Extending Metric-Based Surface Meshing to G1-Connected Patch Complexes

2nd Symposium on Trends in Unstructured Mesh Generation, University of Colorado, Boulder, August 1999

MESHING
RESEARCH
CORNER

CISE, U of Florida
jorg@cise.ufl.edu

Abstract
Given a surface, or surface piece represented as a map from a two- dimensional domain to 3-space, say a NURBS patch, [Chen,Bishop 97], [Cuillere 98] and [Tristano, Owen, Canann 98] have developed efficient mesh generation techniques and tools based on the metric or first fundamental form of the surface (see also [George, Borouchaki 98]).

A recurring challenge in NURBS-based CAD systems is the smooth stitching together of such NURBS patches, each being supported on a *separate* domain, e.g. when more or less than four tensor-product patches join at a point to form a tangent-continuous surface. (Already due to the Euler number such points must occur if the free-form surface has a topological genus other than one.) $G^1$-connected patch complexes address this problem by ensuring the appropriate geometric continuity by local reparametrization (see for example [Peters 95] and references therein).

To allow the application of the efficient advancing front techniques for smooth surfaces across separate domains, it is shown how to locally invert the reparametrizations underlying the geometrically continuous patch complex and join separate domains into larger, overlapping domains analoguous to the classical extension of charts to form an atlas.

References

Hao Chen and Jonathan Bishop (1997) 'Delaunay Triangulation for Curved Surfaces', Proceedings, 6th International Meshing Roundtable, pp.115- 127.

J. C. Cuilliere, (1998) 'An adaptive method for the automatic triangulation of 3D parametric surfaces', Computer-Aided Design, vol 30, no. 2, pp.139-149.

J. Peters, (1995) 'C^1-surface splines' SIAM-J-NUM-ANAL,32 (2) 645-- 666.

Paul-Louis George, and Houman Borouchaki (1998) Delaunay Triangulation and Meshing: Application to Finite Elements, Hermes.

Joseph R. Tristano, Steven J. Owen and Scott A. Canann, (1998) 'Advancing Front Surface Mesh Generation in Parametric Space Using a Riemannian Surface Definition', 7th International Meshing Roundtable.