Albers equal area conic projection. More...

#include <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	Forward (real lon0, real lat, real lon, real &x, real &y, 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, real &gamma, real &k) const

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

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

Inspector functions
Math::real	MajorRadius () const

Math::real	Flattening () const

Math::real	OriginLatitude () const

Math::real	CentralScale () const

Static Public Member Functions
static const AlbersEqualArea &	AzimuthalEqualAreaNorth ()

static const AlbersEqualArea &	AzimuthalEqualAreaSouth ()

static const AlbersEqualArea &	CylindricalEqualArea ()

Private Types
typedef Math::real	real

Private Member Functions
real	atanhee (real x) const

real	Datanhee (real x, real y) const

real	DDatanhee (real x, real y) const

void	Init (real sphi1, real cphi1, real sphi2, real cphi2, real k1)

real	tphif (real txi) const

real	txif (real tphi) const

Static Private Member Functions
static real	atanhxm1 (real x)

static real	Dsn (real x, real y, real sx, real sy)

static real	hyp (real x)

Private Attributes
real	_a

real	_e

real	_e2

real	_e2m

real	_f

real	_fm

real	_k0

real	_k2

real	_lat0

real	_m02

real	_n0

real	_nrho0

real	_qx

real	_qZ

real	_scxi0

real	_sign

real	_sxi0

real	_txi0

real	eps_

real	epsx2_

real	epsx_

real	tol0_

real	tol_

Static Private Attributes
static const int	numit0_ = 20

static const int	numit_ = 5

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.

Member Typedef Documentation

typedef Math::real GeographicLib::AlbersEqualArea::real

private

Definition at line 62 of file AlbersEqualArea.hpp.

Constructor & Destructor Documentation

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 src/AlbersEqualArea.cpp.

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 49 of file src/AlbersEqualArea.cpp.

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 81 of file src/AlbersEqualArea.cpp.

Member Function Documentation

real GeographicLib::AlbersEqualArea::atanhee ( real x ) const

inlineprivate

Definition at line 73 of file AlbersEqualArea.hpp.

Math::real GeographicLib::AlbersEqualArea::atanhxm1 ( real x )

staticprivate

Definition at line 348 of file src/AlbersEqualArea.cpp.

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 284 of file src/AlbersEqualArea.cpp.

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 291 of file src/AlbersEqualArea.cpp.

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 285 of file AlbersEqualArea.hpp.

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 277 of file src/AlbersEqualArea.cpp.

real GeographicLib::AlbersEqualArea::Datanhee	(	real	x,
		real	y
	)		const

inlineprivate

Definition at line 109 of file AlbersEqualArea.hpp.

Math::real GeographicLib::AlbersEqualArea::DDatanhee	(	real	x,
		real	y
	)		const

private

Definition at line 366 of file src/AlbersEqualArea.cpp.

static real GeographicLib::AlbersEqualArea::Dsn	(	real	x,
		real	y,
		real	sx,
		real	sy
	)

inlinestaticprivate

Definition at line 102 of file AlbersEqualArea.hpp.

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 270 of file AlbersEqualArea.hpp.

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 386 of file src/AlbersEqualArea.cpp.

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 241 of file AlbersEqualArea.hpp.

static real GeographicLib::AlbersEqualArea::hyp ( real x )

inlinestaticprivate

Definition at line 69 of file AlbersEqualArea.hpp.

void GeographicLib::AlbersEqualArea::Init	(	real	sphi1,
		real	cphi1,
		real	sphi2,
		real	cphi2,
		real	k1
	)

private

Definition at line 119 of file src/AlbersEqualArea.cpp.

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

inline

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

Definition at line 264 of file AlbersEqualArea.hpp.

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 279 of file AlbersEqualArea.hpp.

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 411 of file src/AlbersEqualArea.cpp.

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 251 of file AlbersEqualArea.hpp.

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 433 of file src/AlbersEqualArea.cpp.

Math::real GeographicLib::AlbersEqualArea::tphif ( real txi ) const

private

Definition at line 325 of file src/AlbersEqualArea.cpp.

Math::real GeographicLib::AlbersEqualArea::txif ( real tphi ) const

private

Definition at line 298 of file src/AlbersEqualArea.cpp.

Friends And Related Function Documentation

friend class Ellipsoid

friend

Definition at line 119 of file AlbersEqualArea.hpp.

Member Data Documentation

real GeographicLib::AlbersEqualArea::_a

private

Definition at line 64 of file AlbersEqualArea.hpp.

real GeographicLib::AlbersEqualArea::_e

private

Definition at line 64 of file AlbersEqualArea.hpp.

real GeographicLib::AlbersEqualArea::_e2

private

Definition at line 64 of file AlbersEqualArea.hpp.

real GeographicLib::AlbersEqualArea::_e2m

private

Definition at line 64 of file AlbersEqualArea.hpp.

real GeographicLib::AlbersEqualArea::_f

private

Definition at line 64 of file AlbersEqualArea.hpp.

real GeographicLib::AlbersEqualArea::_fm

private

Definition at line 64 of file AlbersEqualArea.hpp.

real GeographicLib::AlbersEqualArea::_k0

private

Definition at line 65 of file AlbersEqualArea.hpp.

real GeographicLib::AlbersEqualArea::_k2

private

Definition at line 66 of file AlbersEqualArea.hpp.

real GeographicLib::AlbersEqualArea::_lat0

private

Definition at line 65 of file AlbersEqualArea.hpp.

real GeographicLib::AlbersEqualArea::_m02

private

Definition at line 66 of file AlbersEqualArea.hpp.

real GeographicLib::AlbersEqualArea::_n0

private

Definition at line 66 of file AlbersEqualArea.hpp.

real GeographicLib::AlbersEqualArea::_nrho0

private

Definition at line 66 of file AlbersEqualArea.hpp.

real GeographicLib::AlbersEqualArea::_qx

private

Definition at line 64 of file AlbersEqualArea.hpp.

real GeographicLib::AlbersEqualArea::_qZ

private

Definition at line 64 of file AlbersEqualArea.hpp.

real GeographicLib::AlbersEqualArea::_scxi0

private

Definition at line 66 of file AlbersEqualArea.hpp.

real GeographicLib::AlbersEqualArea::_sign

private

Definition at line 65 of file AlbersEqualArea.hpp.

real GeographicLib::AlbersEqualArea::_sxi0

private

Definition at line 66 of file AlbersEqualArea.hpp.

real GeographicLib::AlbersEqualArea::_txi0

private

Definition at line 66 of file AlbersEqualArea.hpp.

real GeographicLib::AlbersEqualArea::eps_

private

Definition at line 63 of file AlbersEqualArea.hpp.

real GeographicLib::AlbersEqualArea::epsx2_

private

Definition at line 63 of file AlbersEqualArea.hpp.

real GeographicLib::AlbersEqualArea::epsx_

private

Definition at line 63 of file AlbersEqualArea.hpp.

const int GeographicLib::AlbersEqualArea::numit0_ = 20

staticprivate

Definition at line 68 of file AlbersEqualArea.hpp.

const int GeographicLib::AlbersEqualArea::numit_ = 5

staticprivate

Definition at line 67 of file AlbersEqualArea.hpp.

real GeographicLib::AlbersEqualArea::tol0_

private

Definition at line 63 of file AlbersEqualArea.hpp.

real GeographicLib::AlbersEqualArea::tol_

private

Definition at line 63 of file AlbersEqualArea.hpp.

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

Public Member Functions

Static Public Member Functions

Private Types

Private Member Functions

Static Private Member Functions

Private Attributes

Static Private Attributes

Friends

Detailed Description

Member Typedef Documentation

Constructor & Destructor Documentation

Member Function Documentation

Friends And Related Function Documentation

Member Data Documentation