SGGeodesy.cxx

// Copyright (C) 2006  Mathias Froehlich - Mathias.Froehlich@web.de
//
// This library is free software; you can redistribute it and/or
// modify it under the terms of the GNU Library General Public
// License as published by the Free Software Foundation; either
// version 2 of the License, or (at your option) any later version.
//
// This library is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
// Library General Public License for more details.
//
// You should have received a copy of the GNU General Public License
// along with this program; if not, write to the Free Software
// Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.
//

#ifdef HAVE_CONFIG_H
#  include <simgear_config.h>
#endif

#include <cmath>

#include <simgear/structure/exception.hxx>
#include "SGMath.hxx"

// These are hard numbers from the WGS84 standard.  DON'T MODIFY
// unless you want to change the datum.
#define _EQURAD 6378137.0
#define _FLATTENING 298.257223563

// These are derived quantities more useful to the code:
#if 0
#define _SQUASH (1 - 1/_FLATTENING)
#define _STRETCH (1/_SQUASH)
#define _POLRAD (EQURAD * _SQUASH)
#else
// High-precision versions of the above produced with an arbitrary
// precision calculator (the compiler might lose a few bits in the FPU
// operations).  These are specified to 81 bits of mantissa, which is
// higher than any FPU known to me:
#define _SQUASH    0.9966471893352525192801545
#define _STRETCH   1.0033640898209764189003079
#define _POLRAD    6356752.3142451794975639668
#endif

// The constants from the WGS84 standard
const double SGGeodesy::EQURAD = _EQURAD;
const double SGGeodesy::iFLATTENING = _FLATTENING;
const double SGGeodesy::SQUASH = _SQUASH;
const double SGGeodesy::STRETCH = _STRETCH;
const double SGGeodesy::POLRAD = _POLRAD;

// additional derived and precomputable ones
// for the geodetic conversion algorithm

#define E2 fabs(1 - _SQUASH*_SQUASH)
static double a = _EQURAD;
static double ra2 = 1/(_EQURAD*_EQURAD);
//static double e = sqrt(E2);
static double e2 = E2;
static double e4 = E2*E2;

#undef _EQURAD
#undef _FLATTENING
#undef _SQUASH
#undef _STRETCH
#undef _POLRAD
#undef E2

void
SGGeodesy::SGCartToGeod(const SGVec3<double>& cart, SGGeod& geod)
{
  // according to
  // H. Vermeille,
  // Direct transformation from geocentric to geodetic ccordinates,
  // Journal of Geodesy (2002) 76:451-454
  double X = cart(0);
  double Y = cart(1);
  double Z = cart(2);
  double XXpYY = X*X+Y*Y;
  if( XXpYY + Z*Z < 25 ) {
    // This function fails near the geocenter region, so catch that special case here.
    // Define the innermost sphere of small radius as earth center and return the 
    // coordinates 0/0/-EQURAD. It may be any other place on geoide's surface,
    // the Northpole, Hawaii or Wentorf. This one was easy to code ;-)
    geod.setLongitudeRad( 0.0 );
    geod.setLongitudeRad( 0.0 );
    geod.setElevationM( -EQURAD );
    return;
  }
    
  double sqrtXXpYY = sqrt(XXpYY);
  double p = XXpYY*ra2;
  double q = Z*Z*(1-e2)*ra2;
  double r = 1/6.0*(p+q-e4);
  double s = e4*p*q/(4*r*r*r);
/* 
  s*(2+s) is negative for s = [-2..0]
  slightly negative values for s due to floating point rounding errors
  cause nan for sqrt(s*(2+s))
  We can probably clamp the resulting parable to positive numbers
*/
  if( s >= -2.0 && s <= 0.0 )
    s = 0.0;
  double t = pow(1+s+sqrt(s*(2+s)), 1/3.0);
  double u = r*(1+t+1/t);
  double v = sqrt(u*u+e4*q);
  double w = e2*(u+v-q)/(2*v);
  double k = sqrt(u+v+w*w)-w;
  double D = k*sqrtXXpYY/(k+e2);
  geod.setLongitudeRad(2*atan2(Y, X+sqrtXXpYY));
  double sqrtDDpZZ = sqrt(D*D+Z*Z);
  geod.setLatitudeRad(2*atan2(Z, D+sqrtDDpZZ));
  geod.setElevationM((k+e2-1)*sqrtDDpZZ/k);
}

void
SGGeodesy::SGGeodToCart(const SGGeod& geod, SGVec3<double>& cart)
{
  // according to
  // H. Vermeille,
  // Direct transformation from geocentric to geodetic ccordinates,
  // Journal of Geodesy (2002) 76:451-454
  double lambda = geod.getLongitudeRad();
  double phi = geod.getLatitudeRad();
  double h = geod.getElevationM();
  double sphi = sin(phi);
  double n = a/sqrt(1-e2*sphi*sphi);
  double cphi = cos(phi);
  double slambda = sin(lambda);
  double clambda = cos(lambda);
  cart(0) = (h+n)*cphi*clambda;
  cart(1) = (h+n)*cphi*slambda;
  cart(2) = (h+n-e2*n)*sphi;
}

double
SGGeodesy::SGGeodToSeaLevelRadius(const SGGeod& geod)
{
  // this is just a simplified version of the SGGeodToCart function above,
  // substitute h = 0, take the 2-norm of the cartesian vector and simplify
  double phi = geod.getLatitudeRad();
  double sphi = sin(phi);
  double sphi2 = sphi*sphi;
  return a*sqrt((1 + (e4 - 2*e2)*sphi2)/(1 - e2*sphi2));
}

void
SGGeodesy::SGCartToGeoc(const SGVec3<double>& cart, SGGeoc& geoc)
{
  double minVal = SGLimits<double>::min();
  if (fabs(cart(0)) < minVal && fabs(cart(1)) < minVal)
    geoc.setLongitudeRad(0);
  else
    geoc.setLongitudeRad(atan2(cart(1), cart(0)));

  double nxy = sqrt(cart(0)*cart(0) + cart(1)*cart(1));
  if (fabs(nxy) < minVal && fabs(cart(2)) < minVal)
    geoc.setLatitudeRad(0);
  else
    geoc.setLatitudeRad(atan2(cart(2), nxy));

  geoc.setRadiusM(norm(cart));
}

void
SGGeodesy::SGGeocToCart(const SGGeoc& geoc, SGVec3<double>& cart)
{
  double lat = geoc.getLatitudeRad();
  double lon = geoc.getLongitudeRad();
  double slat = sin(lat);
  double clat = cos(lat);
  double slon = sin(lon);
  double clon = cos(lon);
  cart = geoc.getRadiusM()*SGVec3<double>(clat*clon, clat*slon, slat);
}

// Notes:
//
// The XYZ/cartesian coordinate system in use puts the X axis through
// zero lat/lon (off west Africa), the Z axis through the north pole,
// and the Y axis through 90 degrees longitude (in the Indian Ocean).
//
// All latitude and longitude values are in radians.  Altitude is in
// meters, with zero on the WGS84 ellipsoid.
//
// The code below makes use of the notion of "squashed" space.  This
// is a 2D cylindrical coordinate system where the radius from the Z
// axis is multiplied by SQUASH; the earth in this space is a perfect
// circle with a radius of POLRAD.

//
// Direct and inverse distance functions 
//
// Proceedings of the 7th International Symposium on Geodetic
// Computations, 1985
//
// "The Nested Coefficient Method for Accurate Solutions of Direct and
// Inverse Geodetic Problems With Any Length"
//
// Zhang Xue-Lian
// pp 747-763
//
// modified for FlightGear to use WGS84 only -- Norman Vine

static inline double M0( double e2 ) {
    //double e4 = e2*e2;
  return SGMiscd::pi()*0.5*(1.0 - e2*( 1.0/4.0 + e2*( 3.0/64.0 + 
                                                  e2*(5.0/256.0) )));
}


// given, lat1, lon1, az1 and distance (s), calculate lat2, lon2
// and az2.  Lat, lon, and azimuth are in degrees.  distance in meters
static int _geo_direct_wgs_84 ( double lat1, double lon1, double az1,
                        double s, double *lat2, double *lon2,
                        double *az2 )
{
    double a = SGGeodesy::EQURAD, rf = SGGeodesy::iFLATTENING;
    double testv = 1.0E-10;
    double f = ( rf > 0.0 ? 1.0/rf : 0.0 );
    double b = a*(1.0-f);
    double e2 = f*(2.0-f);
    double phi1 = SGMiscd::deg2rad(lat1), lam1 = SGMiscd::deg2rad(lon1);
    double sinphi1 = sin(phi1), cosphi1 = cos(phi1);
    double azm1 = SGMiscd::deg2rad(az1);
    double sinaz1 = sin(azm1), cosaz1 = cos(azm1);
        
        
    if( fabs(s) < 0.01 ) {      // distance < centimeter => congruency
        *lat2 = lat1;
        *lon2 = lon1;
        *az2 = 180.0 + az1;
        if( *az2 > 360.0 ) *az2 -= 360.0;
        return 0;
    } else if( SGLimitsd::min() < fabs(cosphi1) ) {     // non-polar origin
        // u1 is reduced latitude
        double tanu1 = sqrt(1.0-e2)*sinphi1/cosphi1;
        double sig1 = atan2(tanu1,cosaz1);
        double cosu1 = 1.0/sqrt( 1.0 + tanu1*tanu1 ), sinu1 = tanu1*cosu1;
        double sinaz =  cosu1*sinaz1, cos2saz = 1.0-sinaz*sinaz;
        double us = cos2saz*e2/(1.0-e2);

        // Terms
        double  ta = 1.0+us*(4096.0+us*(-768.0+us*(320.0-175.0*us)))/16384.0,
            tb = us*(256.0+us*(-128.0+us*(74.0-47.0*us)))/1024.0,
            tc = 0;

        // FIRST ESTIMATE OF SIGMA (SIG)
        double first = s/(b*ta);  // !!
        double sig = first;
        double c2sigm, sinsig,cossig, temp,denom,rnumer, dlams, dlam;
        do {
            c2sigm = cos(2.0*sig1+sig);
            sinsig = sin(sig); cossig = cos(sig);
            temp = sig;
            sig = first + 
                tb*sinsig*(c2sigm+tb*(cossig*(-1.0+2.0*c2sigm*c2sigm) - 
                                      tb*c2sigm*(-3.0+4.0*sinsig*sinsig)
                                      *(-3.0+4.0*c2sigm*c2sigm)/6.0)
                           /4.0);
        } while( fabs(sig-temp) > testv);

        // LATITUDE OF POINT 2
        // DENOMINATOR IN 2 PARTS (TEMP ALSO USED LATER)
        temp = sinu1*sinsig-cosu1*cossig*cosaz1;
        denom = (1.0-f)*sqrt(sinaz*sinaz+temp*temp);

        // NUMERATOR
        rnumer = sinu1*cossig+cosu1*sinsig*cosaz1;
        *lat2 = SGMiscd::rad2deg(atan2(rnumer,denom));

        // DIFFERENCE IN LONGITUDE ON AUXILARY SPHERE (DLAMS )
        rnumer = sinsig*sinaz1;
        denom = cosu1*cossig-sinu1*sinsig*cosaz1;
        dlams = atan2(rnumer,denom);

        // TERM C
        tc = f*cos2saz*(4.0+f*(4.0-3.0*cos2saz))/16.0;

        // DIFFERENCE IN LONGITUDE
        dlam = dlams-(1.0-tc)*f*sinaz*(sig+tc*sinsig*
                                       (c2sigm+
                                        tc*cossig*(-1.0+2.0*
                                                   c2sigm*c2sigm)));
        *lon2 = SGMiscd::rad2deg(lam1+dlam);
        if (*lon2 > 180.0  ) *lon2 -= 360.0;
        if (*lon2 < -180.0 ) *lon2 += 360.0;

        // AZIMUTH - FROM NORTH
        *az2 = SGMiscd::rad2deg(atan2(-sinaz,temp));
        if ( fabs(*az2) < testv ) *az2 = 0.0;
        if( *az2 < 0.0) *az2 += 360.0;
        return 0;
    } else {                    // phi1 == 90 degrees, polar origin
        double dM = a*M0(e2) - s;
        double paz = ( phi1 < 0.0 ? 180.0 : 0.0 );
        double zero = 0.0f;
        return _geo_direct_wgs_84( zero, lon1, paz, dM, lat2, lon2, az2 );
    } 
}

bool
SGGeodesy::direct(const SGGeod& p1, double course1,
                  double distance, SGGeod& p2, double& course2)
{
  double lat2, lon2;
  int ret = _geo_direct_wgs_84(p1.getLatitudeDeg(), p1.getLongitudeDeg(),
                               course1, distance, &lat2, &lon2, &course2);
  p2.setLatitudeDeg(lat2);
  p2.setLongitudeDeg(lon2);
  p2.setElevationM(0);
  return ret == 0;
}

// given lat1, lon1, lat2, lon2, calculate starting and ending
// az1, az2 and distance (s).  Lat, lon, and azimuth are in degrees.
// distance in meters
static int _geo_inverse_wgs_84( double lat1, double lon1, double lat2,
                        double lon2, double *az1, double *az2,
                        double *s )
{
    double a = SGGeodesy::EQURAD, rf = SGGeodesy::iFLATTENING;
    int iter=0;
    double testv = 1.0E-10;
    double f = ( rf > 0.0 ? 1.0/rf : 0.0 );
    double b = a*(1.0-f);
    // double e2 = f*(2.0-f); // unused in this routine
    double phi1 = SGMiscd::deg2rad(lat1), lam1 = SGMiscd::deg2rad(lon1);
    double sinphi1 = sin(phi1), cosphi1 = cos(phi1);
    double phi2 = SGMiscd::deg2rad(lat2), lam2 = SGMiscd::deg2rad(lon2);
    double sinphi2 = sin(phi2), cosphi2 = cos(phi2);
        
    if( (fabs(lat1-lat2) < testv && 
         ( fabs(lon1-lon2) < testv)) || (fabs(lat1-90.0) < testv ) )
    {   
        // TWO STATIONS ARE IDENTICAL : SET DISTANCE & AZIMUTHS TO ZERO */
        *az1 = 0.0; *az2 = 0.0; *s = 0.0;
        return 0;
    } else if(  fabs(cosphi1) < testv ) {
        // initial point is polar
        int k = _geo_inverse_wgs_84( lat2,lon2,lat1,lon1, az1,az2,s );
        k = k; // avoid compiler error since return result is unused
        b = *az1; *az1 = *az2; *az2 = b;
        return 0;
    } else if( fabs(cosphi2) < testv ) {
        // terminal point is polar
        double _lon1 = lon1 + 180.0f;
        int k = _geo_inverse_wgs_84( lat1, lon1, lat1, _lon1, 
                                    az1, az2, s );
        k = k; // avoid compiler error since return result is unused
        *s /= 2.0;
        *az2 = *az1 + 180.0;
        if( *az2 > 360.0 ) *az2 -= 360.0; 
        return 0;
    } else if( (fabs( fabs(lon1-lon2) - 180 ) < testv) && 
               (fabs(lat1+lat2) < testv) ) 
    {
        // Geodesic passes through the pole (antipodal)
        double s1,s2;
        _geo_inverse_wgs_84( lat1,lon1, lat1,lon2, az1,az2, &s1 );
        _geo_inverse_wgs_84( lat2,lon2, lat1,lon2, az1,az2, &s2 );
        *az2 = *az1;
        *s = s1 + s2;
        return 0;
    } else {
        // antipodal and polar points don't get here
        double dlam = lam2 - lam1, dlams = dlam;
        double sdlams,cdlams, sig,sinsig,cossig, sinaz,
            cos2saz, c2sigm;
        double tc,temp, us,rnumer,denom, ta,tb;
        double cosu1,sinu1, sinu2,cosu2;

        // Reduced latitudes
        temp = (1.0-f)*sinphi1/cosphi1;
        cosu1 = 1.0/sqrt(1.0+temp*temp);
        sinu1 = temp*cosu1;
        temp = (1.0-f)*sinphi2/cosphi2;
        cosu2 = 1.0/sqrt(1.0+temp*temp);
        sinu2 = temp*cosu2;
    
        do {
            sdlams = sin(dlams), cdlams = cos(dlams);
            sinsig = sqrt(cosu2*cosu2*sdlams*sdlams+
                          (cosu1*sinu2-sinu1*cosu2*cdlams)*
                          (cosu1*sinu2-sinu1*cosu2*cdlams));
            cossig = sinu1*sinu2+cosu1*cosu2*cdlams;
            
            sig = atan2(sinsig,cossig);
            sinaz = cosu1*cosu2*sdlams/sinsig;
            cos2saz = 1.0-sinaz*sinaz;
            c2sigm = (sinu1 == 0.0 || sinu2 == 0.0 ? cossig : 
                      cossig-2.0*sinu1*sinu2/cos2saz);
            tc = f*cos2saz*(4.0+f*(4.0-3.0*cos2saz))/16.0;
            temp = dlams;
            dlams = dlam+(1.0-tc)*f*sinaz*
                (sig+tc*sinsig*
                 (c2sigm+tc*cossig*(-1.0+2.0*c2sigm*c2sigm)));
            if (fabs(dlams) > SGMiscd::pi() && iter++ > 50) {
                return iter;
            }
        } while ( fabs(temp-dlams) > testv);

        us = cos2saz*(a*a-b*b)/(b*b); // !!
        // BACK AZIMUTH FROM NORTH
        rnumer = -(cosu1*sdlams);
        denom = sinu1*cosu2-cosu1*sinu2*cdlams;
        *az2 = SGMiscd::rad2deg(atan2(rnumer,denom));
        if( fabs(*az2) < testv ) *az2 = 0.0;
        if(*az2 < 0.0) *az2 += 360.0;

        // FORWARD AZIMUTH FROM NORTH
        rnumer = cosu2*sdlams;
        denom = cosu1*sinu2-sinu1*cosu2*cdlams;
        *az1 = SGMiscd::rad2deg(atan2(rnumer,denom));
        if( fabs(*az1) < testv ) *az1 = 0.0;
        if(*az1 < 0.0) *az1 += 360.0;

        // Terms a & b
        ta = 1.0+us*(4096.0+us*(-768.0+us*(320.0-175.0*us)))/
            16384.0;
        tb = us*(256.0+us*(-128.0+us*(74.0-47.0*us)))/1024.0;

        // GEODETIC DISTANCE
        *s = b*ta*(sig-tb*sinsig*
                   (c2sigm+tb*(cossig*(-1.0+2.0*c2sigm*c2sigm)-tb*
                               c2sigm*(-3.0+4.0*sinsig*sinsig)*
                               (-3.0+4.0*c2sigm*c2sigm)/6.0)/
                    4.0));
        return 0;
    }
}

bool
SGGeodesy::inverse(const SGGeod& p1, const SGGeod& p2, double& course1,
                   double& course2, double& distance)
{
  int ret = _geo_inverse_wgs_84(p1.getLatitudeDeg(), p1.getLongitudeDeg(),
                                p2.getLatitudeDeg(), p2.getLongitudeDeg(),
                                &course1, &course2, &distance);
  return ret == 0;
}

double
SGGeodesy::courseDeg(const SGGeod& p1, const SGGeod& p2)
{
  double course1, course2, distance;
  int r = _geo_inverse_wgs_84(p1.getLatitudeDeg(), p1.getLongitudeDeg(),
                                p2.getLatitudeDeg(), p2.getLongitudeDeg(),
                                &course1, &course2, &distance);
  if (r != 0) {
    throw sg_exception("SGGeodesy::courseDeg, unable to compute course");
  }
  
  return course1;
}

double
SGGeodesy::distanceM(const SGGeod& p1, const SGGeod& p2)
{
  double course1, course2, distance;
  int r = _geo_inverse_wgs_84(p1.getLatitudeDeg(), p1.getLongitudeDeg(),
                                p2.getLatitudeDeg(), p2.getLongitudeDeg(),
                                &course1, &course2, &distance);
  if (r != 0) {
    throw sg_exception("SGGeodesy::distanceM, unable to compute distance");
  }
  
  return distance;
}

double
SGGeodesy::distanceNm(const SGGeod& from, const SGGeod& to)
{
  return distanceM(from, to) * SG_METER_TO_NM;
}


void
SGGeodesy::advanceRadM(const SGGeoc& geoc, double course, double distance,
                       SGGeoc& result)
{
  result.setRadiusM(geoc.getRadiusM());

  // lat=asin(sin(lat1)*cos(d)+cos(lat1)*sin(d)*cos(tc))
  // IF (cos(lat)=0)
  //   lon=lon1      // endpoint a pole
  // ELSE
  //   lon=mod(lon1-asin(sin(tc)*sin(d)/cos(lat))+pi,2*pi)-pi
  // ENDIF
  
  distance *= SG_METER_TO_NM * SG_NM_TO_RAD;
  
  double sinDistance = sin(distance);
  double cosDistance = cos(distance);

  double sinLat = sin(geoc.getLatitudeRad()) * cosDistance +
    cos(geoc.getLatitudeRad()) * sinDistance * cos(course);
  sinLat = SGMiscd::clip(sinLat, -1, 1);
  result.setLatitudeRad(asin(sinLat));
  double cosLat = cos(result.getLatitudeRad());
  
  
  if (cosLat <= SGLimitsd::min()) {
    // endpoint a pole
    result.setLongitudeRad(geoc.getLongitudeRad());
  } else {
    double tmp = SGMiscd::clip(sin(course) * sinDistance / cosLat, -1, 1);
    double lon = SGMiscd::normalizeAngle(geoc.getLongitudeRad() - asin( tmp ));
    result.setLongitudeRad(lon);
  }
}

double
SGGeodesy::courseRad(const SGGeoc& from, const SGGeoc& to)
{
  double diffLon = to.getLongitudeRad() - from.getLongitudeRad();

  double sinLatFrom = sin(from.getLatitudeRad());
  double cosLatFrom = cos(from.getLatitudeRad());

  double sinLatTo = sin(to.getLatitudeRad());
  double cosLatTo = cos(to.getLatitudeRad());

  double x = cosLatTo*sin(diffLon);
  double y = cosLatFrom*sinLatTo - sinLatFrom*cosLatTo*cos(diffLon);

  // guard atan2 returning NaN's
  if (fabs(x) <= SGLimitsd::min() && fabs(y) <= SGLimitsd::min())
    return 0;

  double c = atan2(x, y);
  if (c >= 0)
    return SGMiscd::twopi() - c;
  else
    return -c;
}

double
SGGeodesy::distanceRad(const SGGeoc& from, const SGGeoc& to)
{
  // d = 2*asin(sqrt((sin((lat1-lat2)/2))^2 +
  //            cos(lat1)*cos(lat2)*(sin((lon1-lon2)/2))^2))
  double cosLatFrom = cos(from.getLatitudeRad());
  double cosLatTo = cos(to.getLatitudeRad());
  double tmp1 = sin(0.5*(from.getLatitudeRad() - to.getLatitudeRad()));
  double tmp2 = sin(0.5*(from.getLongitudeRad() - to.getLongitudeRad()));
  double square = tmp1*tmp1 + cosLatFrom*cosLatTo*tmp2*tmp2;
  double s = SGMiscd::min(sqrt(SGMiscd::max(square, 0)), 1);
  return 2 * asin(s);
}


double
SGGeodesy::distanceM(const SGGeoc& from, const SGGeoc& to)
{
  return distanceRad(from, to) * SG_RAD_TO_NM * SG_NM_TO_METER;
}

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