## Fast Rendering of Subdivision Surfaces: Slide 13 of 15.

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Earlier I said that we want to partition the full control mesh into
pairs of adjacent triangles. We want to do this pairing fast, and we
want to minimize the number of triangles that cannot find a unique
pair.

The only way that we are going to end up with unpaired triangles is if
we separate the mesh into disjoint parts, so we want to avoid doing
that. If we try to keep the mesh as compact as possible by carving
evenly from the boundaries, we have fewer chances of breaking the
mesh. Trying to minimize the number of boundary edges automatically
keeps the mesh compact.

This series of images shows the general flavor how the algorithm
proceeds. First we randomly remove a pair of triangles, creating a
boundary. Then we start carving away at the boundary. We continue
carving, and since we avoid breaking things apart, we may in this case
end up with a ring of triangles. At some stage we have to break this
ring, and the remaining triangles can then be paired starting from the
ends.