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/**
* \file AlbersEqualArea.hpp
* \brief Header for GeographicLib::AlbersEqualArea class
*
* Copyright (c) Charles Karney (2010-2023) <karney@alum.mit.edu> and licensed
* under the MIT/X11 License. For more information, see
* https://geographiclib.sourceforge.io/
**********************************************************************/
#if !defined(GEOGRAPHICLIB_ALBERSEQUALAREA_HPP)
#define GEOGRAPHICLIB_ALBERSEQUALAREA_HPP 1
#include <GeographicLib/Constants.hpp>
namespace GeographicLib {
/**
* \brief Albers equal area conic projection
*
* Implementation taken from the report,
* - J. P. Snyder,
* <a href="https://pubs.usgs.gov/publication/pp1395"> Map Projections: A
* Working Manual</a>, 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 \e lon0 argument of the
* AlbersEqualArea::Forward and AlbersEqualArea::Reverse functions.
* AlbersEqualArea::Forward and AlbersEqualArea::Reverse also return the
* meridian convergence, γ, and azimuthal scale, \e k. A small square
* aligned with the cardinal directions is projected to a rectangle with
* dimensions \e k (in the E-W direction) and 1/\e k (in the N-S direction).
* The E-W sides of the rectangle are oriented γ degrees
* counter-clockwise from the \e 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:
* \include example-AlbersEqualArea.cpp
*
* <a href="ConicProj.1.html">ConicProj</a> is a command-line utility
* providing access to the functionality of LambertConformalConic and
* AlbersEqualArea.
**********************************************************************/
class GEOGRAPHICLIB_EXPORT AlbersEqualArea {
private:
typedef Math::real real;
real eps_, epsx_, epsx2_, tol_, tol0_;
real _a, _f, _fm, _e2, _e, _e2m, _qZ, _qx;
real _sign, _lat0, _k0;
real _n0, _m02, _nrho0, _k2, _txi0, _scxi0, _sxi0;
static const int numit_ = 5; // Newton iterations in Reverse
static const int numit0_ = 20; // Newton iterations in Init
static real hyp(real x) {
using std::hypot;
return hypot(real(1), x);
}
// atanh( e * x)/ e if f > 0
// atan (sqrt(-e2) * x)/sqrt(-e2) if f < 0
// x if f = 0
real atanhee(real x) const {
using std::atan; using std::atanh;
return _f > 0 ? atanh(_e * x)/_e : (_f < 0 ? (atan(_e * x)/_e) : x);
}
// return atanh(sqrt(x))/sqrt(x) - 1, accurate for small x
static real atanhxm1(real x);
// Divided differences
// Definition: Df(x,y) = (f(x)-f(y))/(x-y)
// See:
// W. M. Kahan and R. J. Fateman,
// Symbolic computation of divided differences,
// SIGSAM Bull. 33(2), 7-28 (1999)
// https://doi.org/10.1145/334714.334716
// http://www.cs.berkeley.edu/~fateman/papers/divdiff.pdf
//
// General rules
// h(x) = f(g(x)): Dh(x,y) = Df(g(x),g(y))*Dg(x,y)
// h(x) = f(x)*g(x):
// Dh(x,y) = Df(x,y)*g(x) + Dg(x,y)*f(y)
// = Df(x,y)*g(y) + Dg(x,y)*f(x)
// = Df(x,y)*(g(x)+g(y))/2 + Dg(x,y)*(f(x)+f(y))/2
//
// sn(x) = x/sqrt(1+x^2): Dsn(x,y) = (x+y)/((sn(x)+sn(y))*(1+x^2)*(1+y^2))
static real Dsn(real x, real y, real sx, real sy) {
// sx = x/hyp(x)
real t = x * y;
return t > 0 ? (x + y) * Math::sq( (sx * sy)/t ) / (sx + sy) :
(x - y != 0 ? (sx - sy) / (x - y) : 1);
}
// Datanhee(x,y) = (atanee(x)-atanee(y))/(x-y)
// = atanhee((x-y)/(1-e^2*x*y))/(x-y)
real Datanhee(real x, real y) const {
real t = x - y, d = 1 - _e2 * x * y;
return t == 0 ? 1 / d :
(x*y < 0 ? atanhee(x) - atanhee(y) : atanhee(t / d)) / t;
}
// DDatanhee(x,y) = (Datanhee(1,y) - Datanhee(1,x))/(y-x)
real DDatanhee(real x, real y) const;
real DDatanhee0(real x, real y) const;
real DDatanhee1(real x, real y) const;
real DDatanhee2(real x, real y) const;
void Init(real sphi1, real cphi1, real sphi2, real cphi2, real k1);
real txif(real tphi) const;
real tphif(real txi) const;
public:
/**
* Constructor with a single standard parallel.
*
* @param[in] a equatorial radius of ellipsoid (meters).
* @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere.
* Negative \e f gives a prolate ellipsoid.
* @param[in] stdlat standard parallel (degrees), the circle of tangency.
* @param[in] k0 azimuthal scale on the standard parallel.
* @exception GeographicErr if \e a, (1 − \e f) \e a, or \e k0 is
* not positive.
* @exception GeographicErr if \e stdlat is not in [−90°,
* 90°].
**********************************************************************/
AlbersEqualArea(real a, real f, real stdlat, real k0);
/**
* Constructor with two standard parallels.
*
* @param[in] a equatorial radius of ellipsoid (meters).
* @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere.
* Negative \e f gives a prolate ellipsoid.
* @param[in] stdlat1 first standard parallel (degrees).
* @param[in] stdlat2 second standard parallel (degrees).
* @param[in] k1 azimuthal scale on the standard parallels.
* @exception GeographicErr if \e a, (1 − \e f) \e a, or \e k1 is
* not positive.
* @exception GeographicErr if \e stdlat1 or \e stdlat2 is not in
* [−90°, 90°], or if \e stdlat1 and \e stdlat2 are
* opposite poles.
**********************************************************************/
AlbersEqualArea(real a, real f, real stdlat1, real stdlat2, real k1);
/**
* Constructor with two standard parallels specified by sines and cosines.
*
* @param[in] a equatorial radius of ellipsoid (meters).
* @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere.
* Negative \e f gives a prolate ellipsoid.
* @param[in] sinlat1 sine of first standard parallel.
* @param[in] coslat1 cosine of first standard parallel.
* @param[in] sinlat2 sine of second standard parallel.
* @param[in] coslat2 cosine of second standard parallel.
* @param[in] k1 azimuthal scale on the standard parallels.
* @exception GeographicErr if \e a, (1 − \e f) \e a, or \e k1 is
* not positive.
* @exception GeographicErr if \e stdlat1 or \e stdlat2 is not in
* [−90°, 90°], or if \e stdlat1 and \e 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 \e dlat = abs(\e lat2 − \e lat1) ≤ 160°,
* then the error in the latitude of origin is less than 4.5 ×
* 10<sup>−14</sup>d;.
**********************************************************************/
AlbersEqualArea(real a, real f,
real sinlat1, real coslat1,
real sinlat2, real coslat2,
real k1);
/**
* Set the azimuthal scale for the projection.
*
* @param[in] lat (degrees).
* @param[in] k azimuthal scale at latitude \e lat (default 1).
* @exception GeographicErr \e k is not positive.
* @exception GeographicErr if \e lat is not in (−90°,
* 90°).
*
* This allows a "latitude of conformality" to be specified.
**********************************************************************/
void SetScale(real lat, real k = real(1));
/**
* Forward projection, from geographic to Lambert conformal conic.
*
* @param[in] lon0 central meridian longitude (degrees).
* @param[in] lat latitude of point (degrees).
* @param[in] lon longitude of point (degrees).
* @param[out] x easting of point (meters).
* @param[out] y northing of point (meters).
* @param[out] gamma meridian convergence at point (degrees).
* @param[out] k azimuthal scale of projection at point; the radial
* scale is the 1/\e k.
*
* The latitude origin is given by AlbersEqualArea::LatitudeOrigin(). No
* false easting or northing is added and \e lat should be in the range
* [−90°, 90°]. The values of \e x and \e y returned for
* points which project to infinity (i.e., one or both of the poles) will
* be large but finite.
**********************************************************************/
void Forward(real lon0, real lat, real lon,
real& x, real& y, real& gamma, real& k) const;
/**
* Reverse projection, from Lambert conformal conic to geographic.
*
* @param[in] lon0 central meridian longitude (degrees).
* @param[in] x easting of point (meters).
* @param[in] y northing of point (meters).
* @param[out] lat latitude of point (degrees).
* @param[out] lon longitude of point (degrees).
* @param[out] gamma meridian convergence at point (degrees).
* @param[out] k azimuthal scale of projection at point; the radial
* scale is the 1/\e k.
*
* The latitude origin is given by AlbersEqualArea::LatitudeOrigin(). No
* false easting or northing is added. The value of \e lon returned is in
* the range [−180°, 180°]. The value of \e lat returned is
* in the range [−90°, 90°]. If the input point is outside
* the legal projected space the nearest pole is returned.
**********************************************************************/
void Reverse(real lon0, real x, real y,
real& lat, real& lon, real& gamma, real& k) const;
/**
* AlbersEqualArea::Forward without returning the convergence and
* scale.
**********************************************************************/
void Forward(real lon0, real lat, real lon,
real& x, real& y) const {
real gamma, k;
Forward(lon0, lat, lon, x, y, gamma, k);
}
/**
* AlbersEqualArea::Reverse without returning the convergence and
* scale.
**********************************************************************/
void Reverse(real lon0, real x, real y,
real& lat, real& lon) const {
real gamma, k;
Reverse(lon0, x, y, lat, lon, gamma, k);
}
/** \name Inspector functions
**********************************************************************/
///@{
/**
* @return \e a the equatorial radius of the ellipsoid (meters). This is
* the value used in the constructor.
**********************************************************************/
Math::real EquatorialRadius() const { return _a; }
/**
* @return \e f the flattening of the ellipsoid. This is the value used in
* the constructor.
**********************************************************************/
Math::real Flattening() const { return _f; }
/**
* @return latitude of the origin for the projection (degrees).
*
* This is the latitude of minimum azimuthal scale and equals the \e stdlat
* in the 1-parallel constructor and lies between \e stdlat1 and \e stdlat2
* in the 2-parallel constructors.
**********************************************************************/
Math::real OriginLatitude() const { return _lat0; }
/**
* @return central scale for the projection. This is the azimuthal scale
* on the latitude of origin.
**********************************************************************/
Math::real CentralScale() const { return _k0; }
///@}
/**
* A global instantiation of AlbersEqualArea with the WGS84 ellipsoid, \e
* stdlat = 0, and \e k0 = 1. This degenerates to the cylindrical equal
* area projection.
**********************************************************************/
static const AlbersEqualArea& CylindricalEqualArea();
/**
* A global instantiation of AlbersEqualArea with the WGS84 ellipsoid, \e
* stdlat = 90°, and \e k0 = 1. This degenerates to the
* Lambert azimuthal equal area projection.
**********************************************************************/
static const AlbersEqualArea& AzimuthalEqualAreaNorth();
/**
* A global instantiation of AlbersEqualArea with the WGS84 ellipsoid, \e
* stdlat = −90°, and \e k0 = 1. This degenerates to the
* Lambert azimuthal equal area projection.
**********************************************************************/
static const AlbersEqualArea& AzimuthalEqualAreaSouth();
};
} // namespace GeographicLib
#endif // GEOGRAPHICLIB_ALBERSEQUALAREA_HPP