GeographicLib: GeographicLib::AuxLatitude Class Reference (original) (raw)

Conversions between auxiliary latitudes. More...

#include <[GeographicLib/AuxLatitude.hpp](AuxLatitude%5F8hpp%5Fsource.html)>

Conversions between auxiliary latitudes.

This class is an implementation of the methods described in

• C. F. F. Karney, On auxiliary latitudes, Survey Review 56(395), 165–180 (2024); preprint arXiv:2212.05818.

The provides accurate conversions between geographic (phi, φ), parametric (beta, β), geocentric (theta, θ), rectifying (mu, μ), conformal (chi, χ), and authalic (xi, ξ) latitudes for an ellipsoid of revolution. A latitude is represented by the class AuxAngle in order to maintain precision close to the poles.

The class implements two methods for the conversion:

• Direct evaluation of the defining equations, the exact method. These equations are formulated so as to preserve relative accuracy of the tangent of the latitude, ensuring high accuracy near the equator and the poles. Newton's method is used for those conversions that can't be expressed in closed form.
• Expansions in powers of n, the third flattening, the series method. This delivers full accuracy for abs(f) ≤ 1/150. Here, f is the flattening of the ellipsoid.

The series method is the preferred method of conversion for any conversion involving μ, χ, or ξ, with abs(f) ≤ 1/150. The equations for the conversions between φ, β, and θ are sufficiently simple that the exact method should be used for such conversions and also for conversions with with abs(f) > 1/150.

Example of use:

#include

#include

#include

int main(int argc, const char* const argv[]) {

try {

if (argc != 3) {

std::cerr << "Usage: example-AuxLatitude \n";

return 1;

}

double n = GeographicLib::Utility::fract(std::string(argv[1]));

int auxin = GeographicLib::Utility::val(std::string(argv[2]));

double a = 1+n, b = 1-n;

latitude aux(latitude::axes(a, b));

bool series = false;

std::cout << std::setprecision(9) << std::fixed;

int m = 1;

for (int l = 0; l < 90*m; ++l) {

angle phi(angle::degrees((l+0.5)/m));

for (int auxout = 0; auxout < latitude::AUXNUMBER; ++auxout) {

angle eta = aux.Convert(auxin, auxout, phi, series);

std::cout << (auxout ? " " : "") << eta.degrees();

}

std::cout << "\n";

}

}

catch (const std::exception& e) {

std::cerr << "Caught exception: " << e.what() << "\n";

return 1;

}

}

Header for the GeographicLib::AuxLatitude class.

int main(int argc, const char *const argv[])

Header for GeographicLib::Utility class.

An accurate representation of angles.

Conversions between auxiliary latitudes.

Definition at line 75 of file AuxLatitude.hpp.

◆ aux

The floating-point type for real numbers. This just connects to the template parameters for the class. The different auxiliary latitudes.

Definition at line 86 of file AuxLatitude.hpp.

Constructor

Parameters

Exceptions

Note

the constructor does not precompute the coefficients for the Fourier series for the series conversions. These are computed and saved when first needed.

Definition at line 25 of file AuxLatitude.cpp.

References AUXNUMBER, and Lmax.

◆ axes()

Construct and return an AuxLatitude object specified in terms of the semi-axes

Parameters

Exceptions

This allows a new AuxAngle to be initialized as an angle in radians with

static AuxLatitude axes(real a, real b)

Definition at line 195 of file AuxLatitude.hpp.

◆ Convert() [1/2]

Convert between any two auxiliary latitudes specified as AuxAngle.

Parameters

Returns

the output auxiliary latitude eta as an AuxAngle.

With exact = false, the Fourier coefficients for a specific auxin and auxout are computed and saved on the first call; the saved coefficients are used on subsequent calls. The series method is accurate for abs(f) ≤ 1/150; for other f, the exact method should be used.

Definition at line 297 of file AuxLatitude.cpp.

References Clenshaw(), FromAuxiliary(), Lmax, GeographicLib::AuxAngle::NaN(), GeographicLib::AuxAngle::normalized(), GeographicLib::AuxAngle::radians(), ToAuxiliary(), GeographicLib::AuxAngle::x(), and GeographicLib::AuxAngle::y().

Referenced by Convert(), GeographicLib::Rhumb::GenInverse(), GeographicLib::RhumbLine::GenPosition(), and main().

◆ Convert() [2/2]

Convert between any two auxiliary latitudes specified in degrees.

Parameters

Returns

the output auxiliary latitude eta in degrees.

With exact = false, the Fourier coefficients for a specific auxin and auxout are computed and saved on the first call; the saved coefficients are used on subsequent calls. The series method is accurate for abs(f) ≤ 1/150; for other f, the exact method should be used.

Definition at line 318 of file AuxLatitude.cpp.

References Convert(), GeographicLib::AuxAngle::degrees(), and GeographicLib::Math::td.

◆ ToAuxiliary()

Convert geographic latitude to an auxiliary latitude eta.

Parameters

Returns

the auxiliary latitude eta.

This uses the exact equations.

Definition at line 220 of file AuxLatitude.cpp.

References AUTHALIC, Authalic(), CONFORMAL, Conformal(), GEOCENTRIC, Geocentric(), GEOGRAPHIC, GeographicLib::AuxAngle::NaN(), PARAMETRIC, Parametric(), RECTIFYING, and Rectifying().

Referenced by Convert(), and FromAuxiliary().

◆ FromAuxiliary()

Convert an auxiliary latitude zeta to geographic latitude.

Parameters

Returns

the geographic latitude phi.

This uses the exact equations.

Definition at line 236 of file AuxLatitude.cpp.

References AUTHALIC, CONFORMAL, GeographicLib::AuxAngle::copyquadrant(), GEOCENTRIC, GEOGRAPHIC, GeographicLib::AuxAngle::NaN(), PARAMETRIC, RECTIFYING, GeographicLib::AuxAngle::tan(), ToAuxiliary(), GeographicLib::AuxAngle::x(), and GeographicLib::AuxAngle::y().

Referenced by Convert().

◆ RectifyingRadius()

◆ AuthalicRadiusSquared()

◆ EquatorialRadius()

Returns

a the equatorial radius of the ellipsoid (meters).

Definition at line 284 of file AuxLatitude.hpp.

◆ PolarSemiAxis()

Returns

b the polar semi-axis of the ellipsoid (meters).

Definition at line 288 of file AuxLatitude.hpp.

◆ Flattening()

Returns

f, the flattening of the ellipsoid.

Definition at line 292 of file AuxLatitude.hpp.

◆ Clenshaw()

Use Clenshaw to sum a Fouier series.

Parameters

Returns

the Clenshaw sum.

The result returned is \( \sum_0^{K-1} c_k \sin (2k+2)\zeta \), if sinp is true; with sinp false, replace sin by cos.

Referenced by Convert().

◆ WGS84()

◆ Parametric()

◆ Geocentric()

◆ Rectifying()

Convert geographic latitude to rectifying latitude

Parameters

Returns

mu, the rectifying latitude.

Definition at line 94 of file AuxLatitude.cpp.

References GeographicLib::AuxAngle::normalized(), Parametric(), GeographicLib::Math::pi(), GeographicLib::EllipticFunction::RD(), GeographicLib::EllipticFunction::RF(), GeographicLib::Math::sq(), std::swap(), GeographicLib::AuxAngle::tan(), GeographicLib::AuxAngle::x(), and GeographicLib::AuxAngle::y().

Referenced by GeographicLib::DAuxLatitude::DRectifying(), and ToAuxiliary().

◆ Conformal()

◆ Authalic()

◆ Lmax

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

• AuxLatitude.hpp
• AuxLatitude.cpp