I guess you are in the GPS business, aren't you ?
I think that Vincenty 1975 paper is what you are looking for. At least it should be a starting point for a bibliographic search.
Let me add a few remarks. Fortunately, free motion on the ellipsoid is an integrable system. Which means (loosely) that you can explicitely solve the equations of the trajectory using just a few integrals. This was done by Jacobi (1838).
So if you are not happy with Vincenty approach, there are two paths you can follow. Either look in a book (or click here) for the differential equations satisfied by the geodesics, and do a numerical integration. Or you can start from the solutions of these equations, which are given by elliptic functions. There are standard libraries in C for computing numerical values for these functions.
As a reference, I recommend the book "Elliptic functions and applications" by Derek F. Lawden. As far as I recall, the problem is solved in the book (I hope my memory is not betraying me). And I should add, this is a great book for everybody interested in making the connection between elliptic functions and classical mechanics.
By the way, if you are interested in the following question: on which manifold is the geodesic flow is integrable ? then
you can have a look at a short survey by Andre Miller.
And if you are interested in a clever proof of the integrability of the geodesic flow that works in any dimension, there is an online paper by S. Tabachnikov.
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