Albers equal area conic projection. More...

#include <GeographicLib/AlbersEqualArea.hpp>

Public Member Functions
	AlbersEqualArea (real a, real f, real stdlat, real k0)

	AlbersEqualArea (real a, real f, real stdlat1, real stdlat2, real k1)

	AlbersEqualArea (real a, real f, real sinlat1, real coslat1, real sinlat2, real coslat2, real k1)

void	SetScale (real lat, real k=real(1))

void	Forward (real lon0, real lat, real lon, real &x, real &y, real &gamma, real &k) const

void	Reverse (real lon0, real x, real y, real &lat, real &lon, real &gamma, real &k) const

void	Forward (real lon0, real lat, real lon, real &x, real &y) const

void	Reverse (real lon0, real x, real y, real &lat, real &lon) const

Inspector functions
Math::real	EquatorialRadius () const

Math::real	Flattening () const

Math::real	OriginLatitude () const

Math::real	CentralScale () const

Static Public Member Functions
static const AlbersEqualArea &	CylindricalEqualArea ()

static const AlbersEqualArea &	AzimuthalEqualAreaNorth ()

static const AlbersEqualArea &	AzimuthalEqualAreaSouth ()

Friends
class	Ellipsoid

Detailed Description

Albers equal area conic projection.

Implementation taken from the report,

J. P. Snyder, Map Projections: A Working Manual, USGS Professional Paper 1395 (1987), pp. 101–102.

This is a implementation of the equations in Snyder except that divided differences will be [have been] used to transform the expressions into ones which may be evaluated accurately. [In this implementation, the projection correctly becomes the cylindrical equal area or the azimuthal equal area projection when the standard latitude is the equator or a pole.]

The ellipsoid parameters, the standard parallels, and the scale on the standard parallels are set in the constructor. Internally, the case with two standard parallels is converted into a single standard parallel, the latitude of minimum azimuthal scale, with an azimuthal scale specified on this parallel. This latitude is also used as the latitude of origin which is returned by AlbersEqualArea::OriginLatitude. The azimuthal scale on the latitude of origin is given by AlbersEqualArea::CentralScale. The case with two standard parallels at opposite poles is singular and is disallowed. The central meridian (which is a trivial shift of the longitude) is specified as the lon0 argument of the AlbersEqualArea::Forward and AlbersEqualArea::Reverse functions. AlbersEqualArea::Forward and AlbersEqualArea::Reverse also return the meridian convergence, γ, and azimuthal scale, k. A small square aligned with the cardinal directions is projected to a rectangle with dimensions k (in the E-W direction) and 1/k (in the N-S direction). The E-W sides of the rectangle are oriented γ degrees counter-clockwise from the x axis. There is no provision in this class for specifying a false easting or false northing or a different latitude of origin.

Example of use:

// Example of using the GeographicLib::AlbersEqualArea class
 
#include <iostream>
#include <exception>
#include <GeographicLib/AlbersEqualArea.hpp>
 
using namespace std;
using namespace GeographicLib;
 
int main() {
  try {
   const double
     a = Constants::WGS84_a(),
     f = Constants::WGS84_f(),
     lat1 = 40 + 58/60.0, lat2 = 39 + 56/60.0, // standard parallels
     k1 = 1,                                   // scale
     lon0 = -77 - 45/60.0;                     // Central meridian
   // Set up basic projection
   const AlbersEqualArea albers(a, f, lat1, lat2, k1);
   {
     // Sample conversion from geodetic to Albers Equal Area
     double lat = 39.95, lon = -75.17;    // Philadelphia
     double x, y;
     albers.Forward(lon0, lat, lon, x, y);
     cout << x << " " << y << "\n";
   }
   {
     // Sample conversion from Albers Equal Area grid to geodetic
     double x = 220e3, y = -53e3;
     double lat, lon;
     albers.Reverse(lon0, x, y, lat, lon);
     cout << lat << " " << lon << "\n";
   }
  }
  catch (const exception& e) {
    cerr << "Caught exception: " << e.what() << "\n";
    return 1;
  }
}

ConicProj is a command-line utility providing access to the functionality of LambertConformalConic and AlbersEqualArea.

Definition at line 60 of file AlbersEqualArea.hpp.

Constructor & Destructor Documentation

◆ AlbersEqualArea() [1/3]

GeographicLib::AlbersEqualArea::AlbersEqualArea	(	real	a,
		real	f,
		real	stdlat,
		real	k0
	)

Constructor with a single standard parallel.

Parameters

[in]	a	equatorial radius of ellipsoid (meters).
[in]	f	flattening of ellipsoid. Setting f = 0 gives a sphere. Negative f gives a prolate ellipsoid.
[in]	stdlat	standard parallel (degrees), the circle of tangency.
[in]	k0	azimuthal scale on the standard parallel.

Exceptions

GeographicErr	if a, (1 − f) a, or k0 is not positive.
GeographicErr	if stdlat is not in [−90°, 90°].

Definition at line 21 of file AlbersEqualArea.cpp.

References GeographicLib::Math::qd, and GeographicLib::Math::sincosd().

◆ AlbersEqualArea() [2/3]

GeographicLib::AlbersEqualArea::AlbersEqualArea	(	real	a,
		real	f,
		real	stdlat1,
		real	stdlat2,
		real	k1
	)

Constructor with two standard parallels.

Parameters

[in]	a	equatorial radius of ellipsoid (meters).
[in]	f	flattening of ellipsoid. Setting f = 0 gives a sphere. Negative f gives a prolate ellipsoid.
[in]	stdlat1	first standard parallel (degrees).
[in]	stdlat2	second standard parallel (degrees).
[in]	k1	azimuthal scale on the standard parallels.

Exceptions

GeographicErr	if a, (1 − f) a, or k1 is not positive.
GeographicErr	if stdlat1 or stdlat2 is not in [−90°, 90°], or if stdlat1 and stdlat2 are opposite poles.

Definition at line 50 of file AlbersEqualArea.cpp.

References GeographicLib::Math::qd, and GeographicLib::Math::sincosd().

◆ AlbersEqualArea() [3/3]

GeographicLib::AlbersEqualArea::AlbersEqualArea	(	real	a,
		real	f,
		real	sinlat1,
		real	coslat1,
		real	sinlat2,
		real	coslat2,
		real	k1
	)

Constructor with two standard parallels specified by sines and cosines.

Parameters

[in]	a	equatorial radius of ellipsoid (meters).
[in]	f	flattening of ellipsoid. Setting f = 0 gives a sphere. Negative f gives a prolate ellipsoid.
[in]	sinlat1	sine of first standard parallel.
[in]	coslat1	cosine of first standard parallel.
[in]	sinlat2	sine of second standard parallel.
[in]	coslat2	cosine of second standard parallel.
[in]	k1	azimuthal scale on the standard parallels.

Exceptions

GeographicErr	if a, (1 − f) a, or k1 is not positive.
GeographicErr	if stdlat1 or stdlat2 is not in [−90°, 90°], or if stdlat1 and stdlat2 are opposite poles.

This allows parallels close to the poles to be specified accurately. This routine computes the latitude of origin and the azimuthal scale at this latitude. If dlat = abs(lat2 − lat1) ≤ 160°, then the error in the latitude of origin is less than 4.5 × 10⁻¹⁴d;.

Definition at line 86 of file AlbersEqualArea.cpp.

References GeographicLib::Math::qd.

Member Function Documentation

◆ SetScale()

void GeographicLib::AlbersEqualArea::SetScale	(	real	lat,
		real	k = `real(1)`
	)

Set the azimuthal scale for the projection.

Parameters

[in]	lat	(degrees).
[in]	k	azimuthal scale at latitude lat (default 1).

Exceptions

GeographicErr	k is not positive.
GeographicErr	if lat is not in (−90°, 90°).

This allows a "latitude of conformality" to be specified.

Definition at line 542 of file AlbersEqualArea.cpp.

References Forward(), GeographicLib::Math::qd, and GeographicLib::Math::sq().

◆ Forward() [1/2]

void GeographicLib::AlbersEqualArea::Forward	(	real	lon0,
		real	lat,
		real	lon,
		real &	x,
		real &	y,
		real &	gamma,
		real &	k
	)		const

Forward projection, from geographic to Lambert conformal conic.

Parameters

[in]	lon0	central meridian longitude (degrees).
[in]	lat	latitude of point (degrees).
[in]	lon	longitude of point (degrees).
[out]	x	easting of point (meters).
[out]	y	northing of point (meters).
[out]	gamma	meridian convergence at point (degrees).
[out]	k	azimuthal scale of projection at point; the radial scale is the 1/k.

The latitude origin is given by AlbersEqualArea::LatitudeOrigin(). No false easting or northing is added and lat should be in the range [−90°, 90°]. The values of x and y returned for points which project to infinity (i.e., one or both of the poles) will be large but finite.

Definition at line 495 of file AlbersEqualArea.cpp.

References GeographicLib::Math::AngDiff(), GeographicLib::Math::degree(), GeographicLib::Math::LatFix(), GeographicLib::Math::sincosd(), and GeographicLib::Math::sq().

Referenced by main(), and SetScale().

◆ Reverse() [1/2]

void GeographicLib::AlbersEqualArea::Reverse	(	real	lon0,
		real	x,
		real	y,
		real &	lat,
		real &	lon,
		real &	gamma,
		real &	k
	)		const

Reverse projection, from Lambert conformal conic to geographic.

Parameters

[in]	lon0	central meridian longitude (degrees).
[in]	x	easting of point (meters).
[in]	y	northing of point (meters).
[out]	lat	latitude of point (degrees).
[out]	lon	longitude of point (degrees).
[out]	gamma	meridian convergence at point (degrees).
[out]	k	azimuthal scale of projection at point; the radial scale is the 1/k.

The latitude origin is given by AlbersEqualArea::LatitudeOrigin(). No false easting or northing is added. The value of lon returned is in the range [−180°, 180°]. The value of lat returned is in the range [−90°, 90°]. If the input point is outside the legal projected space the nearest pole is returned.

Definition at line 520 of file AlbersEqualArea.cpp.

References GeographicLib::Math::AngNormalize(), GeographicLib::Math::atand(), GeographicLib::Math::degree(), and GeographicLib::Math::sq().

Referenced by main().

◆ Forward() [2/2]

void GeographicLib::AlbersEqualArea::Forward	(	real	lon0,
		real	lat,
		real	lon,
		real &	x,
		real &	y
	)		const

inline

AlbersEqualArea::Forward without returning the convergence and scale.

Definition at line 244 of file AlbersEqualArea.hpp.

◆ Reverse() [2/2]

void GeographicLib::AlbersEqualArea::Reverse	(	real	lon0,
		real	x,
		real	y,
		real &	lat,
		real &	lon
	)		const

inline

AlbersEqualArea::Reverse without returning the convergence and scale.

Definition at line 254 of file AlbersEqualArea.hpp.

◆ EquatorialRadius()

Math::real GeographicLib::AlbersEqualArea::EquatorialRadius ( ) const

inline

Returns: a the equatorial radius of the ellipsoid (meters). This is the value used in the constructor.

Definition at line 267 of file AlbersEqualArea.hpp.

◆ Flattening()

Math::real GeographicLib::AlbersEqualArea::Flattening ( ) const

inline

Returns: f the flattening of the ellipsoid. This is the value used in the constructor.

Definition at line 273 of file AlbersEqualArea.hpp.

◆ OriginLatitude()

Math::real GeographicLib::AlbersEqualArea::OriginLatitude ( ) const

inline

Returns: latitude of the origin for the projection (degrees).

This is the latitude of minimum azimuthal scale and equals the stdlat in the 1-parallel constructor and lies between stdlat1 and stdlat2 in the 2-parallel constructors.

Definition at line 282 of file AlbersEqualArea.hpp.

◆ CentralScale()

Math::real GeographicLib::AlbersEqualArea::CentralScale ( ) const

inline

Returns: central scale for the projection. This is the azimuthal scale on the latitude of origin.

Definition at line 288 of file AlbersEqualArea.hpp.

◆ CylindricalEqualArea()

const AlbersEqualArea & GeographicLib::AlbersEqualArea::CylindricalEqualArea ( )

static

A global instantiation of AlbersEqualArea with the WGS84 ellipsoid, stdlat = 0, and k0 = 1. This degenerates to the cylindrical equal area projection.

Definition at line 286 of file AlbersEqualArea.cpp.

References GeographicLib::Constants::WGS84_a(), and GeographicLib::Constants::WGS84_f().

◆ AzimuthalEqualAreaNorth()

const AlbersEqualArea & GeographicLib::AlbersEqualArea::AzimuthalEqualAreaNorth ( )

static

A global instantiation of AlbersEqualArea with the WGS84 ellipsoid, stdlat = 90°, and k0 = 1. This degenerates to the Lambert azimuthal equal area projection.

Definition at line 293 of file AlbersEqualArea.cpp.

References GeographicLib::Constants::WGS84_a(), and GeographicLib::Constants::WGS84_f().

◆ AzimuthalEqualAreaSouth()

const AlbersEqualArea & GeographicLib::AlbersEqualArea::AzimuthalEqualAreaSouth ( )

static

A global instantiation of AlbersEqualArea with the WGS84 ellipsoid, stdlat = −90°, and k0 = 1. This degenerates to the Lambert azimuthal equal area projection.

Definition at line 300 of file AlbersEqualArea.cpp.

References GeographicLib::Constants::WGS84_a(), and GeographicLib::Constants::WGS84_f().

Friends And Related Function Documentation

◆ Ellipsoid

friend class Ellipsoid

friend

Definition at line 122 of file AlbersEqualArea.hpp.

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

Public Member Functions

Static Public Member Functions

Friends

Detailed Description

Constructor & Destructor Documentation

◆ AlbersEqualArea() [1/3]

◆ AlbersEqualArea() [2/3]

◆ AlbersEqualArea() [3/3]

Member Function Documentation

◆ SetScale()

◆ Forward() [1/2]

◆ Reverse() [1/2]

◆ Forward() [2/2]

◆ Reverse() [2/2]

◆ EquatorialRadius()

◆ Flattening()

◆ OriginLatitude()

◆ CentralScale()

◆ CylindricalEqualArea()

◆ AzimuthalEqualAreaNorth()

◆ AzimuthalEqualAreaSouth()

Friends And Related Function Documentation

◆ Ellipsoid