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A 2.5D meshing technique for triangulation of spatial surfaces has been
presented. The suggested approach consists in approximation of a given
surface by a set of Bezier bicubic patches, their separate triangulation in
parametric space controlled by a mesh control function extracted from the
original surface and the mapping of individual elements from the parametric
space back onto the original surface. The proposed methodology for
triangulation of surfaces proved to be vital and enables generation of
graded meshes on a wide class of surfaces. The crucial point of the whole
algorithm is the availability of such a planar triangulator which supports
generation of pre-stretched and pre-oriented triangles. The experience of
the authors revealed that the meshing algorithm based on the advancing
front technique is very suitable for this purpose. The advantage of the
suggested methodology lies in the possibility to extend this technique to
other tensor product polynomial surfaces which might be more suitable for
approximation and manipulation. Two major drawbacks have been identified so
far. Firstly, there is a tendency to generate at most two triangles sharing
the corner of the parametric space even if this is a reentrant corner on
the original surface. This results apparently in (sometimes very) badly
shaped triangles around this corner. The second one is tightly connected to
the extension of this discretization technique to other tensor product
polynomial surfaces which can potentially uncover some other kinds of
singularities (e.g. inappropriate weight selection for rational Bezier
surfaces). Further research is to be carried out to overcome these
problems.

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*Daniel Rypl *

2005-12-03