00001 /** 00002 * \file CassiniSoldner.hpp 00003 * \brief Header for GeographicLib::CassiniSoldner class 00004 * 00005 * Copyright (c) Charles Karney (2009-2011) <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_CASSINISOLDNER_HPP) 00011 #define GEOGRAPHICLIB_CASSINISOLDNER_HPP 1 00012 00013 #include <GeographicLib/Geodesic.hpp> 00014 #include <GeographicLib/GeodesicLine.hpp> 00015 #include <GeographicLib/Constants.hpp> 00016 00017 namespace GeographicLib { 00018 00019 /** 00020 * \brief Cassini-Soldner projection 00021 * 00022 * Cassini-Soldner projection centered at an arbitrary position, \e lat0, \e 00023 * lon0, on the ellipsoid. This projection is a transverse cylindrical 00024 * equidistant projection. The projection from (\e lat, \e lon) to easting 00025 * and northing (\e x, \e y) is defined by geodesics as follows. Go north 00026 * along a geodesic a distance \e y from the central point; then turn 00027 * clockwise 90° and go a distance \e x along a geodesic. 00028 * (Although the initial heading is north, this changes to south if the pole 00029 * is crossed.) This procedure uniquely defines the reverse projection. The 00030 * forward projection is constructed as follows. Find the point (\e lat1, \e 00031 * lon1) on the meridian closest to (\e lat, \e lon). Here we consider the 00032 * full meridian so that \e lon1 may be either \e lon0 or \e lon0 + 00033 * 180°. \e x is the geodesic distance from (\e lat1, \e lon1) to 00034 * (\e lat, \e lon), appropriately signed according to which side of the 00035 * central meridian (\e lat, \e lon) lies. \e y is the shortest distance 00036 * along the meridian from (\e lat0, \e lon0) to (\e lat1, \e lon1), again, 00037 * appropriately signed according to the initial heading. [Note that, in the 00038 * case of prolate ellipsoids, the shortest meridional path from (\e lat0, \e 00039 * lon0) to (\e lat1, \e lon1) may not be the shortest path.] This procedure 00040 * uniquely defines the forward projection except for a small class of points 00041 * for which there may be two equally short routes for either leg of the 00042 * path. 00043 * 00044 * Because of the properties of geodesics, the (\e x, \e y) grid is 00045 * orthogonal. The scale in the easting direction is unity. The scale, \e 00046 * k, in the northing direction is unity on the central meridian and 00047 * increases away from the central meridian. The projection routines return 00048 * \e azi, the true bearing of the easting direction, and \e rk = 1/\e k, the 00049 * reciprocal of the scale in the northing direction. 00050 * 00051 * The conversions all take place using a Geodesic object (by default 00052 * Geodesic::WGS84()). For more information on geodesics see \ref geodesic. 00053 * The determination of (\e lat1, \e lon1) in the forward projection is by 00054 * solving the inverse geodesic problem for (\e lat, \e lon) and its twin 00055 * obtained by reflection in the meridional plane. The scale is found by 00056 * determining where two neighboring geodesics intersecting the central 00057 * meridian at \e lat1 and \e lat1 + \e dlat1 intersect and taking the ratio 00058 * of the reduced lengths for the two geodesics between that point and, 00059 * respectively, (\e lat1, \e lon1) and (\e lat, \e lon). 00060 * 00061 * Example of use: 00062 * \include example-CassiniSoldner.cpp 00063 * 00064 * <a href="GeodesicProj.1.html">GeodesicProj</a> is a command-line utility 00065 * providing access to the functionality of AzimuthalEquidistant, Gnomonic, 00066 * and CassiniSoldner. 00067 **********************************************************************/ 00068 00069 class GEOGRAPHICLIB_EXPORT CassiniSoldner { 00070 private: 00071 typedef Math::real real; 00072 real eps1_, tiny_; 00073 Geodesic _earth; 00074 GeodesicLine _meridian; 00075 real _sbet0, _cbet0; 00076 static const unsigned maxit_ = 10; 00077 00078 // The following private helper functions are copied from Geodesic. 00079 static inline real AngRound(real x) { 00080 // The makes the smallest gap in x = 1/16 - nextafter(1/16, 0) = 1/2^57 00081 // for reals = 0.7 pm on the earth if x is an angle in degrees. (This 00082 // is about 1000 times more resolution than we get with angles around 90 00083 // degrees.) We use this to avoid having to deal with near singular 00084 // cases when x is non-zero but tiny (e.g., 1.0e-200). 00085 using std::abs; 00086 const real z = 1/real(16); 00087 GEOGRAPHICLIB_VOLATILE real y = abs(x); 00088 // The compiler mustn't "simplify" z - (z - y) to y 00089 y = y < z ? z - (z - y) : y; 00090 return x < 0 ? -y : y; 00091 } 00092 static inline void SinCosNorm(real& sinx, real& cosx) { 00093 real r = Math::hypot(sinx, cosx); 00094 sinx /= r; 00095 cosx /= r; 00096 } 00097 public: 00098 00099 /** 00100 * Constructor for CassiniSoldner. 00101 * 00102 * @param[in] earth the Geodesic object to use for geodesic calculations. 00103 * By default this uses the WGS84 ellipsoid. 00104 * 00105 * This constructor makes an "uninitialized" object. Call Reset to set the 00106 * central latitude and longitude, prior to calling Forward and Reverse. 00107 **********************************************************************/ 00108 explicit CassiniSoldner(const Geodesic& earth = Geodesic::WGS84()); 00109 00110 /** 00111 * Constructor for CassiniSoldner specifying a center point. 00112 * 00113 * @param[in] lat0 latitude of center point of projection (degrees). 00114 * @param[in] lon0 longitude of center point of projection (degrees). 00115 * @param[in] earth the Geodesic object to use for geodesic calculations. 00116 * By default this uses the WGS84 ellipsoid. 00117 * 00118 * \e lat0 should be in the range [−90°, 90°] and \e 00119 * lon0 should be in the range [−540°, 540°). 00120 **********************************************************************/ 00121 CassiniSoldner(real lat0, real lon0, 00122 const Geodesic& earth = Geodesic::WGS84()); 00123 00124 /** 00125 * Set the central point of the projection 00126 * 00127 * @param[in] lat0 latitude of center point of projection (degrees). 00128 * @param[in] lon0 longitude of center point of projection (degrees). 00129 * 00130 * \e lat0 should be in the range [−90°, 90°] and \e 00131 * lon0 should be in the range [−540°, 540°). 00132 **********************************************************************/ 00133 void Reset(real lat0, real lon0); 00134 00135 /** 00136 * Forward projection, from geographic to Cassini-Soldner. 00137 * 00138 * @param[in] lat latitude of point (degrees). 00139 * @param[in] lon longitude of point (degrees). 00140 * @param[out] x easting of point (meters). 00141 * @param[out] y northing of point (meters). 00142 * @param[out] azi azimuth of easting direction at point (degrees). 00143 * @param[out] rk reciprocal of azimuthal northing scale at point. 00144 * 00145 * \e lat should be in the range [−90°, 90°] and \e 00146 * lon should be in the range [−540°, 540°). A call 00147 * to Forward followed by a call to Reverse will return the original (\e 00148 * lat, \e lon) (to within roundoff). The routine does nothing if the 00149 * origin has not been set. 00150 **********************************************************************/ 00151 void Forward(real lat, real lon, 00152 real& x, real& y, real& azi, real& rk) const; 00153 00154 /** 00155 * Reverse projection, from Cassini-Soldner to geographic. 00156 * 00157 * @param[in] x easting of point (meters). 00158 * @param[in] y northing of point (meters). 00159 * @param[out] lat latitude of point (degrees). 00160 * @param[out] lon longitude of point (degrees). 00161 * @param[out] azi azimuth of easting direction at point (degrees). 00162 * @param[out] rk reciprocal of azimuthal northing scale at point. 00163 * 00164 * A call to Reverse followed by a call to Forward will return the original 00165 * (\e x, \e y) (to within roundoff), provided that \e x and \e y are 00166 * sufficiently small not to "wrap around" the earth. The routine does 00167 * nothing if the origin has not been set. 00168 **********************************************************************/ 00169 void Reverse(real x, real y, 00170 real& lat, real& lon, real& azi, real& rk) const; 00171 00172 /** 00173 * CassiniSoldner::Forward without returning the azimuth and scale. 00174 **********************************************************************/ 00175 void Forward(real lat, real lon, 00176 real& x, real& y) const { 00177 real azi, rk; 00178 Forward(lat, lon, x, y, azi, rk); 00179 } 00180 00181 /** 00182 * CassiniSoldner::Reverse without returning the azimuth and scale. 00183 **********************************************************************/ 00184 void Reverse(real x, real y, 00185 real& lat, real& lon) const { 00186 real azi, rk; 00187 Reverse(x, y, lat, lon, azi, rk); 00188 } 00189 00190 /** \name Inspector functions 00191 **********************************************************************/ 00192 ///@{ 00193 /** 00194 * @return true if the object has been initialized. 00195 **********************************************************************/ 00196 bool Init() const { return _meridian.Init(); } 00197 00198 /** 00199 * @return \e lat0 the latitude of origin (degrees). 00200 **********************************************************************/ 00201 Math::real LatitudeOrigin() const 00202 { return _meridian.Latitude(); } 00203 00204 /** 00205 * @return \e lon0 the longitude of origin (degrees). 00206 **********************************************************************/ 00207 Math::real LongitudeOrigin() const 00208 { return _meridian.Longitude(); } 00209 00210 /** 00211 * @return \e a the equatorial radius of the ellipsoid (meters). This is 00212 * the value inherited from the Geodesic object used in the constructor. 00213 **********************************************************************/ 00214 Math::real MajorRadius() const { return _earth.MajorRadius(); } 00215 00216 /** 00217 * @return \e f the flattening of the ellipsoid. This is the value 00218 * inherited from the Geodesic object used in the constructor. 00219 **********************************************************************/ 00220 Math::real Flattening() const { return _earth.Flattening(); } 00221 ///@} 00222 00223 /// \cond SKIP 00224 /** 00225 * <b>DEPRECATED</b> 00226 * @return \e r the inverse flattening of the ellipsoid. 00227 **********************************************************************/ 00228 Math::real InverseFlattening() const 00229 { return _earth.InverseFlattening(); } 00230 /// \endcond 00231 }; 00232 00233 } // namespace GeographicLib 00234 00235 #endif // GEOGRAPHICLIB_CASSINISOLDNER_HPP