/// <summary>
/// Solves the inverse geodetic problem.
/// Calculates the geodetic curve between two points on the ellipsoid.
/// </summary>
/// <param name="ellipsoid">An ellipsoid</param>
/// <param name="start">A start point</param>
/// <param name="end">An end point</param>
/// <returns>A geodetic curve between the start and the end points</returns>
public GeodeticCurve CalculateGeodeticCurve(Ellipsoid ellipsoid, GlobalCoordinates start, GlobalCoordinates end)
{
// http://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf
double a = ellipsoid.SemiMajorAxis;
double b = ellipsoid.SemiMinorAxis;
double f = ellipsoid.IsInvFDefinitive ? 1 / ellipsoid.InverseFlattening : 0;
// get values ??in radians
double phi1 = start.Latitude.Radians;
double lambda1 = start.Longitude.Radians;
double phi2 = end.Latitude.Radians;
double lambda2 = end.Longitude.Radians;
// computing
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 < 1e-13))
{
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.Zero;
}
else if (phi1 < phi2)
{
alpha1 = Angle.Zero;
alpha2 = Angle.Angle180;
}
else
{
alpha1 = new Angle(Double.NaN);
alpha2 = new Angle(Double.NaN);
}
}
// else, it converged, so do the math
else
{
double radians;
alpha1 = new Angle();
alpha2 = new Angle();
// eq. 20
radians = Math.Atan2(cosU2 * Math.Sin(lambda), (cosU1sinU2 - sinU1cosU2 * Math.Cos(lambda)));
if (radians < 0.0)
{
radians += TwoPi;
}
alpha1.Radians = radians;
// eq. 21
radians = Math.Atan2(cosU1 * Math.Sin(lambda), (-sinU1cosU2 + cosU1sinU2 * Math.Cos(lambda))) + Math.PI;
if (radians < 0.0)
{
radians += TwoPi;
}
alpha2.Radians = radians;
}
if (alpha1 >= 360.0)
{
alpha1 -= 360.0;
}
if (alpha2 >= 360.0)
{
alpha2 -= 360.0;
}
return(new GeodeticCurve(s, alpha1, alpha2));
}
/// <summary>
/// Solves the direct geodetic problem.
/// Calculates the destination for a given point, direction and distance.
/// </summary>
/// <param name="ellipsoid">An ellipsoid</param>
/// <param name="start">A start point</param>
/// <param name="startBearing">A start bearing</param>
/// <param name="distance">A distance to target point</param>
/// <returns>A global coordinates of the destination point</returns>
public GlobalCoordinates CalculateEndingGlobalCoordinates(Ellipsoid ellipsoid, GlobalCoordinates start, Angle startBearing, double distance)
{
Angle endBearing = new Angle();
return(CalculateEndingGlobalCoordinates(ellipsoid, start, startBearing, distance, out endBearing));
}
/// <summary>
/// Initializes a new instance of the MapAround.Geography.GlobalPosition with zero elevation.
/// </summary>
/// <param name="coords">A global coordinates</param>
public GlobalPosition(GlobalCoordinates coords)
: this(coords, 0.0)
{
}
/// <summary>
/// Solves the direct geodetic problem.
/// Calculates the destination for a given point, direction and distance.
/// </summary>
/// <param name="ellipsoid">An ellipsoid</param>
/// <param name="start">A start point</param>
/// <param name="startBearing">A start bearing</param>
/// <param name="distance">A distance to target point</param>
/// <param name="endBearing">An end bearing</param>
/// <returns>A global coordinates of the destination point</returns>
public GlobalCoordinates CalculateEndingGlobalCoordinates(Ellipsoid ellipsoid, GlobalCoordinates start, Angle startBearing, double distance, out Angle endBearing)
{
double a = ellipsoid.SemiMajorAxis;
double b = ellipsoid.SemiMinorAxis;
double aSquared = a * a;
double bSquared = b * b;
double f = ellipsoid.IsInvFDefinitive ? 1 / ellipsoid.InverseFlattening : 0;
double phi1 = start.Latitude.Radians;
double alpha1 = startBearing.Radians;
double cosAlpha1 = Math.Cos(alpha1);
double sinAlpha1 = Math.Sin(alpha1);
double s = distance;
double tanU1 = (1.0 - f) * Math.Tan(phi1);
double cosU1 = 1.0 / Math.Sqrt(1.0 + tanU1 * tanU1);
double sinU1 = tanU1 * cosU1;
// eq. 1
double sigma1 = Math.Atan2(tanU1, cosAlpha1);
// eq. 2
double sinAlpha = cosU1 * sinAlpha1;
double sin2Alpha = sinAlpha * sinAlpha;
double cos2Alpha = 1 - sin2Alpha;
double uSquared = cos2Alpha * (aSquared - bSquared) / bSquared;
// eq. 3
double A = 1 + (uSquared / 16384) * (4096 + uSquared * (-768 + uSquared * (320 - 175 * uSquared)));
// eq. 4
double B = (uSquared / 1024) * (256 + uSquared * (-128 + uSquared * (74 - 47 * uSquared)));
// iterate until there is a negligible change in sigma
double deltaSigma;
double sOverbA = s / (b * A);
double sigma = sOverbA;
double sinSigma;
double prevSigma = sOverbA;
double sigmaM2;
double cosSigmaM2;
double cos2SigmaM2;
while (true)
{
// eq. 5
sigmaM2 = 2.0 * sigma1 + sigma;
cosSigmaM2 = Math.Cos(sigmaM2);
cos2SigmaM2 = cosSigmaM2 * cosSigmaM2;
sinSigma = Math.Sin(sigma);
double cosSignma = Math.Cos(sigma);
// eq. 6
deltaSigma = B * sinSigma * (cosSigmaM2 + (B / 4.0) * (cosSignma * (-1 + 2 * cos2SigmaM2)
- (B / 6.0) * cosSigmaM2 * (-3 + 4 * sinSigma * sinSigma) * (-3 + 4 * cos2SigmaM2)));
// eq. 7
sigma = sOverbA + deltaSigma;
// break after converging to tolerance
if (Math.Abs(sigma - prevSigma) < 1e-13)
{
break;
}
prevSigma = sigma;
}
sigmaM2 = 2.0 * sigma1 + sigma;
cosSigmaM2 = Math.Cos(sigmaM2);
cos2SigmaM2 = cosSigmaM2 * cosSigmaM2;
double cosSigma = Math.Cos(sigma);
sinSigma = Math.Sin(sigma);
// eq. 8
double phi2 = Math.Atan2(sinU1 * cosSigma + cosU1 * sinSigma * cosAlpha1,
(1.0 - f) * Math.Sqrt(sin2Alpha + Math.Pow(sinU1 * sinSigma - cosU1 * cosSigma * cosAlpha1, 2.0)));
// eq. 9
// This fixes the pole crossing defect spotted by Matt Feemster. When a path
// passes a pole and essentially crosses a line of latitude twice - once in
// each direction - the longitude calculation got messed up. Using Atan2
// instead of Atan fixes the defect. The change is in the next 3 lines.
//double tanLambda = sinSigma * sinAlpha1 / (cosU1 * cosSigma - sinU1*sinSigma*cosAlpha1);
//double lambda = Math.Atan(tanLambda);
double lambda = Math.Atan2(sinSigma * sinAlpha1, cosU1 * cosSigma - sinU1 * sinSigma * cosAlpha1);
// eq. 10
double C = (f / 16) * cos2Alpha * (4 + f * (4 - 3 * cos2Alpha));
// eq. 11
double L = lambda - (1 - C) * f * sinAlpha * (sigma + C * sinSigma * (cosSigmaM2 + C * cosSigma * (-1 + 2 * cos2SigmaM2)));
// eq. 12
double alpha2 = Math.Atan2(sinAlpha, -sinU1 * sinSigma + cosU1 * cosSigma * cosAlpha1);
// build result
Angle latitude = new Angle();
Angle longitude = new Angle();
latitude.Radians = phi2;
longitude.Radians = start.Longitude.Radians + L;
endBearing = new Angle();
endBearing.Radians = alpha2;
return(new GlobalCoordinates(latitude, longitude));
}
/// <summary>
/// Initializes a new instance of the MapAround.Geography.GlobalPosition.
/// </summary>
/// <param name="coords">A global coordinates</param>
/// <param name="elevation">An elevation obove the surface of the ellipsoid (in units of the ellipsoid's axes)</param>
public GlobalPosition(GlobalCoordinates coords, double elevation)
{
_coordinates = coords;
_elevation = elevation;
}