/// <summary>
/// Given different v1 and v2, which are both unit vectors on
/// sphere, we can get a great circle path from v1 to v2 (choose the
/// shortest great circle path). We walk the path by angle alpha from
/// v1 towards v2. This returns the point we end up with, which is
/// an unit vector.
/// If v1 == v2, an exception is thrown.
/// If v1 == -v2, the chosen path is the one that goes through
/// the north pole, if none of v1, v2 is north pole. Otherwise,
/// the point with lat:0, lon:0.
/// </summary>
public static Vector3D GetV(Vector3D v1, Vector3D v2, double alpha)
{
double t = v1.Dot(v2);
if (t >= 1.0)
{
throw new ArgumentException();
}
if (t <= -1.0)
{
if (v1.Equals(NorthPole) || v2.Equals(NorthPole))
{
return(GetV(v1, Lat0Lon0, alpha));
}
return(GetV(v1, NorthPole, alpha));
}
var matrix = new Matrix2by2(1.0, t, t, 1.0);
double beta = Acos(t);
var b = new Vector2D(Cos(alpha), Cos(beta - alpha));
var a = matrix.Inverse().Multiply(b);
return(v1 * a.X + v2 * a.Y);
}
public Matrix2by2 Inverse()
{
var det = Determinant;
if (det == 0.0)
{
throw new InvalidOperationException();
}
var B = new Matrix2by2(A22, -A12, -A21, A11);
B.Multiply(1.0 / det);
return(B);
}