Examples

The capabilities of the indirect 3D surface mesh generation technique
described in preceding sections are presented on a set of
examples. The meshing of the parametric space was performed by
an external planar anisotropic mesh generator^{3} based
on the AFT with a quadtree based control space.

The first example, the uniform mesh of a hemisphere, has been chosen to present the overall quality of the mesh for different decompositions of the hemisphere into Bezier bicubic patches. Three cases are investigated. In the first one, the hemisphere is approximated by a single patch with two strong singularities as it is displayed in Figure 10 (left). The parametric space with outlined principal directions and associated stretches is presented in Figure 13. The final mesh in the real space is presented in Figure 10, while the anisotropic mesh in the parametric space is displayed in Figure 13 demonstrating the power of the modified AFT to generate anisotropic meshes with significant stretches. Shaded triangles mark the subregions used to eliminate evaluation of infinite stretches around singular points. An alternative representation of the hemisphere by a single patch but without any stretch singularity is visualized in Figure 11. Note that even this representation exhibits some kind of singularity in terms of the normal evaluation (namely at four corner points of the control polygon with collinear components of the surface gradient). However, this type of singularity is not relevant for the presented technique. The triangulation resulting from this approximation is displayed in Figure 11 on the right. The associated mesh in the parametric space and the principal stretches are visualized in Figure 14. The third case deals with the hemisphere decomposed into four identical patches cut off from the hemisphere by four octants (Fig. 12). Each of the patches has a singular point which corresponds to the top of the hemisphere. The final mesh of the whole hemisphere and the mesh of one of the patches in the parametric space are depicted in Figures 12 and 15, respectively, the latter one demonstrating again the generation of significantly stretched triangles. The positioning of control space points at the nodes of a quadtree in the parametric space may me distinct from the figure of principal directions and stretches presented in Figure 15. It can be seen that there are no significant differences in the mesh quality for the individual decompositions of the hemisphere into patches. However, each of the meshes still contains some elements rather far from the optimal shape (equilateral triangle). These disturbances can be typically encountered around the singular points and along the boundary of the patch, where the efficiency of the Laplacian smoothing is considerably reduced due to the fixed nodes on the boundary. This could be alleviated by topological transformations (e.g. the edge swapping). The other problem can be identified in the fact that there is a tendency to generate at most two triangles sharing the corner of the parametric space even if this is a point with angle close to on the original surface (see Fig. 14). This results apparently in badly shaped triangles around this point. This feature can be eliminated by using the anisotropic metric for the evaluation of angles.

The second example presents a graded mesh on a hyperbolic paraboloid approximated by a single patch without any singularity (Fig. 18). The final mesh in the real space is depicted in Figure 18 on the right, while the corresponding mesh in the parametric space is visualized in Figure 16.

In the final example, a surface in the form of a twisted rectangle with a hole in the middle is triangulated (Fig. 19). The surface is not decomposed into more patches, as it could be expected, only a single patch is used. The opening is simply specified in the parametric space that is meshed as a domain containing one void subregion, as it is presented in Figure 17. The resulting mesh is depicted in Figure 19 on the right.

- ... generator
^{3} - Thanks to Petr Krysl for providing his planar anisotropic mesh generator Targe2.

*Daniel Rypl
2005-12-07*