/// <summary> /// Geodetic curve between two points on a specified reference ellipsoid. /// This is the solution to the inverse geodetic problem. /// </summary> /// <param name="start">starting coordinates</param> /// <param name="end">ending coordinates </param> /// <param name="referenceDatum">reference datum to use, default="WGS84"</param> /// <returns></returns> public GeodeticCurve(LatLonCoordinates start, LatLonCoordinates end, string referenceDatum = "WGS84") { // // All equation numbers refer back to Vincenty's publication: // See http://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf // // get constants var datum = Datum.GetInstance(referenceDatum); double a = datum.a; double b = datum.a * Math.Sqrt(1 - datum.e2); double f = (a - b) / a; // get parameters as radians double phi1 = start.Latitude.Radians; double lambda1 = start.Longitude.Radians; double phi2 = end.Latitude.Radians; double lambda2 = end.Longitude.Radians; // calculations double a2 = a * a; double b2 = b * b; double a2b2b2 = (a2 - b2) / b2; double omega = lambda2 - lambda1; double tanphi1 = Math.Tan(phi1); double tanU1 = (1.0 - f) * tanphi1; double U1 = Math.Atan(tanU1); double sinU1 = Math.Sin(U1); double cosU1 = Math.Cos(U1); double tanphi2 = Math.Tan(phi2); double tanU2 = (1.0 - f) * tanphi2; double U2 = Math.Atan(tanU2); double sinU2 = Math.Sin(U2); double cosU2 = Math.Cos(U2); double sinU1sinU2 = sinU1 * sinU2; double cosU1sinU2 = cosU1 * sinU2; double sinU1cosU2 = sinU1 * cosU2; double cosU1cosU2 = cosU1 * cosU2; // eq. 13 double lambda = omega; // intermediates we'll need to compute 's' double A = 0.0; double B = 0.0; double sigma = 0.0; double deltasigma = 0.0; double lambda0; bool converged = false; for (int i = 0; i < 20; i++) { lambda0 = lambda; double sinlambda = Math.Sin(lambda); double coslambda = Math.Cos(lambda); // eq. 14 double sin2sigma = (cosU2 * sinlambda * cosU2 * sinlambda) + Math.Pow(cosU1sinU2 - sinU1cosU2 * coslambda, 2.0); double sinsigma = Math.Sqrt(sin2sigma); // eq. 15 double cossigma = sinU1sinU2 + (cosU1cosU2 * coslambda); // eq. 16 sigma = Math.Atan2(sinsigma, cossigma); // eq. 17 Careful! sin2sigma might be almost 0! double sinalpha = (sin2sigma == 0) ? 0.0 : cosU1cosU2 * sinlambda / sinsigma; double alpha = Math.Asin(sinalpha); double cosalpha = Math.Cos(alpha); double cos2alpha = cosalpha * cosalpha; // eq. 18 Careful! cos2alpha might be almost 0! double cos2sigmam = cos2alpha == 0.0 ? 0.0 : cossigma - 2 * sinU1sinU2 / cos2alpha; double u2 = cos2alpha * a2b2b2; double cos2sigmam2 = cos2sigmam * cos2sigmam; // eq. 3 A = 1.0 + u2 / 16384 * (4096 + u2 * (-768 + u2 * (320 - 175 * u2))); // eq. 4 B = u2 / 1024 * (256 + u2 * (-128 + u2 * (74 - 47 * u2))); // eq. 6 deltasigma = B * sinsigma * (cos2sigmam + B / 4 * (cossigma * (-1 + 2 * cos2sigmam2) - B / 6 * cos2sigmam * (-3 + 4 * sin2sigma) * (-3 + 4 * cos2sigmam2))); // eq. 10 double C = f / 16 * cos2alpha * (4 + f * (4 - 3 * cos2alpha)); // eq. 11 (modified) lambda = omega + (1 - C) * f * sinalpha * (sigma + C * sinsigma * (cos2sigmam + C * cossigma * (-1 + 2 * cos2sigmam2))); // see how much improvement we got double change = Math.Abs((lambda - lambda0) / lambda); if ((i > 1) && (change < 0.0000000000001)) { converged = true; break; } } // eq. 19 double s = b * A * (sigma - deltasigma); Angle alpha1; Angle alpha2; // didn't converge? must be N/S if (!converged) { if (phi1 > phi2) { alpha1 = Angle.Angle180; alpha2 = Angle.Angle0; } else if (phi1 < phi2) { alpha1 = Angle.Angle0; alpha2 = Angle.Angle180; } else { alpha1 = Angle.NaA; alpha2 = Angle.NaA; } } // else, it converged, so do the math else { double radians; // eq. 20 radians = Math.Atan2(cosU2 * Math.Sin(lambda), (cosU1sinU2 - sinU1cosU2 * Math.Cos(lambda))); if (radians < 0.0) { radians += Angle.TWOPI; } alpha1 = new Angle() { Radians = radians }; // eq. 21 radians = Math.Atan2(cosU1 * Math.Sin(lambda), (-sinU1cosU2 + cosU1sinU2 * Math.Cos(lambda))) + Math.PI; if (radians < 0.0) { radians += Angle.TWOPI; } alpha2 = new Angle() { Radians = radians }; } distance = s; azimuth = alpha1; reverseAzimuth = alpha2; }