How do I check if a set of plane polygones create a watertight polyhedra

Question

I am currently wondering if there is a common algorithm to check whether a set of plane polygones, not nescessarily triangles, contruct a watertight polyhedra. Each polygon has an oriantation (normal vector). A simple solution would just be to say yes or no. A more advanced version would be to point out the edges, where the polyhedron is "open". I am not really interesed on how to close to polyhedra.

I would like to point out, that my "holes" are not nescessarily small, e.g., one face of a cube might be missing. Thus, the "undersampling correction" algorithms dont seem to be the correct approach. Furthermore, I am talking of about 100 - 1000, not 1000000 polygons, so computation time should not really be a problem.

Any hints or tips?

kind regards, curator

Answer 1

I believe you can use a simple topological test -- count the number of times each edge appears in the full list of polygons.

If the set of polygons define the surface of a closed volume, each edge should have count>=2, indicating that each edge is shared by (at least) two adjacent polygons. If the surface is manifold count==2 exactly.

Edges with count==1 indicate open regions of the surface.

Answer 2

The above answer does not cover many cases. A more correct (but not necessarily complete: I wouldn't know) algorithm is to ensure that every edge of every polygon (or of the mesh/polyhedron) has an even number of faces connected to it. Consider the following mesh:

The segment (line) between the closest vertex and the one below is attached to 3 faces (one one of the outer triangle and two of the inner triangle), which is greater than two faces. However this is clearly not closed.