Global parameterization of a topological surface defined as a collection of trimmed bi-parametric patches: Application to automatic mesh construction

We describe a new geometric algorithm to map surfaces into a plane convex area. The mapping transformation is bijective; it rede nes the whole surface as a unique bi-parametric patch. Thus this mapping provides a global parametrization of the surface. The surfaces are issued from industrial CAD software; they are usually described by a large number of patches and there are many shortcomings. Indeed, the decomposition into patches depends on the algorithm of the geometric modelling system used for design and usually has no meaning for any technological application. Moreover, in many cases, the surface de nition is not compatible, i.e. patches are not well connected, some patches are self-intersecting or intersect each other. Many applications are hard to address because of these defects. In this paper we show how patch-independent meshing techniques may be easily automated using a unique metric in a plane parametric space. Thus we provide an automatic procedure to build valid meshes over free-form surfaces issued from industrial CAD software (Computer Aided Design: this terminology should refer to a large amount of software. For the scope of this paper we only refer to geometric modelling systems. Indeed geometric modelling systems remain the kernel of many CAD software).