SimGear: SGGeodesy.cxx Source File

00001 // Copyright (C) 2006  Mathias Froehlich - Mathias.Froehlich@web.de
00002 //
00003 // This library is free software; you can redistribute it and/or
00004 // modify it under the terms of the GNU Library General Public
00005 // License as published by the Free Software Foundation; either
00006 // version 2 of the License, or (at your option) any later version.
00007 //
00008 // This library is distributed in the hope that it will be useful,
00009 // but WITHOUT ANY WARRANTY; without even the implied warranty of
00010 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
00011 // Library General Public License for more details.
00012 //
00013 // You should have received a copy of the GNU General Public License
00014 // along with this program; if not, write to the Free Software
00015 // Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.
00016 //
00017 
00018 #ifdef HAVE_CONFIG_H
00019 #  include <simgear_config.h>
00020 #endif
00021 
00022 #include <cmath>
00023 
00024 #include "SGMath.hxx"
00025 
00026 // These are hard numbers from the WGS84 standard.  DON'T MODIFY
00027 // unless you want to change the datum.
00028 #define _EQURAD 6378137.0
00029 #define _FLATTENING 298.257223563
00030 
00031 // These are derived quantities more useful to the code:
00032 #if 0
00033 #define _SQUASH (1 - 1/_FLATTENING)
00034 #define _STRETCH (1/_SQUASH)
00035 #define _POLRAD (EQURAD * _SQUASH)
00036 #else
00037 // High-precision versions of the above produced with an arbitrary
00038 // precision calculator (the compiler might lose a few bits in the FPU
00039 // operations).  These are specified to 81 bits of mantissa, which is
00040 // higher than any FPU known to me:
00041 #define _SQUASH    0.9966471893352525192801545
00042 #define _STRETCH   1.0033640898209764189003079
00043 #define _POLRAD    6356752.3142451794975639668
00044 #endif
00045 
00046 // The constants from the WGS84 standard
00047 const double SGGeodesy::EQURAD = _EQURAD;
00048 const double SGGeodesy::iFLATTENING = _FLATTENING;
00049 const double SGGeodesy::SQUASH = _SQUASH;
00050 const double SGGeodesy::STRETCH = _STRETCH;
00051 const double SGGeodesy::POLRAD = _POLRAD;
00052 
00053 // additional derived and precomputable ones
00054 // for the geodetic conversion algorithm
00055 
00056 #define E2 fabs(1 - _SQUASH*_SQUASH)
00057 static double a = _EQURAD;
00058 static double ra2 = 1/(_EQURAD*_EQURAD);
00059 static double e = sqrt(E2);
00060 static double e2 = E2;
00061 static double e4 = E2*E2;
00062 
00063 #undef _EQURAD
00064 #undef _FLATTENING
00065 #undef _SQUASH
00066 #undef _STRETCH
00067 #undef _POLRAD
00068 #undef E2
00069 
00070 void
00071 SGGeodesy::SGCartToGeod(const SGVec3<double>& cart, SGGeod& geod)
00072 {
00073   // according to
00074   // H. Vermeille,
00075   // Direct transformation from geocentric to geodetic ccordinates,
00076   // Journal of Geodesy (2002) 76:451-454
00077   double X = cart(0);
00078   double Y = cart(1);
00079   double Z = cart(2);
00080   double XXpYY = X*X+Y*Y;
00081   double sqrtXXpYY = sqrt(XXpYY);
00082   double p = XXpYY*ra2;
00083   double q = Z*Z*(1-e2)*ra2;
00084   double r = 1/6.0*(p+q-e4);
00085   double s = e4*p*q/(4*r*r*r);
00086   double t = pow(1+s+sqrt(s*(2+s)), 1/3.0);
00087   double u = r*(1+t+1/t);
00088   double v = sqrt(u*u+e4*q);
00089   double w = e2*(u+v-q)/(2*v);
00090   double k = sqrt(u+v+w*w)-w;
00091   double D = k*sqrtXXpYY/(k+e2);
00092   geod.setLongitudeRad(2*atan2(Y, X+sqrtXXpYY));
00093   double sqrtDDpZZ = sqrt(D*D+Z*Z);
00094   geod.setLatitudeRad(2*atan2(Z, D+sqrtDDpZZ));
00095   geod.setElevationM((k+e2-1)*sqrtDDpZZ/k);
00096 }
00097 
00098 void
00099 SGGeodesy::SGGeodToCart(const SGGeod& geod, SGVec3<double>& cart)
00100 {
00101   // according to
00102   // H. Vermeille,
00103   // Direct transformation from geocentric to geodetic ccordinates,
00104   // Journal of Geodesy (2002) 76:451-454
00105   double lambda = geod.getLongitudeRad();
00106   double phi = geod.getLatitudeRad();
00107   double h = geod.getElevationM();
00108   double sphi = sin(phi);
00109   double n = a/sqrt(1-e2*sphi*sphi);
00110   double cphi = cos(phi);
00111   double slambda = sin(lambda);
00112   double clambda = cos(lambda);
00113   cart(0) = (h+n)*cphi*clambda;
00114   cart(1) = (h+n)*cphi*slambda;
00115   cart(2) = (h+n-e2*n)*sphi;
00116 }
00117 
00118 double
00119 SGGeodesy::SGGeodToSeaLevelRadius(const SGGeod& geod)
00120 {
00121   // this is just a simplified version of the SGGeodToCart function above,
00122   // substitute h = 0, take the 2-norm of the cartesian vector and simplify
00123   double phi = geod.getLatitudeRad();
00124   double sphi = sin(phi);
00125   double sphi2 = sphi*sphi;
00126   return a*sqrt((1 + (e4 - 2*e2)*sphi2)/(1 - e2*sphi2));
00127 }
00128 
00129 void
00130 SGGeodesy::SGCartToGeoc(const SGVec3<double>& cart, SGGeoc& geoc)
00131 {
00132   double minVal = SGLimits<double>::min();
00133   if (fabs(cart(0)) < minVal && fabs(cart(1)) < minVal)
00134     geoc.setLongitudeRad(0);
00135   else
00136     geoc.setLongitudeRad(atan2(cart(1), cart(0)));
00137 
00138   double nxy = sqrt(cart(0)*cart(0) + cart(1)*cart(1));
00139   if (fabs(nxy) < minVal && fabs(cart(2)) < minVal)
00140     geoc.setLatitudeRad(0);
00141   else
00142     geoc.setLatitudeRad(atan2(cart(2), nxy));
00143 
00144   geoc.setRadiusM(norm(cart));
00145 }
00146 
00147 void
00148 SGGeodesy::SGGeocToCart(const SGGeoc& geoc, SGVec3<double>& cart)
00149 {
00150   double lat = geoc.getLatitudeRad();
00151   double lon = geoc.getLongitudeRad();
00152   double slat = sin(lat);
00153   double clat = cos(lat);
00154   double slon = sin(lon);
00155   double clon = cos(lon);
00156   cart = geoc.getRadiusM()*SGVec3<double>(clat*clon, clat*slon, slat);
00157 }
00158 
00159 // Notes:
00160 //
00161 // The XYZ/cartesian coordinate system in use puts the X axis through
00162 // zero lat/lon (off west Africa), the Z axis through the north pole,
00163 // and the Y axis through 90 degrees longitude (in the Indian Ocean).
00164 //
00165 // All latitude and longitude values are in radians.  Altitude is in
00166 // meters, with zero on the WGS84 ellipsoid.
00167 //
00168 // The code below makes use of the notion of "squashed" space.  This
00169 // is a 2D cylindrical coordinate system where the radius from the Z
00170 // axis is multiplied by SQUASH; the earth in this space is a perfect
00171 // circle with a radius of POLRAD.
00172 
00174 //
00175 // Direct and inverse distance functions 
00176 //
00177 // Proceedings of the 7th International Symposium on Geodetic
00178 // Computations, 1985
00179 //
00180 // "The Nested Coefficient Method for Accurate Solutions of Direct and
00181 // Inverse Geodetic Problems With Any Length"
00182 //
00183 // Zhang Xue-Lian
00184 // pp 747-763
00185 //
00186 // modified for FlightGear to use WGS84 only -- Norman Vine
00187 
00188 static inline double M0( double e2 ) {
00189     //double e4 = e2*e2;
00190   return SGMiscd::pi()*0.5*(1.0 - e2*( 1.0/4.0 + e2*( 3.0/64.0 + 
00191                                                   e2*(5.0/256.0) )));
00192 }
00193 
00194 
00195 // given, lat1, lon1, az1 and distance (s), calculate lat2, lon2
00196 // and az2.  Lat, lon, and azimuth are in degrees.  distance in meters
00197 static int _geo_direct_wgs_84 ( double lat1, double lon1, double az1,
00198                         double s, double *lat2, double *lon2,
00199                         double *az2 )
00200 {
00201     double a = SGGeodesy::EQURAD, rf = SGGeodesy::iFLATTENING;
00202     double testv = 1.0E-10;
00203     double f = ( rf > 0.0 ? 1.0/rf : 0.0 );
00204     double b = a*(1.0-f);
00205     double e2 = f*(2.0-f);
00206     double phi1 = SGMiscd::deg2rad(lat1), lam1 = SGMiscd::deg2rad(lon1);
00207     double sinphi1 = sin(phi1), cosphi1 = cos(phi1);
00208     double azm1 = SGMiscd::deg2rad(az1);
00209     double sinaz1 = sin(azm1), cosaz1 = cos(azm1);
00210         
00211         
00212     if( fabs(s) < 0.01 ) {      // distance < centimeter => congruency
00213         *lat2 = lat1;
00214         *lon2 = lon1;
00215         *az2 = 180.0 + az1;
00216         if( *az2 > 360.0 ) *az2 -= 360.0;
00217         return 0;
00218     } else if( SGLimitsd::min() < fabs(cosphi1) ) {     // non-polar origin
00219         // u1 is reduced latitude
00220         double tanu1 = sqrt(1.0-e2)*sinphi1/cosphi1;
00221         double sig1 = atan2(tanu1,cosaz1);
00222         double cosu1 = 1.0/sqrt( 1.0 + tanu1*tanu1 ), sinu1 = tanu1*cosu1;
00223         double sinaz =  cosu1*sinaz1, cos2saz = 1.0-sinaz*sinaz;
00224         double us = cos2saz*e2/(1.0-e2);
00225 
00226         // Terms
00227         double  ta = 1.0+us*(4096.0+us*(-768.0+us*(320.0-175.0*us)))/16384.0,
00228             tb = us*(256.0+us*(-128.0+us*(74.0-47.0*us)))/1024.0,
00229             tc = 0;
00230 
00231         // FIRST ESTIMATE OF SIGMA (SIG)
00232         double first = s/(b*ta);  // !!
00233         double sig = first;
00234         double c2sigm, sinsig,cossig, temp,denom,rnumer, dlams, dlam;
00235         do {
00236             c2sigm = cos(2.0*sig1+sig);
00237             sinsig = sin(sig); cossig = cos(sig);
00238             temp = sig;
00239             sig = first + 
00240                 tb*sinsig*(c2sigm+tb*(cossig*(-1.0+2.0*c2sigm*c2sigm) - 
00241                                       tb*c2sigm*(-3.0+4.0*sinsig*sinsig)
00242                                       *(-3.0+4.0*c2sigm*c2sigm)/6.0)
00243                            /4.0);
00244         } while( fabs(sig-temp) > testv);
00245 
00246         // LATITUDE OF POINT 2
00247         // DENOMINATOR IN 2 PARTS (TEMP ALSO USED LATER)
00248         temp = sinu1*sinsig-cosu1*cossig*cosaz1;
00249         denom = (1.0-f)*sqrt(sinaz*sinaz+temp*temp);
00250 
00251         // NUMERATOR
00252         rnumer = sinu1*cossig+cosu1*sinsig*cosaz1;
00253         *lat2 = SGMiscd::rad2deg(atan2(rnumer,denom));
00254 
00255         // DIFFERENCE IN LONGITUDE ON AUXILARY SPHERE (DLAMS )
00256         rnumer = sinsig*sinaz1;
00257         denom = cosu1*cossig-sinu1*sinsig*cosaz1;
00258         dlams = atan2(rnumer,denom);
00259 
00260         // TERM C
00261         tc = f*cos2saz*(4.0+f*(4.0-3.0*cos2saz))/16.0;
00262 
00263         // DIFFERENCE IN LONGITUDE
00264         dlam = dlams-(1.0-tc)*f*sinaz*(sig+tc*sinsig*
00265                                        (c2sigm+
00266                                         tc*cossig*(-1.0+2.0*
00267                                                    c2sigm*c2sigm)));
00268         *lon2 = SGMiscd::rad2deg(lam1+dlam);
00269         if (*lon2 > 180.0  ) *lon2 -= 360.0;
00270         if (*lon2 < -180.0 ) *lon2 += 360.0;
00271 
00272         // AZIMUTH - FROM NORTH
00273         *az2 = SGMiscd::rad2deg(atan2(-sinaz,temp));
00274         if ( fabs(*az2) < testv ) *az2 = 0.0;
00275         if( *az2 < 0.0) *az2 += 360.0;
00276         return 0;
00277     } else {                    // phi1 == 90 degrees, polar origin
00278         double dM = a*M0(e2) - s;
00279         double paz = ( phi1 < 0.0 ? 180.0 : 0.0 );
00280         double zero = 0.0f;
00281         return _geo_direct_wgs_84( zero, lon1, paz, dM, lat2, lon2, az2 );
00282     } 
00283 }
00284 
00285 bool
00286 SGGeodesy::direct(const SGGeod& p1, double course1,
00287                   double distance, SGGeod& p2, double& course2)
00288 {
00289   double lat2, lon2;
00290   int ret = _geo_direct_wgs_84(p1.getLatitudeDeg(), p1.getLongitudeDeg(),
00291                                course1, distance, &lat2, &lon2, &course2);
00292   p2.setLatitudeDeg(lat2);
00293   p2.setLongitudeDeg(lon2);
00294   p2.setElevationM(0);
00295   return ret == 0;
00296 }
00297 
00298 // given lat1, lon1, lat2, lon2, calculate starting and ending
00299 // az1, az2 and distance (s).  Lat, lon, and azimuth are in degrees.
00300 // distance in meters
00301 static int _geo_inverse_wgs_84( double lat1, double lon1, double lat2,
00302                         double lon2, double *az1, double *az2,
00303                         double *s )
00304 {
00305     double a = SGGeodesy::EQURAD, rf = SGGeodesy::iFLATTENING;
00306     int iter=0;
00307     double testv = 1.0E-10;
00308     double f = ( rf > 0.0 ? 1.0/rf : 0.0 );
00309     double b = a*(1.0-f);
00310     // double e2 = f*(2.0-f); // unused in this routine
00311     double phi1 = SGMiscd::deg2rad(lat1), lam1 = SGMiscd::deg2rad(lon1);
00312     double sinphi1 = sin(phi1), cosphi1 = cos(phi1);
00313     double phi2 = SGMiscd::deg2rad(lat2), lam2 = SGMiscd::deg2rad(lon2);
00314     double sinphi2 = sin(phi2), cosphi2 = cos(phi2);
00315         
00316     if( (fabs(lat1-lat2) < testv && 
00317          ( fabs(lon1-lon2) < testv) || fabs(lat1-90.0) < testv ) )
00318     {   
00319         // TWO STATIONS ARE IDENTICAL : SET DISTANCE & AZIMUTHS TO ZERO */
00320         *az1 = 0.0; *az2 = 0.0; *s = 0.0;
00321         return 0;
00322     } else if(  fabs(cosphi1) < testv ) {
00323         // initial point is polar
00324         int k = _geo_inverse_wgs_84( lat2,lon2,lat1,lon1, az1,az2,s );
00325         k = k; // avoid compiler error since return result is unused
00326         b = *az1; *az1 = *az2; *az2 = b;
00327         return 0;
00328     } else if( fabs(cosphi2) < testv ) {
00329         // terminal point is polar
00330         double _lon1 = lon1 + 180.0f;
00331         int k = _geo_inverse_wgs_84( lat1, lon1, lat1, _lon1, 
00332                                     az1, az2, s );
00333         k = k; // avoid compiler error since return result is unused
00334         *s /= 2.0;
00335         *az2 = *az1 + 180.0;
00336         if( *az2 > 360.0 ) *az2 -= 360.0; 
00337         return 0;
00338     } else if( (fabs( fabs(lon1-lon2) - 180 ) < testv) && 
00339                (fabs(lat1+lat2) < testv) ) 
00340     {
00341         // Geodesic passes through the pole (antipodal)
00342         double s1,s2;
00343         _geo_inverse_wgs_84( lat1,lon1, lat1,lon2, az1,az2, &s1 );
00344         _geo_inverse_wgs_84( lat2,lon2, lat1,lon2, az1,az2, &s2 );
00345         *az2 = *az1;
00346         *s = s1 + s2;
00347         return 0;
00348     } else {
00349         // antipodal and polar points don't get here
00350         double dlam = lam2 - lam1, dlams = dlam;
00351         double sdlams,cdlams, sig,sinsig,cossig, sinaz,
00352             cos2saz, c2sigm;
00353         double tc,temp, us,rnumer,denom, ta,tb;
00354         double cosu1,sinu1, sinu2,cosu2;
00355 
00356         // Reduced latitudes
00357         temp = (1.0-f)*sinphi1/cosphi1;
00358         cosu1 = 1.0/sqrt(1.0+temp*temp);
00359         sinu1 = temp*cosu1;
00360         temp = (1.0-f)*sinphi2/cosphi2;
00361         cosu2 = 1.0/sqrt(1.0+temp*temp);
00362         sinu2 = temp*cosu2;
00363     
00364         do {
00365             sdlams = sin(dlams), cdlams = cos(dlams);
00366             sinsig = sqrt(cosu2*cosu2*sdlams*sdlams+
00367                           (cosu1*sinu2-sinu1*cosu2*cdlams)*
00368                           (cosu1*sinu2-sinu1*cosu2*cdlams));
00369             cossig = sinu1*sinu2+cosu1*cosu2*cdlams;
00370             
00371             sig = atan2(sinsig,cossig);
00372             sinaz = cosu1*cosu2*sdlams/sinsig;
00373             cos2saz = 1.0-sinaz*sinaz;
00374             c2sigm = (sinu1 == 0.0 || sinu2 == 0.0 ? cossig : 
00375                       cossig-2.0*sinu1*sinu2/cos2saz);
00376             tc = f*cos2saz*(4.0+f*(4.0-3.0*cos2saz))/16.0;
00377             temp = dlams;
00378             dlams = dlam+(1.0-tc)*f*sinaz*
00379                 (sig+tc*sinsig*
00380                  (c2sigm+tc*cossig*(-1.0+2.0*c2sigm*c2sigm)));
00381             if (fabs(dlams) > SGMiscd::pi() && iter++ > 50) {
00382                 return iter;
00383             }
00384         } while ( fabs(temp-dlams) > testv);
00385 
00386         us = cos2saz*(a*a-b*b)/(b*b); // !!
00387         // BACK AZIMUTH FROM NORTH
00388         rnumer = -(cosu1*sdlams);
00389         denom = sinu1*cosu2-cosu1*sinu2*cdlams;
00390         *az2 = SGMiscd::rad2deg(atan2(rnumer,denom));
00391         if( fabs(*az2) < testv ) *az2 = 0.0;
00392         if(*az2 < 0.0) *az2 += 360.0;
00393 
00394         // FORWARD AZIMUTH FROM NORTH
00395         rnumer = cosu2*sdlams;
00396         denom = cosu1*sinu2-sinu1*cosu2*cdlams;
00397         *az1 = SGMiscd::rad2deg(atan2(rnumer,denom));
00398         if( fabs(*az1) < testv ) *az1 = 0.0;
00399         if(*az1 < 0.0) *az1 += 360.0;
00400 
00401         // Terms a & b
00402         ta = 1.0+us*(4096.0+us*(-768.0+us*(320.0-175.0*us)))/
00403             16384.0;
00404         tb = us*(256.0+us*(-128.0+us*(74.0-47.0*us)))/1024.0;
00405 
00406         // GEODETIC DISTANCE
00407         *s = b*ta*(sig-tb*sinsig*
00408                    (c2sigm+tb*(cossig*(-1.0+2.0*c2sigm*c2sigm)-tb*
00409                                c2sigm*(-3.0+4.0*sinsig*sinsig)*
00410                                (-3.0+4.0*c2sigm*c2sigm)/6.0)/
00411                     4.0));
00412         return 0;
00413     }
00414 }
00415 
00416 bool
00417 SGGeodesy::inverse(const SGGeod& p1, const SGGeod& p2, double& course1,
00418                    double& course2, double& distance)
00419 {
00420   int ret = _geo_inverse_wgs_84(p1.getLatitudeDeg(), p1.getLongitudeDeg(),
00421                                 p2.getLatitudeDeg(), p2.getLongitudeDeg(),
00422                                 &course1, &course2, &distance);
00423   return ret == 0;
00424 }
00425 
00427 
00428 void
00429 SGGeodesy::advanceRadM(const SGGeoc& geoc, double course, double distance,
00430                        SGGeoc& result)
00431 {
00432   result.setRadiusM(geoc.getRadiusM());
00433 
00434   // lat=asin(sin(lat1)*cos(d)+cos(lat1)*sin(d)*cos(tc))
00435   // IF (cos(lat)=0)
00436   //   lon=lon1      // endpoint a pole
00437   // ELSE
00438   //   lon=mod(lon1-asin(sin(tc)*sin(d)/cos(lat))+pi,2*pi)-pi
00439   // ENDIF
00440   
00441   distance *= SG_METER_TO_NM * SG_NM_TO_RAD;
00442   
00443   double sinDistance = sin(distance);
00444   double cosDistance = cos(distance);
00445 
00446   double sinLat = sin(geoc.getLatitudeRad()) * cosDistance +
00447     cos(geoc.getLatitudeRad()) * sinDistance * cos(course);
00448   sinLat = SGMiscd::clip(sinLat, -1, 1);
00449   result.setLatitudeRad(asin(sinLat));
00450   double cosLat = cos(result.getLatitudeRad());
00451   
00452   
00453   if (cosLat <= SGLimitsd::min()) {
00454     // endpoint a pole
00455     result.setLongitudeRad(geoc.getLongitudeRad());
00456   } else {
00457     double tmp = SGMiscd::clip(sin(course) * sinDistance / cosLat, -1, 1);
00458     double lon = SGMiscd::normalizeAngle(geoc.getLongitudeRad() - asin( tmp ));
00459     result.setLongitudeRad(lon);
00460   }
00461 }
00462 
00463 double
00464 SGGeodesy::courseRad(const SGGeoc& from, const SGGeoc& to)
00465 {
00466   double diffLon = to.getLongitudeRad() - from.getLongitudeRad();
00467 
00468   double sinLatFrom = sin(from.getLatitudeRad());
00469   double cosLatFrom = cos(from.getLatitudeRad());
00470 
00471   double sinLatTo = sin(to.getLatitudeRad());
00472   double cosLatTo = cos(to.getLatitudeRad());
00473 
00474   double x = cosLatTo*sin(diffLon);
00475   double y = cosLatFrom*sinLatTo - sinLatFrom*cosLatTo*cos(diffLon);
00476 
00477   // guard atan2 returning NaN's
00478   if (fabs(x) <= SGLimitsd::min() && fabs(y) <= SGLimitsd::min())
00479     return 0;
00480 
00481   double c = atan2(x, y);
00482   if (c >= 0)
00483     return SGMiscd::twopi() - c;
00484   else
00485     return -c;
00486 }
00487 
00488 double
00489 SGGeodesy::distanceM(const SGGeoc& from, const SGGeoc& to)
00490 {
00491   // d = 2*asin(sqrt((sin((lat1-lat2)/2))^2 +
00492   //            cos(lat1)*cos(lat2)*(sin((lon1-lon2)/2))^2))
00493   double cosLatFrom = cos(from.getLatitudeRad());
00494   double cosLatTo = cos(to.getLatitudeRad());
00495   double tmp1 = sin(0.5*(from.getLatitudeRad() - to.getLatitudeRad()));
00496   double tmp2 = sin(0.5*(from.getLongitudeRad() - to.getLongitudeRad()));
00497   double square = tmp1*tmp1 + cosLatFrom*cosLatTo*tmp2*tmp2;
00498   double s = SGMiscd::min(sqrt(SGMiscd::max(square, 0)), 1);
00499   return 2 * asin(s) * SG_RAD_TO_NM * SG_NM_TO_METER;
00500 }