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simgear/simgear/math/SGGeodesy.cxx
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// 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 <simgear/debug/logstream.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;
}
/// Geocentric routines
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 diffLon = from.getLongitudeRad() - to.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;
}
bool
SGGeodesy::radialIntersection(const SGGeoc& a, double r1,
const SGGeoc& b, double r2, SGGeoc& result)
{
// implementation of
// http://williams.best.vwh.net/avform.htm#Intersection
double crs13 = r1 * SG_DEGREES_TO_RADIANS;
double crs23 = r2 * SG_DEGREES_TO_RADIANS;
double dst12 = SGGeodesy::distanceRad(a, b);
//IF sin(lon2-lon1)<0
// crs12=acos((sin(lat2)-sin(lat1)*cos(dst12))/(sin(dst12)*cos(lat1)))
// crs21=2.*pi-acos((sin(lat1)-sin(lat2)*cos(dst12))/(sin(dst12)*cos(lat2)))
// ELSE
// crs12=2.*pi-acos((sin(lat2)-sin(lat1)*cos(dst12))/(sin(dst12)*cos(lat1)))
// crs21=acos((sin(lat1)-sin(lat2)*cos(dst12))/(sin(dst12)*cos(lat2)))
// ENDIF
double crs12 = SGGeodesy::courseRad(a, b),
crs21 = SGGeodesy::courseRad(b, a);
double sinLat1 = sin(a.getLatitudeRad());
double cosLat1 = cos(a.getLatitudeRad());
double sinDst12 = sin(dst12);
double cosDst12 = cos(dst12);
double ang1 = SGMiscd::normalizeAngle2(crs13-crs12);
double ang2 = SGMiscd::normalizeAngle2(crs21-crs23);
if ((sin(ang1) == 0.0) && (sin(ang2) == 0.0)) {
SG_LOG(SG_GENERAL, SG_WARN, "SGGeodesy::radialIntersection: infinity of intersections");
return false;
}
if ((sin(ang1)*sin(ang2))<0.0) {
SG_LOG(SG_GENERAL, SG_WARN, "SGGeodesy::radialIntersection: intersection ambiguous");
return false;
}
ang1 = fabs(ang1);
ang2 = fabs(ang2);
//ang3=acos(-cos(ang1)*cos(ang2)+sin(ang1)*sin(ang2)*cos(dst12))
//dst13=atan2(sin(dst12)*sin(ang1)*sin(ang2),cos(ang2)+cos(ang1)*cos(ang3))
//lat3=asin(sin(lat1)*cos(dst13)+cos(lat1)*sin(dst13)*cos(crs13))
//lon3=mod(lon1-dlon+pi,2*pi)-pi
double ang3 = acos(-cos(ang1) * cos(ang2) + sin(ang1) * sin(ang2) * cosDst12);
double dst13 = atan2(sinDst12 * sin(ang1) * sin(ang2), cos(ang2) + cos(ang1)*cos(ang3));
double lat3 = asin(sinLat1 * cos(dst13) + cosLat1 * sin(dst13) * cos(crs13));
//dlon=atan2(sin(crs13)*sin(dst13)*cos(lat1),cos(dst13)-sin(lat1)*sin(lat3))
double dlon = atan2(sin(crs13)*sin(dst13)*cosLat1, cos(dst13)- (sinLat1 * sin(lat3)));
double lon3 = SGMiscd::normalizeAngle(-a.getLongitudeRad()-dlon);
result = SGGeoc::fromRadM(-lon3, lat3, a.getRadiusM());
return true;
}
bool
SGGeodesy::radialIntersection(const SGGeod& a, double aRadial,
const SGGeod& b, double bRadial, SGGeod& result)
{
SGGeoc r;
bool ok = radialIntersection(SGGeoc::fromGeod(a), aRadial,
SGGeoc::fromGeod(b), bRadial, r);
if (!ok) {
return false;
}
result = SGGeod::fromGeoc(r);
return true;
}
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