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I was wondering if there is a robust mathematical definition for the 'deepest point of overlap' of ellipsoid (or, equally as good, spheroid) 1 that has overlapped with ellipsoid 2. For non-overlapping ellipsoids the point of nearest approach is defined as the points that share a common tangent plane normal to the vector joining the surfaces. This breaks down for overlapping surfaces.

I'm trying to develop some intuition and knowledge about the 'deepest points' such that I can eventually design a decent algorithm for finding these points. However, in the literature the phrase in quotation marks seems to be used quite loosely without being properly defined anywhere - at least that I have seen.

Many thanks if anyone has any insight...! :)

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Your overlap being convex (as intersection of two convex sets), a thorough mathematical definition is the center of the maximal sphere included in the overlap.

It is a particular case of what is called the "Utimate eroded set" in the context of the domain called mathematical morphology ; see for example fig. 13.33 page 67 in this document ; but in your case (where the overlap is convex), you aren't faced with such issues of local minima.

The next step is developing an algorithm for finding this point.

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  • $\begingroup$ very interesting, I've never heard of this branch of mathematics haha. I will look into existing algorithms in this domain, thanks. The application is collisions between rigid body particles. The currently used approach is what's known as a geometric potential, not sure if you're familiar. :) $\endgroup$
    – JPA Physics
    Commented Oct 24, 2023 at 19:07
  • $\begingroup$ I have heard a little only about geometric potential. $\endgroup$
    – Jean Marie
    Commented Oct 24, 2023 at 23:23

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