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I need to identify certain topological entities (circular holes) in a given geometry, that is, given the boundary representation of a solid, I need to identify the exact coordinates of all holes in the solid. I have been told to use a topological sweep, yet the information available is not clear to me as in how it fits in the solution of such problem. Does anyone have a recommendation or more information about what exactly is a topological sweep?

Christian Rau
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jcp1966
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  • You're looking for circular holes (a two-dimensional thing) in the boundary of a solid? Are you looking for where the boundary has a hole? Or are you looking for topological handles, like a torus? – gilgamec Mar 14 '18 at 08:14
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    To count number of holes you want to compute Betti numbers. To get actual holes, you want to get generators of the corresponding homology groups. Although, they are not unique and you might have to add some condition. Anyway what do you mean by "exact coordinate of a hole"? How do you define a hole? (Btw. I do not have the time to give a full answer, so Im posting only a comment.) – tom Apr 05 '18 at 20:13

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Since nobody else has provided a more useful and to the point solution, I'll offer some crude theoretical grounds to a solution of a simpler but related problem. The issue of knowing at least if there are holes (let alone describing them). You could use the methods employed to solve Poincaré conjecture but for 3d sphere.

If the 3d mesh can be suitably represented by a nice enough manifold, Ricci flow can be used. So in theory it is a matter of turning the discrete polygons into curved surfaces, and matching their curvature at boundaries. A suitable algorithm should exist for this, then solving roughly a diffusion differential equation once the continuous surface has been correctly parameterized.

Intuitively I can only wager in cases that the surface is not homomorphic to a sphere, the solution would be the minimum radius of the hole, thus describing the hole.

Also I'm not sure that in cases there are more than one whole you would have multiple unique solutions.

I'm not sure either if there are discretized theorems rather that go the other way and probably are better in terms of performance.

Simon F
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marshal craft
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  • It won't be as hard as the Poincare conjecture, since that's a question about homeomorphism type assuming already a certain homotopy type. I would love to see an implementation of the Ricci flow that I can watch by the way. Do you know of one? – user404153 Mar 26 '18 at 02:20
  • I may have overstepped my areas of knowledge. I was simply wagering a known minimum point to cover his solution, there may be better points also considering it is polygon mesh. Also we don't know the performance requirements. There may be some approximations which work for real time processing, or more than likely the result is not time dependent and the holes can be stored before run time or included with the models. I'll try to look further into this. Most likely there is probably a best solution available in academics for 3d graphics. – marshal craft Mar 26 '18 at 06:32