Tuesday, April 22, 2014
Good Intersection Point for a Tangent Plane
While trying to minimize the map distortions that result from treating the lines of latitude and longitude as straight lines in a plane I decided to do the 2D intersect in the plane tangent to the unit sphere at the coordinates of the War Memorial. The directions of east, E, north, N, and zenith, Z, there define this plane and the projections onto it. The deviation of intersection point for the tangent plane from that found for the intersecting arcs on the surface of the sphere was measurable in seconds of arc. Finding the required coordinate rotation can be a little tricky. Rotating the radial line through M to the z-axis in the plane defined by these two directions and then rotating the line in the direction of E to the x-axis while keeping the z-axis fixed works fairly well.
Supplemental (Apr 22): The positions of x'X and y'X were found for the intersection of the two straight lines determined by the heading angles for the two known positions. The third coordinate, z'X can be computed with the Pythagorean Theorem. The following calculation allows us to compare the components of X' with those of X and also the new coordinates. The basis B is the set of coordinate axes found for station M and acting on it with rotation R shows that they map onto the xyz-axes. D gives the 3D coordinates of the known positions, M and R, and the calculated positions, X and X', for comparison.
A good reference for finding the rotation R is Kuipers, Quaternions and Rotation Sequences (1999), Sect. 4.6 Great Circle Navigation (p. 91).