# 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!

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Your question isn't at the right level for this site. To give you some pointers, what you want is to calculate the "geodesics" on the sphere or ellipsoid. For spheres, these are simply the great circles. You should probably consult a book on differential geometry of curves and surfaces e.g. Do Carmo's, which will give you the background to calculate these curves. – j.c. May 21 '10 at 21:55
And how exactly do you propose to find the shortest curve between two points on the ellipsoid numerically? Note also that unlike the sphere case, the situation when you have several geodesics joining two points on the ellipsoid is not exceptional. I think the level of the question is OK and that consulting a textbook on differential geometry won't be really helpful here. – fedja May 21 '10 at 22:17
The tag "elliptic curves" may have been slightly appropriate. After all, elliptic integrals may pertain... – Steve Huntsman May 21 '10 at 22:36
@fedja Good points, I admit I didn't think very hard about the question before my response and hereby retract my statement that it may be at the wrong level. I only mentioned the textbook as it seemed the asker wasn't using the standard terminology, and so I assumed perhaps didn't have differential geometry background. @Steve, I think it's a stretch considering the other questions that are tagged elliptic-curves. Feel free to add back the tag if you feel strongly about it. – j.c. May 21 '10 at 22:45
You can get a geodesic on the ellipsoid by exploiting the fact that the problem is underspecified. We have two points. Work in the plane containing them and the origin. Fit an ellipse. We can fatten this up to an ellipsoid with a principal axis perpendicular to the plane. Symmetry ensures the shortest path on the ellipse is a geodesic on the ellipsoid, at least of the ellipsoid is fat enough along the perpendicular. From the description it's hard to know if this is what the asker wants. – Dan Piponi May 22 '10 at 0:48