/** * \file AlbersEqualArea.hpp * \brief Header for GeographicLib::AlbersEqualArea class * * Copyright (c) Charles Karney (2010-2019) 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 namespace GeographicLib { /** * \brief 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 \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 * * ConicProj 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) { return Math::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::atan2; using std::abs; return _f > 0 ? Math::atanh(_e * x)/_e : // We only invoke atanhee in txif for positive latitude. Then x is // only negative for very prolate ellipsoids (_b/_a >= sqrt(2)) and we // still need to return a positive result in this case; hence the need // for the call to atan2. (_f < 0 ? (atan2(_e * abs(x), real(x < 0 ? -1 : 1))/_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(3), 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) = 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 ? atanhee(t / d) / t : 1 / d; } // DDatanhee(x,y) = (Datanhee(1,y) - Datanhee(1,x))/(y-x) real DDatanhee(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; friend class Ellipsoid; // For access to txif, tphif, etc. 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 × * 10−14d;. **********************************************************************/ 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; } /** * \deprecated An old name for EquatorialRadius(). **********************************************************************/ // GEOGRAPHICLIB_DEPRECATED("Use EquatorialRadius()") Math::real MajorRadius() const { return EquatorialRadius(); } ///@} /** * 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