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As mentioned above, the high complexity of the mathematical
description of various surfaces complicates seriously the meshing
procedure with surface constraint. Therefore the proper surface
representation is essential for the efficiency of the method. To make
the algorithm applicable to a wide variety of surfaces with a
reasonable level of complexity the family of tensor product polynomial
surfaces^{1}(e.g., rational Bezier surface, B-spline surface, NURBS, etc.)
is considered in the presented approach. This brings in several
advantages. Since these surfaces are widely used in CAD and modeling
systems a natural and consistent interface (at least in terms of
geometric description of the surface) between the modeler and the
mesh generator is ensured. All these surfaces enable relatively simple
evaluation of surface normal and gradients at discrete
locations on the surface. And finally, the parametric space of these
surfaces allows for some procedures to be performed more efficiently
than in 3D.

In the presented study, only the set of rational Bezier surfaces has
been implemented. Each surface to be discretized is bounded by four
rational Bezier curves from which at most two (but not the two
adjacent ones) may be collapsed into a single point. This allows for
modeling of degenerated surfaces often used as transition
patches. This simple data structure, which can be easily extracted
from modeler allowing to handle rational Bezier entities, may be
extended in terms of additional curves (again, reducible to a single
point) fixed to the surface, for example to specify the mesh density
source or to constrain the triangulation.

#### Footnotes

- ...
surfaces
^{1}
- Note, the same class of surfaces is usually used in
the indirect algorithms using the parametric space for actual grid generation.

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*Daniel Rypl *

2005-12-03