322 lines
14 KiB
C++
322 lines
14 KiB
C++
/**
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* \file AlbersEqualArea.hpp
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* \brief Header for GeographicLib::AlbersEqualArea class
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*
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* Copyright (c) Charles Karney (2010-2021) <charles@karney.com> and licensed
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* under the MIT/X11 License. For more information, see
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* https://geographiclib.sourceforge.io/
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**********************************************************************/
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#if !defined(GEOGRAPHICLIB_ALBERSEQUALAREA_HPP)
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#define GEOGRAPHICLIB_ALBERSEQUALAREA_HPP 1
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#include <GeographicLib/Constants.hpp>
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namespace GeographicLib {
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/**
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* \brief Albers equal area conic projection
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*
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* Implementation taken from the report,
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* - J. P. Snyder,
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* <a href="http://pubs.er.usgs.gov/usgspubs/pp/pp1395"> Map Projections: A
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* Working Manual</a>, USGS Professional Paper 1395 (1987),
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* pp. 101--102.
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*
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* This is a implementation of the equations in Snyder except that divided
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* differences will be [have been] used to transform the expressions into
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* ones which may be evaluated accurately. [In this implementation, the
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* projection correctly becomes the cylindrical equal area or the azimuthal
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* equal area projection when the standard latitude is the equator or a
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* pole.]
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*
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* The ellipsoid parameters, the standard parallels, and the scale on the
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* standard parallels are set in the constructor. Internally, the case with
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* two standard parallels is converted into a single standard parallel, the
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* latitude of minimum azimuthal scale, with an azimuthal scale specified on
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* this parallel. This latitude is also used as the latitude of origin which
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* is returned by AlbersEqualArea::OriginLatitude. The azimuthal scale on
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* the latitude of origin is given by AlbersEqualArea::CentralScale. The
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* case with two standard parallels at opposite poles is singular and is
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* disallowed. The central meridian (which is a trivial shift of the
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* longitude) is specified as the \e lon0 argument of the
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* AlbersEqualArea::Forward and AlbersEqualArea::Reverse functions.
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* AlbersEqualArea::Forward and AlbersEqualArea::Reverse also return the
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* meridian convergence, γ, and azimuthal scale, \e k. A small square
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* aligned with the cardinal directions is projected to a rectangle with
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* dimensions \e k (in the E-W direction) and 1/\e k (in the N-S direction).
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* The E-W sides of the rectangle are oriented γ degrees
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* counter-clockwise from the \e x axis. There is no provision in this class
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* for specifying a false easting or false northing or a different latitude
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* of origin.
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*
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* Example of use:
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* \include example-AlbersEqualArea.cpp
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*
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* <a href="ConicProj.1.html">ConicProj</a> is a command-line utility
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* providing access to the functionality of LambertConformalConic and
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* AlbersEqualArea.
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**********************************************************************/
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class GEOGRAPHICLIB_EXPORT AlbersEqualArea {
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private:
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typedef Math::real real;
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real eps_, epsx_, epsx2_, tol_, tol0_;
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real _a, _f, _fm, _e2, _e, _e2m, _qZ, _qx;
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real _sign, _lat0, _k0;
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real _n0, _m02, _nrho0, _k2, _txi0, _scxi0, _sxi0;
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static const int numit_ = 5; // Newton iterations in Reverse
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static const int numit0_ = 20; // Newton iterations in Init
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static real hyp(real x) {
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using std::hypot;
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return hypot(real(1), x);
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}
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// atanh( e * x)/ e if f > 0
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// atan (sqrt(-e2) * x)/sqrt(-e2) if f < 0
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// x if f = 0
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real atanhee(real x) const {
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using std::atan; using std::abs; using std::atanh;
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return _f > 0 ? atanh(_e * x)/_e : (_f < 0 ? (atan(_e * x)/_e) : x);
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}
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// return atanh(sqrt(x))/sqrt(x) - 1, accurate for small x
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static real atanhxm1(real x);
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// Divided differences
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// Definition: Df(x,y) = (f(x)-f(y))/(x-y)
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// See:
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// W. M. Kahan and R. J. Fateman,
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// Symbolic computation of divided differences,
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// SIGSAM Bull. 33(3), 7-28 (1999)
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// https://doi.org/10.1145/334714.334716
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// http://www.cs.berkeley.edu/~fateman/papers/divdiff.pdf
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//
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// General rules
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// h(x) = f(g(x)): Dh(x,y) = Df(g(x),g(y))*Dg(x,y)
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// h(x) = f(x)*g(x):
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// Dh(x,y) = Df(x,y)*g(x) + Dg(x,y)*f(y)
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// = Df(x,y)*g(y) + Dg(x,y)*f(x)
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// = Df(x,y)*(g(x)+g(y))/2 + Dg(x,y)*(f(x)+f(y))/2
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//
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// sn(x) = x/sqrt(1+x^2): Dsn(x,y) = (x+y)/((sn(x)+sn(y))*(1+x^2)*(1+y^2))
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static real Dsn(real x, real y, real sx, real sy) {
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// sx = x/hyp(x)
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real t = x * y;
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return t > 0 ? (x + y) * Math::sq( (sx * sy)/t ) / (sx + sy) :
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(x - y != 0 ? (sx - sy) / (x - y) : 1);
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}
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// Datanhee(x,y) = (atanee(x)-atanee(y))/(x-y)
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// = atanhee((x-y)/(1-e^2*x*y))/(x-y)
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real Datanhee(real x, real y) const {
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real t = x - y, d = 1 - _e2 * x * y;
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return t == 0 ? 1 / d :
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(x*y < 0 ? atanhee(x) - atanhee(y) : atanhee(t / d)) / t;
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}
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// DDatanhee(x,y) = (Datanhee(1,y) - Datanhee(1,x))/(y-x)
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real DDatanhee(real x, real y) const;
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real DDatanhee0(real x, real y) const;
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real DDatanhee1(real x, real y) const;
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real DDatanhee2(real x, real y) const;
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void Init(real sphi1, real cphi1, real sphi2, real cphi2, real k1);
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real txif(real tphi) const;
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real tphif(real txi) const;
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friend class Ellipsoid; // For access to txif, tphif, etc.
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public:
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/**
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* Constructor with a single standard parallel.
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*
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* @param[in] a equatorial radius of ellipsoid (meters).
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* @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere.
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* Negative \e f gives a prolate ellipsoid.
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* @param[in] stdlat standard parallel (degrees), the circle of tangency.
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* @param[in] k0 azimuthal scale on the standard parallel.
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* @exception GeographicErr if \e a, (1 − \e f) \e a, or \e k0 is
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* not positive.
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* @exception GeographicErr if \e stdlat is not in [−90°,
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* 90°].
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**********************************************************************/
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AlbersEqualArea(real a, real f, real stdlat, real k0);
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/**
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* Constructor with two standard parallels.
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*
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* @param[in] a equatorial radius of ellipsoid (meters).
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* @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere.
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* Negative \e f gives a prolate ellipsoid.
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* @param[in] stdlat1 first standard parallel (degrees).
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* @param[in] stdlat2 second standard parallel (degrees).
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* @param[in] k1 azimuthal scale on the standard parallels.
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* @exception GeographicErr if \e a, (1 − \e f) \e a, or \e k1 is
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* not positive.
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* @exception GeographicErr if \e stdlat1 or \e stdlat2 is not in
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* [−90°, 90°], or if \e stdlat1 and \e stdlat2 are
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* opposite poles.
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**********************************************************************/
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AlbersEqualArea(real a, real f, real stdlat1, real stdlat2, real k1);
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/**
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* Constructor with two standard parallels specified by sines and cosines.
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*
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* @param[in] a equatorial radius of ellipsoid (meters).
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* @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere.
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* Negative \e f gives a prolate ellipsoid.
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* @param[in] sinlat1 sine of first standard parallel.
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* @param[in] coslat1 cosine of first standard parallel.
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* @param[in] sinlat2 sine of second standard parallel.
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* @param[in] coslat2 cosine of second standard parallel.
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* @param[in] k1 azimuthal scale on the standard parallels.
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* @exception GeographicErr if \e a, (1 − \e f) \e a, or \e k1 is
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* not positive.
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* @exception GeographicErr if \e stdlat1 or \e stdlat2 is not in
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* [−90°, 90°], or if \e stdlat1 and \e stdlat2 are
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* opposite poles.
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*
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* This allows parallels close to the poles to be specified accurately.
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* This routine computes the latitude of origin and the azimuthal scale at
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* this latitude. If \e dlat = abs(\e lat2 − \e lat1) ≤ 160°,
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* then the error in the latitude of origin is less than 4.5 ×
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* 10<sup>−14</sup>d;.
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**********************************************************************/
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AlbersEqualArea(real a, real f,
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real sinlat1, real coslat1,
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real sinlat2, real coslat2,
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real k1);
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/**
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* Set the azimuthal scale for the projection.
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*
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* @param[in] lat (degrees).
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* @param[in] k azimuthal scale at latitude \e lat (default 1).
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* @exception GeographicErr \e k is not positive.
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* @exception GeographicErr if \e lat is not in (−90°,
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* 90°).
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*
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* This allows a "latitude of conformality" to be specified.
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**********************************************************************/
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void SetScale(real lat, real k = real(1));
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/**
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* Forward projection, from geographic to Lambert conformal conic.
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*
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* @param[in] lon0 central meridian longitude (degrees).
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* @param[in] lat latitude of point (degrees).
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* @param[in] lon longitude of point (degrees).
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* @param[out] x easting of point (meters).
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* @param[out] y northing of point (meters).
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* @param[out] gamma meridian convergence at point (degrees).
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* @param[out] k azimuthal scale of projection at point; the radial
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* scale is the 1/\e k.
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*
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* The latitude origin is given by AlbersEqualArea::LatitudeOrigin(). No
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* false easting or northing is added and \e lat should be in the range
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* [−90°, 90°]. The values of \e x and \e y returned for
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* points which project to infinity (i.e., one or both of the poles) will
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* be large but finite.
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**********************************************************************/
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void Forward(real lon0, real lat, real lon,
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real& x, real& y, real& gamma, real& k) const;
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/**
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* Reverse projection, from Lambert conformal conic to geographic.
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*
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* @param[in] lon0 central meridian longitude (degrees).
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* @param[in] x easting of point (meters).
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* @param[in] y northing of point (meters).
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* @param[out] lat latitude of point (degrees).
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* @param[out] lon longitude of point (degrees).
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* @param[out] gamma meridian convergence at point (degrees).
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* @param[out] k azimuthal scale of projection at point; the radial
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* scale is the 1/\e k.
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*
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* The latitude origin is given by AlbersEqualArea::LatitudeOrigin(). No
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* false easting or northing is added. The value of \e lon returned is in
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* the range [−180°, 180°]. The value of \e lat returned is
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* in the range [−90°, 90°]. If the input point is outside
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* the legal projected space the nearest pole is returned.
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**********************************************************************/
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void Reverse(real lon0, real x, real y,
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real& lat, real& lon, real& gamma, real& k) const;
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/**
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* AlbersEqualArea::Forward without returning the convergence and
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* scale.
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**********************************************************************/
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void Forward(real lon0, real lat, real lon,
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real& x, real& y) const {
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real gamma, k;
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Forward(lon0, lat, lon, x, y, gamma, k);
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}
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/**
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* AlbersEqualArea::Reverse without returning the convergence and
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* scale.
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**********************************************************************/
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void Reverse(real lon0, real x, real y,
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real& lat, real& lon) const {
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real gamma, k;
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Reverse(lon0, x, y, lat, lon, gamma, k);
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}
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/** \name Inspector functions
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**********************************************************************/
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///@{
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/**
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* @return \e a the equatorial radius of the ellipsoid (meters). This is
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* the value used in the constructor.
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**********************************************************************/
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Math::real EquatorialRadius() const { return _a; }
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/**
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* @return \e f the flattening of the ellipsoid. This is the value used in
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* the constructor.
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**********************************************************************/
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Math::real Flattening() const { return _f; }
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/**
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* @return latitude of the origin for the projection (degrees).
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*
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* This is the latitude of minimum azimuthal scale and equals the \e stdlat
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* in the 1-parallel constructor and lies between \e stdlat1 and \e stdlat2
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* in the 2-parallel constructors.
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**********************************************************************/
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Math::real OriginLatitude() const { return _lat0; }
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/**
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* @return central scale for the projection. This is the azimuthal scale
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* on the latitude of origin.
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**********************************************************************/
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Math::real CentralScale() const { return _k0; }
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/**
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* \deprecated An old name for EquatorialRadius().
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**********************************************************************/
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GEOGRAPHICLIB_DEPRECATED("Use EquatorialRadius()")
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Math::real MajorRadius() const { return EquatorialRadius(); }
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///@}
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/**
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* A global instantiation of AlbersEqualArea with the WGS84 ellipsoid, \e
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* stdlat = 0, and \e k0 = 1. This degenerates to the cylindrical equal
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* area projection.
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**********************************************************************/
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static const AlbersEqualArea& CylindricalEqualArea();
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/**
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* A global instantiation of AlbersEqualArea with the WGS84 ellipsoid, \e
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* stdlat = 90°, and \e k0 = 1. This degenerates to the
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* Lambert azimuthal equal area projection.
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**********************************************************************/
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static const AlbersEqualArea& AzimuthalEqualAreaNorth();
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/**
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* A global instantiation of AlbersEqualArea with the WGS84 ellipsoid, \e
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* stdlat = −90°, and \e k0 = 1. This degenerates to the
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* Lambert azimuthal equal area projection.
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**********************************************************************/
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static const AlbersEqualArea& AzimuthalEqualAreaSouth();
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};
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} // namespace GeographicLib
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#endif // GEOGRAPHICLIB_ALBERSEQUALAREA_HPP
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