GeographicLib
1.21
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00001 /** 00002 * \file AlbersEqualArea.hpp 00003 * \brief Header for GeographicLib::AlbersEqualArea class 00004 * 00005 * Copyright (c) Charles Karney (2010, 2011) <charles@karney.com> and licensed 00006 * under the MIT/X11 License. For more information, see 00007 * http://geographiclib.sourceforge.net/ 00008 **********************************************************************/ 00009 00010 #if !defined(GEOGRAPHICLIB_ALBERSEQUALAREA_HPP) 00011 #define GEOGRAPHICLIB_ALBERSEQUALAREA_HPP \ 00012 "$Id: d17f37d1bec84543dc3753e882d8e95f1c1d5a1b $" 00013 00014 #include <algorithm> 00015 #include <GeographicLib/Constants.hpp> 00016 00017 namespace GeographicLib { 00018 00019 /** 00020 * \brief Albers Equal Area Conic Projection 00021 * 00022 * Implementation taken from the report, 00023 * - J. P. Snyder, 00024 * <a href="http://pubs.er.usgs.gov/usgspubs/pp/pp1395"> Map Projections: A 00025 * Working Manual</a>, USGS Professional Paper 1395 (1987), 00026 * pp. 101–102. 00027 * 00028 * This is a implementation of the equations in Snyder except that divided 00029 * differences will be [have been] used to transform the expressions into 00030 * ones which may be evaluated accurately. [In this implementation, the 00031 * projection correctly becomes the cylindrical equal area or the azimuthal 00032 * equal area projection when the standard latitude is the equator or a 00033 * pole.] 00034 * 00035 * The ellipsoid parameters, the standard parallels, and the scale on the 00036 * standard parallels are set in the constructor. Internally, the case with 00037 * two standard parallels is converted into a single standard parallel, the 00038 * latitude of minimum azimuthal scale, with an azimuthal scale specified on 00039 * this parallel. This latitude is also used as the latitude of origin which 00040 * is returned by AlbersEqualArea::OriginLatitude. The azimuthal scale on 00041 * the latitude of origin is given by AlbersEqualArea::CentralScale. The 00042 * case with two standard parallels at opposite poles is singular and is 00043 * disallowed. The central meridian (which is a trivial shift of the 00044 * longitude) is specified as the \e lon0 argument of the 00045 * AlbersEqualArea::Forward and AlbersEqualArea::Reverse functions. 00046 * AlbersEqualArea::Forward and AlbersEqualArea::Reverse also return the 00047 * meridian convergence, \e gamma, and azimuthal scale, \e k. A small square 00048 * aligned with the cardinal directions is projected to a rectangle with 00049 * dimensions \e k (in the E-W direction) and 1/\e k (in the N-S direction). 00050 * The E-W sides of the rectangle are oriented \e gamma degrees 00051 * counter-clockwise from the \e x axis. There is no provision in this class 00052 * for specifying a false easting or false northing or a different latitude 00053 * of origin. 00054 * 00055 * Example of use: 00056 * \include example-AlbersEqualArea.cpp 00057 * 00058 * <a href="ConicProj.1.html">ConicProj</a> is a command-line utility 00059 * providing access to the functionality of LambertConformalConic and 00060 * AlbersEqualArea. 00061 **********************************************************************/ 00062 class GEOGRAPHIC_EXPORT AlbersEqualArea { 00063 private: 00064 typedef Math::real real; 00065 real _a, _f, _fm, _e2, _e, _e2m, _qZ, _qx; 00066 real _sign, _lat0, _k0; 00067 real _n0, _m02, _nrho0, _k2, _txi0, _scxi0, _sxi0; 00068 static const real eps_; 00069 static const real epsx_; 00070 static const real epsx2_; 00071 static const real tol_; 00072 static const real tol0_; 00073 static const real ahypover_; 00074 static const int numit_ = 5; // Newton iterations in Reverse 00075 static const int numit0_ = 20; // Newton iterations in Init 00076 static inline real hyp(real x) throw() { return Math::hypot(real(1), x); } 00077 // atanh( e * x)/ e if f > 0 00078 // atan (sqrt(-e2) * x)/sqrt(-e2) if f < 0 00079 // x if f = 0 00080 inline real atanhee(real x) const throw() { 00081 return _f > 0 ? Math::atanh(_e * x)/_e : 00082 (_f < 0 ? std::atan(_e * x)/_e : x); 00083 } 00084 // return atanh(sqrt(x))/sqrt(x) - 1, accurate for small x 00085 static real atanhxm1(real x) throw(); 00086 00087 // Divided differences 00088 // Definition: Df(x,y) = (f(x)-f(y))/(x-y) 00089 // See: W. M. Kahan and R. J. Fateman, 00090 // Symbolic computation of divided differences, 00091 // SIGSAM Bull. 33(3), 7-28 (1999) 00092 // http://doi.acm.org/10.1145/334714.334716 00093 // http://www.cs.berkeley.edu/~fateman/papers/divdiff.pdf 00094 // 00095 // General rules 00096 // h(x) = f(g(x)): Dh(x,y) = Df(g(x),g(y))*Dg(x,y) 00097 // h(x) = f(x)*g(x): 00098 // Dh(x,y) = Df(x,y)*(g(x)+g(y))/2 + Dg(x,y)*(f(x)+f(y))/2 00099 // 00100 // sn(x) = x/sqrt(1+x^2): Dsn(x,y) = (x+y)/((sn(x)+sn(y))*(1+x^2)*(1+y^2)) 00101 static inline real Dsn(real x, real y, real sx, real sy) throw() { 00102 // sx = x/hyp(x) 00103 real t = x * y; 00104 return t > 0 ? (x + y) * Math::sq( (sx * sy)/t ) / (sx + sy) : 00105 (x - y != 0 ? (sx - sy) / (x - y) : 1); 00106 } 00107 // Datanhee(x,y) = atanhee((x-y)/(1-e^2*x*y))/(x-y) 00108 inline real Datanhee(real x, real y) const throw() { 00109 real t = x - y, d = 1 - _e2 * x * y; 00110 return t != 0 ? atanhee(t / d) / t : 1 / d; 00111 } 00112 // DDatanhee(x,y) = (Datanhee(1,y) - Datanhee(1,x))/(y-x) 00113 real DDatanhee(real x, real y) const throw(); 00114 void Init(real sphi1, real cphi1, real sphi2, real cphi2, real k1) throw(); 00115 real txif(real tphi) const throw(); 00116 real tphif(real txi) const throw(); 00117 public: 00118 00119 /** 00120 * Constructor with a single standard parallel. 00121 * 00122 * @param[in] a equatorial radius of ellipsoid (meters). 00123 * @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere. 00124 * Negative \e f gives a prolate ellipsoid. If \e f > 1, set flattening 00125 * to 1/\e f. 00126 * @param[in] stdlat standard parallel (degrees), the circle of tangency. 00127 * @param[in] k0 azimuthal scale on the standard parallel. 00128 * 00129 * An exception is thrown if \e a or \e k0 is not positive or if \e stdlat 00130 * is not in the range [-90, 90]. 00131 **********************************************************************/ 00132 AlbersEqualArea(real a, real f, real stdlat, real k0); 00133 00134 /** 00135 * Constructor with two standard parallels. 00136 * 00137 * @param[in] a equatorial radius of ellipsoid (meters). 00138 * @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere. 00139 * Negative \e f gives a prolate ellipsoid. If \e f > 1, set flattening 00140 * to 1/\e f. 00141 * @param[in] stdlat1 first standard parallel (degrees). 00142 * @param[in] stdlat2 second standard parallel (degrees). 00143 * @param[in] k1 azimuthal scale on the standard parallels. 00144 * 00145 * An exception is thrown if \e a or \e k0 is not positive or if \e stdlat1 00146 * or \e stdlat2 is not in the range [-90, 90]. In addition, an exception 00147 * is thrown if \e stdlat1 and \e stdlat2 are opposite poles. 00148 **********************************************************************/ 00149 AlbersEqualArea(real a, real f, real stdlat1, real stdlat2, real k1); 00150 00151 /** 00152 * Constructor with two standard parallels specified by sines and cosines. 00153 * 00154 * @param[in] a equatorial radius of ellipsoid (meters). 00155 * @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere. 00156 * Negative \e f gives a prolate ellipsoid. If \e f > 1, set flattening 00157 * to 1/\e f. 00158 * @param[in] sinlat1 sine of first standard parallel. 00159 * @param[in] coslat1 cosine of first standard parallel. 00160 * @param[in] sinlat2 sine of second standard parallel. 00161 * @param[in] coslat2 cosine of second standard parallel. 00162 * @param[in] k1 azimuthal scale on the standard parallels. 00163 * 00164 * This allows parallels close to the poles to be specified accurately. 00165 * This routine computes the latitude of origin and the azimuthal scale at 00166 * this latitude. If \e dlat = abs(\e lat2 - \e lat1) <= 160<sup>o</sup>, 00167 * then the error in the latitude of origin is less than 00168 * 4.5e-14<sup>o</sup>. 00169 **********************************************************************/ 00170 AlbersEqualArea(real a, real f, 00171 real sinlat1, real coslat1, 00172 real sinlat2, real coslat2, 00173 real k1); 00174 00175 /** 00176 * Set the azimuthal scale for the projection. 00177 * 00178 * @param[in] lat (degrees). 00179 * @param[in] k azimuthal scale at latitude \e lat (default 1). 00180 * 00181 * This allows a "latitude of conformality" to be specified. An exception 00182 * is thrown if \e k is not positive or if \e lat is not in the range (-90, 00183 * 90). 00184 **********************************************************************/ 00185 void SetScale(real lat, real k = real(1)); 00186 00187 /** 00188 * Forward projection, from geographic to Lambert conformal conic. 00189 * 00190 * @param[in] lon0 central meridian longitude (degrees). 00191 * @param[in] lat latitude of point (degrees). 00192 * @param[in] lon longitude of point (degrees). 00193 * @param[out] x easting of point (meters). 00194 * @param[out] y northing of point (meters). 00195 * @param[out] gamma meridian convergence at point (degrees). 00196 * @param[out] k azimuthal scale of projection at point; the radial 00197 * scale is the 1/\e k. 00198 * 00199 * The latitude origin is given by AlbersEqualArea::LatitudeOrigin(). No 00200 * false easting or northing is added and \e lat should be in the range 00201 * [-90, 90]; \e lon and \e lon0 should be in the range [-180, 360]. The 00202 * values of \e x and \e y returned for points which project to infinity 00203 * (i.e., one or both of the poles) will be large but finite. 00204 **********************************************************************/ 00205 void Forward(real lon0, real lat, real lon, 00206 real& x, real& y, real& gamma, real& k) const throw(); 00207 00208 /** 00209 * Reverse projection, from Lambert conformal conic to geographic. 00210 * 00211 * @param[in] lon0 central meridian longitude (degrees). 00212 * @param[in] x easting of point (meters). 00213 * @param[in] y northing of point (meters). 00214 * @param[out] lat latitude of point (degrees). 00215 * @param[out] lon longitude of point (degrees). 00216 * @param[out] gamma meridian convergence at point (degrees). 00217 * @param[out] k azimuthal scale of projection at point; the radial 00218 * scale is the 1/\e k. 00219 * 00220 * The latitude origin is given by AlbersEqualArea::LatitudeOrigin(). No 00221 * false easting or northing is added. \e lon0 should be in the range 00222 * [-180, 360]. The value of \e lon returned is in the range [-180, 180). 00223 * The value of \e lat returned is in the range [-90,90]. If the input 00224 * point is outside the legal projected space the nearest pole is returned. 00225 **********************************************************************/ 00226 void Reverse(real lon0, real x, real y, 00227 real& lat, real& lon, real& gamma, real& k) const throw(); 00228 00229 /** 00230 * AlbersEqualArea::Forward without returning the convergence and 00231 * scale. 00232 **********************************************************************/ 00233 void Forward(real lon0, real lat, real lon, 00234 real& x, real& y) const throw() { 00235 real gamma, k; 00236 Forward(lon0, lat, lon, x, y, gamma, k); 00237 } 00238 00239 /** 00240 * AlbersEqualArea::Reverse without returning the convergence and 00241 * scale. 00242 **********************************************************************/ 00243 void Reverse(real lon0, real x, real y, 00244 real& lat, real& lon) const throw() { 00245 real gamma, k; 00246 Reverse(lon0, x, y, lat, lon, gamma, k); 00247 } 00248 00249 /** \name Inspector functions 00250 **********************************************************************/ 00251 ///@{ 00252 /** 00253 * @return \e a the equatorial radius of the ellipsoid (meters). This is 00254 * the value used in the constructor. 00255 **********************************************************************/ 00256 Math::real MajorRadius() const throw() { return _a; } 00257 00258 /** 00259 * @return \e f the flattening of the ellipsoid. This is the value used in 00260 * the constructor. 00261 **********************************************************************/ 00262 Math::real Flattening() const throw() { return _f; } 00263 00264 /// \cond SKIP 00265 /** 00266 * <b>DEPRECATED</b> 00267 * @return \e r the inverse flattening of the ellipsoid. 00268 **********************************************************************/ 00269 Math::real InverseFlattening() const throw() { return 1/_f; } 00270 /// \endcond 00271 00272 /** 00273 * @return latitude of the origin for the projection (degrees). 00274 * 00275 * This is the latitude of minimum azimuthal scale and equals the \e stdlat 00276 * in the 1-parallel constructor and lies between \e stdlat1 and \e stdlat2 00277 * in the 2-parallel constructors. 00278 **********************************************************************/ 00279 Math::real OriginLatitude() const throw() { return _lat0; } 00280 00281 /** 00282 * @return central scale for the projection. This is the azimuthal scale 00283 * on the latitude of origin. 00284 **********************************************************************/ 00285 Math::real CentralScale() const throw() { return _k0; } 00286 ///@} 00287 00288 /** 00289 * A global instantiation of AlbersEqualArea with the WGS84 ellipsoid, \e 00290 * stdlat = 0, and \e k0 = 1. This degenerates to the cylindrical equal 00291 * area projection. 00292 **********************************************************************/ 00293 static const AlbersEqualArea CylindricalEqualArea; 00294 00295 /** 00296 * A global instantiation of AlbersEqualArea with the WGS84 ellipsoid, \e 00297 * stdlat = 90<sup>o</sup>, and \e k0 = 1. This degenerates to the 00298 * Lambert azimuthal equal area projection. 00299 **********************************************************************/ 00300 static const AlbersEqualArea AzimuthalEqualAreaNorth; 00301 00302 /** 00303 * A global instantiation of AlbersEqualArea with the WGS84 ellipsoid, \e 00304 * stdlat = -90<sup>o</sup>, and \e k0 = 1. This degenerates to the 00305 * Lambert azimuthal equal area projection. 00306 **********************************************************************/ 00307 static const AlbersEqualArea AzimuthalEqualAreaSouth; 00308 }; 00309 00310 } // namespace GeographicLib 00311 00312 #endif // GEOGRAPHICLIB_ALBERSEQUALAREA_HPP