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Singularities

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 surface2, 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

... surface2
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