Indirect Discretization of 3D Surfaces

The indirect methods of the discretization of 3D surfaces are generally applicable only to a parametric surface with a bijective mapping between the surface and a planar parametric space. Initially, the parametric space is triangulated using a suitable planar mesh generator and the generated grid is then mapped back onto the original surface and (optionally) optimized. The power of this approach consists in the fact that the mesh generated in the parametric space is planar, which allows to use standard methods for 2D meshing and which ensures desirable robustness and efficiency. However, the mapping is generally not affine. This implies the presence of element distortion and stretching, when elements are being mapped between the real and parametric spaces. The various approaches to the indirect discretization of 3D surfaces then differ in the way how the distortion and stretching induced by the mapping are treated.

The most common and natural approach is to use an anisotropic 2D mesh generator, typically based on the AFT or the DT [79,82]. The anisotropy of the mesh is defined by the surface metric tensor corresponding to the first fundamental form of the surface. The mesh generation is performed in the standard way, however some geometrical criteria are judged with respect to the lengths evaluated in the anisotropic metric. Such an ``exact'' approach is necessary only if the DT is adopted. In the case of the AFT, it is enough to use a suitable heuristic rules designed to force the elements to follow the stretch trends of the mesh defined by the metric tensor [75].

Some of the techniques try to avoid the anisotropic meshing of the parametric space by the reparameterization of the surface [51] or by the application of an inverse mapping [11,47] in the local neighbourhood of a given point on the surface, which makes the anisotropic parametric space look isotropic, thus allowing to use the standard isotropic meshing. However both approaches exhibit intensive computations as they are often highly non-linear and need to be applied for a large number of points.

The other approach [58] avoids the anisotropic meshing by generating an initial coarse mesh in the parametric space which is, after being mapped back to the surface, enriched by inserted points to reflect the required local element density and surface curvature. The point insertion is controlled by the Delaunay property applied to elements in the local neighbourhood projected to the plane of the element into which the inserted point is projected. An interesting feature of the presented implementation of this approach consists in the connection between the modeler and the meshing module using a very limited message passing with just two types of messages.

A similar approach is adopted in [64]. This time, however, the mesh enrichment, employed until the geometric similarity [38] between the model and the mesh is not achieved, is based on a quasi-Delaunay property using the minimum empty circumsphere criterion.

*Daniel Rypl
2005-12-07*