Find latitude of point on great circle given longitude

Question

I am trying to find out wether my implementation has faults and/or if there is a better method.

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The problem:

Given two points (Lat/Long) defined over the WGS84 ellipsoid, find the point lying on the great circle that connect them and has a given Longitude

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My approach:

• Convert the two points to ECEF coordinates
• Derive the orientation of the plane where the great circle lies (cross product of the two ECEF vectors)
• Derive the orientation of the meridian plane containing the point we are looking for (we know the longitude and that is perpendicular to the equator, so we only need sin/cos of the longitude)
• Derive the ECEF vector of the sought point (is the intersection of the two planes, we need only the cross product of the two plane orientations)
• Find the latitude of the point (is the arcsin of the third vector component, when normalized)

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My doubts:

• When converting to ECEF I was using the fact that the local earth radius is not constant, and thus I was computing it to correctly weight the three components. When I tried to plot my computed points vs the great circle on online maps (googlemaps, skyvector) they seemed a little bit off and I solved the discrepancy by removing the weighting (i.e., ECEF on a perfect sphere). Who is right?

• Is there a better way? (that maybe avoids mapping to ECEF?)

Answer 1

See the wikipedia page on great-circle navigation. Following the prescription there, you compute the heading of the great circle at the first point. Then you can compute the intersection of the resulting path with a particular meridian. (This addresses the problem on a sphere. The ellipsoidal problem is a little more involved. A good place to start is geodesics on an ellipsoid.)