It seems like this 'irregularity' of vertices is caused by us turning this triangle mesh (which has a regular degree of 6) to quads (which has a regular degree of 4). This would explain why loop subdivision works for triangle meshes, since the way it cuts up triangles guarantees all vertices are still degree-6 after the transformation.
In this way, Catmull-Clark and loop subdivision both seem to belong to the same "family" of algos that:
- subdivides one shape into 4 pieces
- if the original shape is perfectly regular, the resulting shapes are also regular and the pieces have the same size (and shape)
- retain a regular vertex degree after subdivision
Do similar algorithms exist for higher degree polygon meshes? Or are they not used that often.
@zyx Interesting question. It seems there has been at least a very small amount of work on hexagonal subdivision schemes. Other regular polygons don't tile the Euclidean plane, which would seem to make n-gon subdivision challenging. Perhaps one could consider subdivision schemes in the hyperbolic plane (which can be tiled by other regular n-gons). Or Archimedean tilings in the Euclidean plane. But now we're getting really exotic. :-)