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## Singularities

As mentioned above, the rational Bezier surface may be represented by a matrix of 3D points and corresponding weights. Quite apparently, merging a point of the control grid with its neighbour(s) in the same row or column results in zero component of the surface gradient at that point in the corresponding (degenerated) direction. A typical example may be a biquadratic triangular patch representing one octant of a sphere (Fig. 6a), where three points associated to the top side of the control grid are identical, or a badly parameterized rectangle (Fig. 6b) where two neighbouring control points in each row coincide. While the principal directions at such a point coincide with the directions of parametric curves, no stretch in the degenerated direction can be calculated because of the zero gradient component. This complicates seriously the calculation of the ellipse of stretches not only at these points but also in their vicinity where the values of the stretch in the crucial direction are not infinite but are still too high to be handled numerically.

These difficulties have to be solved by a particular implementation. One possible remedy of this problem arises from the fact that the 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 rigorous reason why to construct the ellipse of stretches just at the base of the tentative triangle. The idea to construct the ellipse somewhere around its barycenter sounds much more reasonable. A simple situation of this type is outlined in Figure 7. The parametric space with tentative triangles (two cases (a) and (b) are considered) is depicted on the left-hand side of the figure. On the right-hand side, the theoretical element spacing in the direction is drawn, which reveals an intensive singularity along the bottom side () of the parametric space (compare with Fig. 6b). Since the stretch and consequently the height of the triangle to be generated are not yet known, an iterative approach is applied. In this approach, the element size in the appropriate principal direction is gradually evaluated (starting at the singular 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. 7). The stretch from this point is then associated with the original point instead of the original infinite stretch. If point is not reached (iteration process propagates through the whole parametric space without convergence to a reasonable stretch), a maximum possible element size is used. Numerical experiments carried out show that rather than the barycenter it is advisable to consider the middle point of the triangle median as the representative point of the triangle (case (b) in Fig. 7), which tends to produce slightly more conservative stretches. This approach proved to be quite satisfactory and it is also adopted for the evaluation of the ellipse of stretches in the vicinity of singular points. However, the same approach cannot work in cases, when all control points corresponding to a side row or side column in the control grid are merged into a single point (Fig. 6a). 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. The solution of this problem consists in the fact 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. It is therefore sufficient to prescribe one (or more) triangle(s) along the whole length of the singular side in the parametric space, for example in the form of an empty subregion that is not subjected to the meshing (Fig. 8). The only care must be taken to prescribe appropriately the height of that triangle and to ensure that its edges not coinciding with the singular side are topologically identical when mapped back to the original surface.

#### Footnotes

... surface,2
A singularity may also appear inside the surface. But if this is the case, the surface suffers from insufficient continuity and is generally not suitable for modeling.

Next: Smoothing Up: Indirect Triangulation of 3D Surfaces Previous: Parametric Space Meshing

Daniel Rypl
2005-12-07