An accurate representation of angles. More...

#include <GeographicLib/AuxAngle.hpp>

Public Member Functions
	AuxAngle (real y=0, real x=1)

Math::real	y () const

Math::real	x () const

Math::real &	y ()

Math::real &	x ()

Math::real	degrees () const

Math::real	radians () const

Math::real	lam () const

Math::real	lamd () const

Math::real	tan () const

AuxAngle	normalized () const

void	normalize ()

AuxAngle	copyquadrant (const AuxAngle &p) const

AuxAngle &	operator+= (const AuxAngle &p)

Static Public Member Functions
static AuxAngle	degrees (real d)

static AuxAngle	radians (real r)

static AuxAngle	lam (real psi)

static AuxAngle	lamd (real psid)

static AuxAngle	NaN ()

Detailed Description

An accurate representation of angles.

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.

An angle is represented be the y and x coordinates of a point in the 2d plane. The two coordinates are proportional to the sine and cosine of the angle. This allows angles close to the cardinal points to be represented accurately. It also saves on unnecessary recomputations of trigonometric functions of the angle. Only angles in [−180°, 180°] can be represented. (A possible extension would be to keep count of the number of turns.)

Example of use:

// Example of using the GeographicLib::AuxAngle class.
 
#include <iostream>
#include <iomanip>
#include <exception>
#include <GeographicLib/Utility.hpp>
#include <GeographicLib/AuxAngle.hpp>
 
int main(int argc, const char* const argv[]) {
  try {
    typedef GeographicLib::AuxAngle angle;
    // Print table of parametric latitudes for f = 0.5
    double f = 0.5;
    std::cout << std::fixed << std::setprecision(4);
    for (int d = 0; d <= 90; d+=10) {
      angle phi(angle::degrees(d)), beta(phi);
      beta.y() *= (1 - f);
      std::cout << d << " " << beta.degrees() << "\n";
    }
  }
  catch (const std::exception& e) {
    std::cerr << "Caught exception: " << e.what() << "\n";
    return 1;
  }
}

Definition at line 47 of file AuxAngle.hpp.

Constructor & Destructor Documentation

◆ AuxAngle()

GeographicLib::AuxAngle::AuxAngle	(	real	y = `0`,
		real	x = `1`
	)

inlineexplicit

The constructor.

Parameters

[in]	y	the y coordinate.
[in]	x	the x coordinate.

Note: the y coordinate is specified first.

Warning: either x or y can be infinite, but not both.

The defaults (x = 1 and y = 0) are such that

no arguments gives an angle of 0;
1 argument specifies the tangent of the angle.

Definition at line 65 of file AuxAngle.hpp.

Member Function Documentation

◆ y() [1/2]

Math::real GeographicLib::AuxAngle::y ( ) const

inline

Returns: the y component. This is the sine of the angle if the AuxAngle has been normalized.

Definition at line 70 of file AuxAngle.hpp.

Referenced by GeographicLib::AuxLatitude::Authalic(), GeographicLib::Ellipsoid::CircleHeight(), GeographicLib::AuxLatitude::Convert(), copyquadrant(), GeographicLib::DAuxLatitude::DConvert(), degrees(), GeographicLib::AuxLatitude::FromAuxiliary(), GeographicLib::AuxLatitude::Geocentric(), normalized(), GeographicLib::AuxLatitude::Parametric(), and GeographicLib::AuxLatitude::Rectifying().

◆ x() [1/2]

Math::real GeographicLib::AuxAngle::x ( ) const

inline

Returns: the x component. This is the cosine of the angle if the AuxAngle has been normalized.

Definition at line 75 of file AuxAngle.hpp.

Referenced by GeographicLib::AuxLatitude::Authalic(), GeographicLib::Ellipsoid::CircleRadius(), GeographicLib::AuxLatitude::Conformal(), GeographicLib::AuxLatitude::Convert(), copyquadrant(), GeographicLib::DAuxLatitude::DConvert(), degrees(), GeographicLib::AuxLatitude::FromAuxiliary(), GeographicLib::AuxLatitude::Geocentric(), normalized(), operator+=(), GeographicLib::AuxLatitude::Parametric(), and GeographicLib::AuxLatitude::Rectifying().

◆ y() [2/2]

Math::real & GeographicLib::AuxAngle::y ( )

inline

Returns: a reference to the y component. This allows this component to be altered.

Definition at line 80 of file AuxAngle.hpp.

◆ x() [2/2]

Math::real & GeographicLib::AuxAngle::x ( )

inline

Returns: a reference to the x component. This allows this component to be altered.

Definition at line 85 of file AuxAngle.hpp.

◆ degrees() [1/2]

Math::real GeographicLib::AuxAngle::degrees ( ) const

inline

Returns: the AuxAngle converted to the conventional angle measured in degrees.

Definition at line 224 of file AuxAngle.hpp.

References GeographicLib::Math::atan2d().

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

◆ radians() [1/2]

Math::real GeographicLib::AuxAngle::radians ( ) const

inline

Returns: the AuxAngle converted to the conventional angle measured in radians.

Definition at line 228 of file AuxAngle.hpp.

Referenced by GeographicLib::AuxLatitude::Convert(), GeographicLib::DAuxLatitude::DConvert(), GeographicLib::DAuxLatitude::DRectifying(), and GeographicLib::Rhumb::GenInverse().

◆ lam() [1/2]

Math::real GeographicLib::AuxAngle::lam ( ) const

inline

Returns: the lambertian of the AuxAngle.

Note: the lambertian of an angle χ is lam(χ) = asinh(tan(χ)).

Definition at line 232 of file AuxAngle.hpp.

References tan().

Referenced by GeographicLib::Rhumb::GenInverse().

◆ lamd() [1/2]

Math::real GeographicLib::AuxAngle::lamd ( ) const

inline

Returns: the lambertian of the AuxAngle in degrees.

Note: the lambertian of an angle χ is lam(χ) = asinh(tan(χ)).

Definition at line 236 of file AuxAngle.hpp.

References GeographicLib::Math::degree(), and tan().

◆ tan()

Math::real GeographicLib::AuxAngle::tan ( ) const

inline

Returns: the tangent of the angle.

Definition at line 113 of file AuxAngle.hpp.

Referenced by GeographicLib::AuxLatitude::Authalic(), GeographicLib::AuxLatitude::Conformal(), GeographicLib::DAuxLatitude::DIsometric(), GeographicLib::DAuxLatitude::DParametric(), GeographicLib::DAuxLatitude::DRectifying(), GeographicLib::AuxLatitude::FromAuxiliary(), GeographicLib::Rhumb::GenInverse(), GeographicLib::RhumbLine::GenPosition(), lam(), lamd(), normalized(), operator+=(), and GeographicLib::AuxLatitude::Rectifying().

◆ normalized()

AuxAngle GeographicLib::AuxAngle::normalized ( ) const

Returns: a new normalized AuxAngle with the point lying on the unit circle and the y and x components are equal to the sine and cosine of the angle.

Definition at line 28 of file AuxAngle.cpp.

References NaN(), tan(), x(), and y().

Referenced by GeographicLib::AuxLatitude::Authalic(), GeographicLib::AuxLatitude::Conformal(), GeographicLib::AuxLatitude::Convert(), GeographicLib::DAuxLatitude::DConvert(), and GeographicLib::AuxLatitude::Rectifying().