Plotting path between sphere or ellipsoid points? Hi,  my apologies if this is not the right place to ask this- I am not a mathematician (I'm a software engineer) and Im working on some 3D applications.
My situation is this- given an origin of 0,0,0 and any two points I need to be able to return the xyz coordinate of any point that is on the line between these two points- with the assumption being that the two points must fall on a symmetrical closed surface surrounding the origin, sphere or ellipsoid.  So the line would follow the surface shortest distance between the two points.
So for example, I would need the xyz point that is 0.1 of the line length... but I have no idea how to get it...
Is this possible from 2 points and an origin only?  I am implementing this in C# if that helps.
Thanks!
 A: 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 explicitly 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 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. 
A: If you are willing to accept an approximation, there is quite a bit of work on
finding shortest paths on convex polyhedra, much of it implemented.
For example, here are two images of shortest paths from one point to all vertices
of a polyhedron inscribed in an ellipsoid:





Images from Biliana Kaneva and J. O'Rourke,
"An Implementation of Chen & Han's Shortest Paths Algorithm"
Proc. of the 12th Canadian Conference on Computational Geometry,
New Brunswick, 2000, pp. 139-146.
Webpage link.


There is a nice Stanford presentation on the topic:


"Approximating Shortest Paths
  on a Convex Polytope in
  Three Dimensions."
  (PDF download.)

Exact shortest paths can even be computed in optimal $O(n \log n)$ time now, but the algorithm is quite complicated:

Yevgeny Schreiber, Micha Sharir. "An Optimal-Time Algorithm for Shortest Paths on a Convex Polytope in Three Dimensions."
  Discrete & Computational Geometry,
  March 2008, Volume 39, Issue 1–3, pp 500–579.
  (Springer link.)

