** Next:** Triangulation in the Parametric Space
**Up:** Top
** Previous:** Mesh Control Function

As mentioned above, the Bezier bicubic surface is defined by
a set of sixteen control points in 3D which constitute
an orthogonal regular 4 4 grid in the parametric space.
While the corner vertices of the
grid are physically fixed with the surface the points on the sides
determine the bow of the surface boundary edges and the inner points
influence the rise of the surface interior. The location of the control
polygon points, which are used typically to influence the geometry and
topology of the surface, is arbitrary, and merging several control points
into a single one leads to the definition of a degenerated surface.
A typical example may be a triangular patch to approximate
one octant of a sphere with a single
Bezier bicubic patch (Fig. 2a), where four points associated to one side of the
control grid are identical, or a badly parameterized rectangle
(Fig. 2b) where three neighboring control points in each row coincide.
Quite apparently, merging a point
of the control grid with its neighbors on the same
parametric curve will result in zero gradient of the surface at point
(pure singular point) in the direction of that parametric curve. While the
principal direction at such a point coincides with the direction of that
parametric curve, no stretch in the direction of zero gradient can be
calculated. This complicates seriously the calculation of the mesh control
function not only at these points but also in their vicinity where pure
singularity does not exist, nevertheless the values of stretch in the
crucial direction are still too high. The basic idea to overcome these
difficulties rises from the fact that the pure singular case usually occurs
only at the boundary of the surface^{2}, it means at the base of triangles to
be generated along this boundary.
However, there is no reason why to
extract mesh control function just at the base of the tentative
triangle. The idea to construct the required information somewhere around
its barycenter sounds much more reasonable.
A simple situation of this type is outlined in Figure 3.
The parametric space with tentative triangles (two cases (a) and (b) are
considered) is depicted in the upper part of the figure. In the lower
part, the theoretical element spacing in the direction is drawn
which reveals a intensive singularity along the right-hand side of the
parametric space (compare with Fig. 2b). Since the stretch and
consequently the height of the triangle to be generated are not yet known,
an iterative approach has been applied. In this approach, the element size
in the appropriate principal direction is gradually evaluated
(starting at point and proceeding inside the domain)
until the calculated element spacing reaches the triple of the distance between
the point and the currently investigated point (ideally the
barycenter of the tentative triangle - point in case (a) in Fig. 3).
Element stretch from this point is then associated with the original
point instead of the original infinite stretch. If the point
is not reached (iteration propagates through the whole parametric
space without convergence to reasonable stretch) a maximum
possible element size is used. Numerical experiments
carried out by the authors showed that rather than the barycenter it is
advisable to consider the middle point of the triangle's median
as the representative point of the triangle (case (b) in Fig. 3)
which tends to
produce slightly more conservative stretches.
The same approach has been adopted
for the calculation of the mesh control function in the vicinity of the
purely singular points. The above described approach proved to be quite
satisfactory but cannot work in cases, when all four control points
corresponding to side row or column in the control grid are merged into
a single position (Fig. 2a). In such a case the zero component of the surface
gradient is constant along the whole row or column, which essentially
eliminates the possibility of the application of the suggested iterative
approach. To tackle this problem let us realize that all triangles
generated in the parametric space from the boundary associated with the
merged control points will be degenerated into a line after being mapped
onto the original surface. This fact brings the authors to the idea to
prescribe one triangle along the whole length of the singular side in the
parametric space, for example in the form of a subregion (Fig. 4). Such triangle
having zero area vanishes from the final mesh and therefore the mesh
control function inside that subregion (including the singular
side - marked in Fig. 4) are
irrelevant and can be generated almost arbitrary. The only attention should
be paid to the interface of the subregion with the rest of the parametric
space, where reasonable values are expected due to the local interpolation.

#### Footnotes

- ... surface
^{2}
- Pure singularity may appear
also inside the surface. But if this is
the case, the surface suffers from insufficient continuity and is not
suitable for modeling.

** Next:** Triangulation in the Parametric Space
**Up:** Top
** Previous:** Mesh Control Function
*Daniel Rypl *

2005-12-03