00001 /** 00002 * \file AlbersEqualArea.hpp 00003 * \brief Header for GeographicLib::AlbersEqualArea class 00004 * 00005 * Copyright (c) Charles Karney (2010) <charles@karney.com> and licensed under 00006 * the LGPL. For more information, see http://geographiclib.sourceforge.net/ 00007 **********************************************************************/ 00008 00009 #if !defined(GEOGRAPHICLIB_ALBERSEQUALAREA_HPP) 00010 #define GEOGRAPHICLIB_ALBERSEQUALAREA_HPP "$Id: AlbersEqualArea.hpp 6919 2010-12-21 13:23:47Z karney $" 00011 00012 #include "GeographicLib/Constants.hpp" 00013 #include <algorithm> 00014 00015 namespace GeographicLib { 00016 00017 /** 00018 * \brief Albers Equal Area Conic Projection 00019 * 00020 * Implementation taken from the report, 00021 * - J. P. Snyder, 00022 * <a href="http://pubs.er.usgs.gov/usgspubs/pp/pp1395"> Map Projections: A 00023 * Working Manual</a>, USGS Professional Paper 1395 (1987), 00024 * pp. 101–102. 00025 * 00026 * This is a implementation of the equations in Snyder except that divided 00027 * differences will be [have been] used to transform the expressions into 00028 * ones which may be evaluated accurately. [In this implementation, the 00029 * projection correctly becomes the cylindrical equal area or the azimuthal 00030 * equal area projection when the standard latitude is the equator or a 00031 * pole.] 00032 * 00033 * The ellipsoid parameters, the standard parallels, and the scale on the 00034 * standard parallels are set in the constructor. Internally, the case with 00035 * two standard parallels is converted into a single standard parallel, the 00036 * latitude of minimum azimuthal scale, with an azimuthal scale specified on 00037 * this parallel. This latitude is also used as the latitude of origin which 00038 * is returned by AlbersEqualArea::OriginLatitude. The azimuthal scale on 00039 * the latitude of origin is given by AlbersEqualArea::CentralScale. The 00040 * case with two standard parallels at opposite poles is singular and is 00041 * disallowed. The central meridian (which is a trivial shift of the 00042 * longitude) is specified as the \e lon0 argument of the 00043 * AlbersEqualArea::Forward and AlbersEqualArea::Reverse functions. 00044 * AlbersEqualArea::Forward and AlbersEqualArea::Reverse also return the 00045 * meridian convergence, \e gamma, and azimuthal scale, \e k. A small square 00046 * aligned with the cardinal directions is projected to a rectangle with 00047 * dimensions \e k (in the E-W direction) and 1/\e k (in the N-S direction). 00048 * The E-W sides of the rectangle are oriented \e gamma degrees 00049 * counter-clockwise from the \e x axis. There is no provision in this class 00050 * for specifying a false easting or false northing or a different latitude 00051 * of origin. 00052 **********************************************************************/ 00053 class AlbersEqualArea { 00054 private: 00055 typedef Math::real real; 00056 const real _a, _r, _f, _fm, _e2, _e, _e2m, _qZ, _qx; 00057 real _sign, _lat0, _k0; 00058 real _n0, _m02, _nrho0, _k2, _txi0, _scxi0, _sxi0; 00059 static const real eps, epsx, epsx2, tol, tol0, ahypover; 00060 static const int numit = 5; // Newton iterations in Reverse 00061 static const int numit0 = 20; // Newton iterations in Init 00062 static inline real sq(real x) throw() { return x * x; } 00063 static inline real hyp(real x) throw() { return Math::hypot(real(1), x); } 00064 // atanh( e * x)/ e if f > 0 00065 // atan (sqrt(-e2) * x)/sqrt(-e2) if f < 0 00066 // x if f = 0 00067 inline real atanhee(real x) const throw() { 00068 return _f > 0 ? Math::atanh(_e * x)/_e : 00069 (_f < 0 ? std::atan(_e * x)/_e : x); 00070 } 00071 // return atanh(sqrt(x))/sqrt(x) - 1, accurate for small x 00072 static real atanhxm1(real x) throw(); 00073 00074 // Divided differences 00075 // Definition: Df(x,y) = (f(x)-f(y))/(x-y) 00076 // See: W. M. Kahan and R. J. Fateman, 00077 // Symbolic computation of divided differences, 00078 // SIGSAM Bull. 33(3), 7-28 (1999) 00079 // http://doi.acm.org/10.1145/334714.334716 00080 // http://www.cs.berkeley.edu/~fateman/papers/divdiff.pdf 00081 // 00082 // General rules 00083 // h(x) = f(g(x)): Dh(x,y) = Df(g(x),g(y))*Dg(x,y) 00084 // h(x) = f(x)*g(x): 00085 // Dh(x,y) = Df(x,y)*(g(x)+g(y))/2 + Dg(x,y)*(f(x)+f(y))/2 00086 // 00087 // sn(x) = x/sqrt(1+x^2): Dsn(x,y) = (x+y)/((sn(x)+sn(y))*(1+x^2)*(1+y^2)) 00088 static inline real Dsn(real x, real y, real sx, real sy) throw() { 00089 // sx = x/hyp(x) 00090 real t = x * y; 00091 return t > 0 ? (x + y) * sq( (sx * sy)/t ) / (sx + sy) : 00092 (x - y != 0 ? (sx - sy) / (x - y) : 1); 00093 } 00094 // Datanhee(x,y) = atanhee((x-y)/(1-e^2*x*y))/(x-y) 00095 inline real Datanhee(real x, real y) const throw() { 00096 real t = x - y, d = 1 - _e2 * x * y; 00097 return t != 0 ? atanhee(t / d) / t : 1 / d; 00098 } 00099 // DDatanhee(x,y) = (Datanhee(1,y) - Datanhee(1,x))/(y-x) 00100 real DDatanhee(real x, real y) const throw(); 00101 void Init(real sphi1, real cphi1, real sphi2, real cphi2, real k1) throw(); 00102 real txif(real tphi) const throw(); 00103 real tphif(real txi) const throw(); 00104 public: 00105 00106 /** 00107 * Constructor with a single standard parallel. 00108 * 00109 * @param[in] a equatorial radius of ellipsoid (meters) 00110 * @param[in] r reciprocal flattening of ellipsoid. Setting \e r = 0 00111 * implies \e r = inf or flattening = 0 (i.e., a sphere). Negative \e r 00112 * indicates a prolate ellipsoid. 00113 * @param[in] stdlat standard parallel (degrees), the circle of tangency. 00114 * @param[in] k0 azimuthal scale on the standard parallel. 00115 * 00116 * An exception is thrown if \e a or \e k0 is not positive or if \e stdlat 00117 * is not in the range [-90, 90]. 00118 **********************************************************************/ 00119 AlbersEqualArea(real a, real r, real stdlat, real k0); 00120 00121 /** 00122 * Constructor with two standard parallels. 00123 * 00124 * @param[in] a equatorial radius of ellipsoid (meters) 00125 * @param[in] r reciprocal flattening of ellipsoid. Setting \e r = 0 00126 * implies \e r = inf or flattening = 0 (i.e., a sphere). Negative \e r 00127 * indicates a prolate ellipsoid. 00128 * @param[in] stdlat1 first standard parallel (degrees). 00129 * @param[in] stdlat2 second standard parallel (degrees). 00130 * @param[in] k1 azimuthal scale on the standard parallels. 00131 * 00132 * An exception is thrown if \e a or \e k0 is not positive or if \e stdlat1 00133 * or \e stdlat2 is not in the range [-90, 90]. In addition, an exception 00134 * is thrown if \e stdlat1 and \e stdlat2 are opposite poles. 00135 **********************************************************************/ 00136 AlbersEqualArea(real a, real r, real stdlat1, real stdlat2, real k1); 00137 00138 /** 00139 * Constructor with two standard parallels specified by sines and cosines. 00140 * 00141 * @param[in] a equatorial radius of ellipsoid (meters) 00142 * @param[in] r reciprocal flattening of ellipsoid. Setting \e r = 0 00143 * implies \e r = inf or flattening = 0 (i.e., a sphere). Negative \e r 00144 * indicates a prolate ellipsoid. 00145 * @param[in] sinlat1 sine of first standard parallel. 00146 * @param[in] coslat1 cosine of first standard parallel. 00147 * @param[in] sinlat2 sine of second standard parallel. 00148 * @param[in] coslat2 cosine of second standard parallel. 00149 * @param[in] k1 azimuthal scale on the standard parallels. 00150 * 00151 * This allows parallels close to the poles to be specified accurately. 00152 * This routine computes the latitude of origin and the azimuthal scale at 00153 * this latitude. If \e dlat = abs(\e lat2 - \e lat1) <= 160<sup>o</sup>, 00154 * then the error in the latitude of origin is less than 00155 * 4.5e-14<sup>o</sup>. 00156 **********************************************************************/ 00157 AlbersEqualArea(real a, real r, 00158 real sinlat1, real coslat1, 00159 real sinlat2, real coslat2, 00160 real k1); 00161 00162 /** 00163 * Set the azimuthal scale for the projection. 00164 * 00165 * @param[in] lat (degrees). 00166 * @param[in] k azimuthal scale at latitude \e lat (default 1). 00167 * 00168 * This allows a "latitude of conformality" to be specified. An exception 00169 * is thrown if \e k is not positive or if \e lat is not in the range (-90, 00170 * 90). 00171 **********************************************************************/ 00172 void SetScale(real lat, real k = real(1)); 00173 00174 /** 00175 * Forward projection, from geographic to Lambert conformal conic. 00176 * 00177 * @param[in] lon0 central meridian longitude (degrees). 00178 * @param[in] lat latitude of point (degrees). 00179 * @param[in] lon longitude of point (degrees). 00180 * @param[out] x easting of point (meters). 00181 * @param[out] y northing of point (meters). 00182 * @param[out] gamma meridian convergence at point (degrees). 00183 * @param[out] k azimuthal scale of projection at point; the radial 00184 * scale is the 1/\e k. 00185 * 00186 * The latitude origin is given by AlbersEqualArea::LatitudeOrigin(). No 00187 * false easting or northing is added and \e lat should be in the range 00188 * [-90, 90]; \e lon and \e lon0 should be in the range [-180, 360]. The 00189 * values of \e x and \e y returned for points which project to infinity 00190 * (i.e., one or both of the poles) will be large but finite. 00191 **********************************************************************/ 00192 void Forward(real lon0, real lat, real lon, 00193 real& x, real& y, real& gamma, real& k) const throw(); 00194 00195 /** 00196 * Reverse projection, from Lambert conformal conic to geographic. 00197 * 00198 * @param[in] lon0 central meridian longitude (degrees). 00199 * @param[in] x easting of point (meters). 00200 * @param[in] y northing of point (meters). 00201 * @param[out] lat latitude of point (degrees). 00202 * @param[out] lon longitude of point (degrees). 00203 * @param[out] gamma meridian convergence at point (degrees). 00204 * @param[out] k azimuthal scale of projection at point; the radial 00205 * scale is the 1/\e k. 00206 * 00207 * The latitude origin is given by AlbersEqualArea::LatitudeOrigin(). No 00208 * false easting or northing is added. \e lon0 should be in the range 00209 * [-180, 360]. The value of \e lon returned is in the range [-180, 180). 00210 * The value of \e lat returned is in the range [-90,90]. If the input 00211 * point is outside the legal projected space the nearest pole is returned. 00212 **********************************************************************/ 00213 void Reverse(real lon0, real x, real y, 00214 real& lat, real& lon, real& gamma, real& k) const throw(); 00215 00216 /** 00217 * AlbersEqualArea::Forward without returning the convergence and 00218 * scale. 00219 **********************************************************************/ 00220 void Forward(real lon0, real lat, real lon, 00221 real& x, real& y) const throw() { 00222 real gamma, k; 00223 Forward(lon0, lat, lon, x, y, gamma, k); 00224 } 00225 00226 /** 00227 * AlbersEqualArea::Reverse without returning the convergence and 00228 * scale. 00229 **********************************************************************/ 00230 void Reverse(real lon0, real x, real y, 00231 real& lat, real& lon) const throw() { 00232 real gamma, k; 00233 Reverse(lon0, x, y, lat, lon, gamma, k); 00234 } 00235 00236 /** \name Inspector functions 00237 **********************************************************************/ 00238 ///@{ 00239 /** 00240 * @return \e a the equatorial radius of the ellipsoid (meters). This is 00241 * the value used in the constructor. 00242 **********************************************************************/ 00243 Math::real MajorRadius() const throw() { return _a; } 00244 00245 /** 00246 * @return \e r the inverse flattening of the ellipsoid. This is the 00247 * value used in the constructor. A value of 0 is returned for a sphere 00248 * (infinite inverse flattening). 00249 **********************************************************************/ 00250 Math::real InverseFlattening() const throw() { return _r; } 00251 00252 /** 00253 * @return latitude of the origin for the projection (degrees). 00254 * 00255 * This is the latitude of minimum azimuthal scale and equals the \e stdlat 00256 * in the 1-parallel constructor and lies between \e stdlat1 and \e stdlat2 00257 * in the 2-parallel constructors. 00258 **********************************************************************/ 00259 Math::real OriginLatitude() const throw() { return _lat0; } 00260 00261 /** 00262 * @return central scale for the projection. This is the azimuthal scale 00263 * on the latitude of origin. 00264 **********************************************************************/ 00265 Math::real CentralScale() const throw() { return _k0; } 00266 ///@} 00267 00268 /** 00269 * A global instantiation of AlbersEqualArea with the WGS84 ellipsoid, \e 00270 * stdlat = 0, and \e k0 = 1. This degenerates to the cylindrical equal 00271 * area projection. 00272 **********************************************************************/ 00273 static const AlbersEqualArea CylindricalEqualArea; 00274 00275 /** 00276 * A global instantiation of AlbersEqualArea with the WGS84 ellipsoid, \e 00277 * stdlat = 90<sup>o</sup>, and \e k0 = 1. This degenerates to the 00278 * Lambert azimuthal equal area projection. 00279 **********************************************************************/ 00280 static const AlbersEqualArea AzimuthalEqualAreaNorth; 00281 00282 /** 00283 * A global instantiation of AlbersEqualArea with the WGS84 ellipsoid, \e 00284 * stdlat = -90<sup>o</sup>, and \e k0 = 1. This degenerates to the 00285 * Lambert azimuthal equal area projection. 00286 **********************************************************************/ 00287 static const AlbersEqualArea AzimuthalEqualAreaSouth; 00288 }; 00289 00290 } // namespace GeographicLib 00291 00292 #endif