19 and http://dlmf.nist.gov/22

C# Example:

using System;
using NETGeographicLib;
namespace example_EllipticFunction
{
    class Program
    {
        static void Main(string[] args)
        {
            try {
                EllipticFunction ell = new EllipticFunction(0.1, 1.0);  // parameter m = 0.1
                // See Abramowitz and Stegun, table 17.1
                Console.WriteLine( String.Format( "{0} {1}", ell.K(), ell.E()));
                double phi = 20 * Math.Acos(-1.0) / 180.0;;
                // See Abramowitz and Stegun, table 17.6 with
                // alpha = asin(sqrt(m)) = 18.43 deg and phi = 20 deg
                Console.WriteLine( String.Format("{0} {1}", ell.E(phi),
                        ell.E(Math.Sin(phi), Math.Cos(phi),
                            Math.Sqrt(1 - ell.k2 * Math.Sin(phi) * Math.Sin(phi))) ) );
                // See Carlson 1995, Sec 3.
                Console.WriteLine(String.Format("RF(1,2,0)      = {0}", EllipticFunction.RF(1,2)));
                Console.WriteLine(String.Format("RF(2,3,4)      = {0}", EllipticFunction.RF(2,3,4)));
                Console.WriteLine(String.Format("RC(0,1/4)      = {0}", EllipticFunction.RC(0,0.25)));
                Console.WriteLine(String.Format("RC(9/4,2)      = {0}", EllipticFunction.RC(2.25,2)));
                Console.WriteLine(String.Format("RC(1/4,-2)     = {0}", EllipticFunction.RC(0.25,-2)));
                Console.WriteLine(String.Format("RJ(0,1,2,3)    = {0}", EllipticFunction.RJ(0,1,2,3)));
                Console.WriteLine(String.Format("RJ(2,3,4,5)    = {0}", EllipticFunction.RJ(2,3,4,5)));
                Console.WriteLine(String.Format("RD(0,2,1)      = {0}", EllipticFunction.RD(0,2,1)));
                Console.WriteLine(String.Format("RD(2,3,4)      = {0}", EllipticFunction.RD(2,3,4)));
                Console.WriteLine(String.Format("RG(0,16,16)    = {0}", EllipticFunction.RG(16,16)));
                Console.WriteLine(String.Format("RG(2,3,4)      = {0}", EllipticFunction.RG(2,3,4)));
                Console.WriteLine(String.Format("RG(0,0.0796,4) = {0}", EllipticFunction.RG(0.0796, 4)));
            }
            catch (GeographicErr e) {
                Console.WriteLine( String.Format( "Caught exception: {0}", e.Message ) );
            }
        }
    }
}

Managed C++ Example:

// 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";
  }
}

Visual Basic Example:

Imports NETGeographicLib
Module example_EllipticFunction
    Sub Main()
        Try
            Dim ell As EllipticFunction = New EllipticFunction(0.1, 1.0)
            ' See Abramowitz and Stegun, table 17.1
            Console.WriteLine(String.Format("{0} {1}", ell.K(), ell.E()))
            Dim phi As Double = 20 * Math.Acos(-1.0) / 180.0
            ' See Abramowitz and Stegun, table 17.6 with
            ' alpha = asin(sqrt(m)) = 18.43 deg and phi = 20 deg
            Console.WriteLine(String.Format("{0} {1}", ell.E(phi),
                    ell.E(Math.Sin(phi), Math.Cos(phi),
                        Math.Sqrt(1 - ell.k2 * Math.Sin(phi) * Math.Sin(phi)))))
            ' See Carlson 1995, Sec 3.
            Console.WriteLine(String.Format("RF(1,2,0)      = {0}", EllipticFunction.RF(1, 2)))
            Console.WriteLine(String.Format("RF(2,3,4)      = {0}", EllipticFunction.RF(2, 3, 4)))
            Console.WriteLine(String.Format("RC(0,1/4)      = {0}", EllipticFunction.RC(0, 0.25)))
            Console.WriteLine(String.Format("RC(9/4,2)      = {0}", EllipticFunction.RC(2.25, 2)))
            Console.WriteLine(String.Format("RC(1/4,-2)     = {0}", EllipticFunction.RC(0.25, -2)))
            Console.WriteLine(String.Format("RJ(0,1,2,3)    = {0}", EllipticFunction.RJ(0, 1, 2, 3)))
            Console.WriteLine(String.Format("RJ(2,3,4,5)    = {0}", EllipticFunction.RJ(2, 3, 4, 5)))
            Console.WriteLine(String.Format("RD(0,2,1)      = {0}", EllipticFunction.RD(0, 2, 1)))
            Console.WriteLine(String.Format("RD(2,3,4)      = {0}", EllipticFunction.RD(2, 3, 4)))
            Console.WriteLine(String.Format("RG(0,16,16)    = {0}", EllipticFunction.RG(16, 16)))
            Console.WriteLine(String.Format("RG(2,3,4)      = {0}", EllipticFunction.RG(2, 3, 4)))
            Console.WriteLine(String.Format("RG(0,0.0796,4) = {0}", EllipticFunction.RG(0.0796, 4)))
        Catch ex As GeographicErr
            Console.WriteLine(String.Format("Caught exception: {0}", ex.Message))
        End Try
    End Sub
End Module

INTERFACE DIFFERENCES:
The k2, kp2, alpha2, and alphap2 functions are implemented as properties.

Definition at line 69 of file EllipticFunction.h.

Constructor & Destructor Documentation

EllipticFunction::!EllipticFunction ( )

private

Definition at line 47 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

EllipticFunction::EllipticFunction	(	double	k2,
		double	alpha2
	)

Constructor specifying the modulus and parameter.

Parameters

[in]	k2	the square of the modulus k². k² must lie in (-∞, 1). (No checking is done.)
[in]	alpha2	the parameter α². α² must lie in (-∞, 1). (No checking is done.)

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 21 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

EllipticFunction::EllipticFunction	(	double	k2,
		double	alpha2,
		double	kp2,
		double	alphap2
	)

Constructor specifying the modulus and parameter and their complements.

Parameters

[in]	k2	the square of the modulus k². k² must lie in (-∞, 1). (No checking is done.)
[in]	alpha2	the parameter α². α² must lie in (-∞, 1). (No checking is done.)
[in]	kp2	the complementary modulus squared k'² = 1 − k².
[in]	alphap2	the complementary parameter α'² = 1 − α².

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 34 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

NETGeographicLib::EllipticFunction::~EllipticFunction ( )

inline

Destructor calls the finalizer.

Definition at line 121 of file EllipticFunction.h.

Member Function Documentation

double EllipticFunction::D ( )

Jahnke's complete integral.

Returns: D(k).

D(k) is defined in http://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 81 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::D ( double phi )

Jahnke's incomplete elliptic integral.

Parameters

[in] phi

Returns: D(φ, k).

D(φ, k) is defined in http://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 141 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::D	(	double	sn,
		double	cn,
		double	dn
	)

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 177 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::Delta	(	double	sn,
		double	cn
	)

The Δ amplitude function.

Parameters

[in]	sn	sinφ
[in]	cn	cosφ

Returns: Δ = sqrt(1 − k² sin²φ)

Definition at line 250 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::deltaD	(	double	sn,
		double	cn,
		double	dn
	)

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 219 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::deltaE	(	double	sn,
		double	cn,
		double	dn
	)

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 201 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::deltaEinv	(	double	stau,
		double	ctau
	)

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 207 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::deltaF	(	double	sn,
		double	cn,
		double	dn
	)

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 195 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::deltaG	(	double	sn,
		double	cn,
		double	dn
	)

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 225 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::deltaH	(	double	sn,
		double	cn,
		double	dn
	)

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 231 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::deltaPi	(	double	sn,
		double	cn,
		double	dn
	)

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, k) / (2 Π(k)) - φ

Definition at line 213 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::E ( )

The complete integral of the second kind.

Returns: E(k)

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

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

Definition at line 75 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::E ( double phi )

The incomplete integral of the second kind.

Parameters

[in] phi

Returns: E(φ, k).

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

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

Definition at line 117 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::E	(	double	sn,
		double	cn,
		double	dn
	)

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 165 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::Ed ( double ang )

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 123 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::Einv ( double x )

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 129 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::F ( double phi )

The incomplete integral of the first kind.

Parameters

[in] phi

Returns: F(φ, k).

F(φ, k) is defined in http://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 111 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::F	(	double	sn,
		double	cn,
		double	dn
	)

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 159 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::G ( )

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 99 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::G ( double phi )

Legendre's geodesic longitude integral.

Parameters

[in] phi

Returns: G(φ, α², k).

G(φ, α², k) is defined by

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

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 geodellip for the expression for the longitude in terms of this function.

Definition at line 147 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::G	(	double	sn,
		double	cn,
		double	dn
	)

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 183 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::H ( )

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 105 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::H ( double phi )

Cayley's geodesic longitude difference integral.

Parameters

[in] phi

Returns: H(φ, α², k).

H(φ, α², k) is defined by

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

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 geodellip for the expression for the longitude in terms of this function.

Definition at line 153 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::H	(	double	sn,
		double	cn,
		double	dn
	)

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 189 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::K ( )

The complete integral of the first kind.

Returns: K(k).

K(k) is defined in http://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 69 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::KE ( )

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

Returns: K(k) − E(k).

Definition at line 87 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::Pi ( )

The complete integral of the third kind.

Returns: Π(α², k)

Π(α², k) is defined in http://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 93 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::Pi ( double phi )

The incomplete integral of the third kind.

Parameters

[in] phi

Returns: Π(φ, α², k).

Π(φ, α², k) is defined in http://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 135 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::Pi	(	double	sn,
		double	cn,
		double	dn
	)

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 171 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::RC	(	double	x,
		double	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 http://dlmf.nist.gov/19.2.E17

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

Definition at line 268 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::RD	(	double	x,
		double	y,
		double	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 http://dlmf.nist.gov/19.16.E5

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

Definition at line 292 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

void EllipticFunction::Reset	(	double	k2,
		double	alpha2
	)

Reset the modulus and parameter.

Parameters

[in]	k2	the new value of square of the modulus k² which must lie in (-∞, 1). (No checking is done.)
[in]	alpha2	the new value of parameter α². α² must lie in (-∞, 1). (No checking is done.)

Definition at line 57 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

void EllipticFunction::Reset	(	double	k2,
		double	alpha2,
		double	kp2,
		double	alphap2
	)

Reset the modulus and parameter supplying also their complements.

Parameters

[in]	k2	the square of the modulus k². k² must lie in (-∞, 1). (No checking is done.)
[in]	alpha2	the parameter α². α² must lie in (-∞, 1). (No checking is done.)
[in]	kp2	the complementary modulus squared k'² = 1 − k².
[in]	alphap2	the complementary parameter α'² = 1 − α².

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 63 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::RF	(	double	x,
		double	y,
		double	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 http://dlmf.nist.gov/19.16.E1

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

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 256 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::RF	(	double	x,
		double	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)

Definition at line 262 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::RG	(	double	x,
		double	y,
		double	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

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

See also http://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.

Definition at line 274 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::RG	(	double	x,
		double	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)

Definition at line 280 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

double EllipticFunction::RJ	(	double	x,
		double	y,
		double	z,
		double	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 http://dlmf.nist.gov/19.16.E2

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

Definition at line 286 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

void EllipticFunction::sncndn	(	double	x,
		[System::Runtime::InteropServices::Out] double%	sn,
		[System::Runtime::InteropServices::Out] double%	cn,
		[System::Runtime::InteropServices::Out] double%	dn
	)

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 237 of file dotnet/NETGeographicLib/EllipticFunction.cpp.

Member Data Documentation

property double NETGeographicLib::EllipticFunction::alpha2 { double get()

Returns: the parameter α².

Definition at line 174 of file EllipticFunction.h.

property double NETGeographicLib::EllipticFunction::alphap2 { double get()

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

Definition at line 180 of file EllipticFunction.h.

property double NETGeographicLib::EllipticFunction::k2 { double get()

Returns: the square of the modulus k².

Definition at line 163 of file EllipticFunction.h.

property double NETGeographicLib::EllipticFunction::kp2 { double get()

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

Definition at line 169 of file EllipticFunction.h.

GeographicLib::EllipticFunction* NETGeographicLib::EllipticFunction::m_pEllipticFunction

private

Definition at line 73 of file EllipticFunction.h.

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

Public Member Functions

Static Public Member Functions

Public Attributes

Private Member Functions

Private Attributes

Detailed Description

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation