how to draw geodesic on the ellipsoid?

Question

I try to simulate geodesic on the ellipsoid recently. I have two points on the ellipsoid. After solving the inverse problem, I can get the distance and two azimuths of two points. As the figure https://i.stack.imgur.com/dJnUj.jpg. (I can obtain S12 and alpha 1 2) But how can I draw the geodesic on the ellipsoid? What I can get from the inverse problem doesn't make sense to me to draw this geodesic path. Can somebody help me with this question? Thanks!!

Answer 1

Once you have s₁₂ and α₁, you can solve a series of direct problems with a fixed φ₁, λ₁ and α₁ and a varying s₁₃ in the range 0 ≤ s₁₃ ≤ s₁₂ to find a sequence of waypoints 3. Join up these points and you have your geodesic.

The GeodSolve utility from GeographicLib will do the job on the command line (scroll down to the examples section). Alternatively use the GeodesicLine class in GeographicLib (see the example C++ code provided). Equivalent functionality is also provided in C by the proj.4 library, version 4.9.1.

Answer 2

If it is a surface of revolution ( two axes of ellipsoid same) by Clairaut's Law it is easy to integrate numerically. The invariant minimum radii are: $r_1 sin \alpha_1 ,r_2 sin \alpha_2 $. If there is no axis-symmetry it is doable, but very cumbersome.