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Topological Cleanup

The topological cleanup consists of several sets of topological operations that are
performed in an appropriate order. The following sets of operations have been implemented:

- merge operations (Fig. 64a),
- swap operations (Fig. 64b),
- refine operations (Fig. 64c),
- coarse operations (Fig. 64d),
- split operations (Fig. 64e), and
- transform operations (Fig. 65).

While the merge and transform operations attempt to remove as many
triangles as possible, the swap, refine, and coarse operations try to
optimize the valence of individual nodes by decreasing the difference
between the actual and the ``optimal'' valence (see Eq. (129)). The
split operations are adopted to remove non-convex quadrilaterals by
splitting them into two triangles. Note that the nodes newly
introduced during the refine operation has to be projected to the
surface to satisfy the surface constraint. A particular topological
transformation is accomplished only if the violation of the following
geometrical criteria does not occur:
- the new quadrilaterals must be convex,
- the new elements must not be inverted,
- the modified or newly created edges must comply (weakly) with
the desired mesh density distribution,
- the angle between the surface normals at the ends of the modified or
newly created edges must not exceed .

Theoretically, if the surface is bounded by an even number of edges,
all triangles could be removed from the mesh. In practice, however, individual
triangles are topologically so far from each other that the adopted
transformation schemes are not capable to remove them. Therefore, even
after the topological cleanup, some triangles remain in the mesh.

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*Daniel Rypl *

2005-12-07