This paper deals with the problem of creating and maintaining a spatial

A planar map is a figure formed by a set of intersecting lines and curves. Such an object captures both the geometrical and the topological information implicitly defined by the data. In the context of 2D drawing, it provides a new interaction paradigm, map sketching, for editing graphic shapes. To build a planar map, one must compute curve intersections and deduce from them the map they define. The computed topology must be consistent with the underlying geometry. Robustness of geometric computations is a key issue in this process. This report presents a robust solution to Bezier curve intersection that uses exact forward differencing and bounded rational arithmetic. Data structures and algorithms supporting incremental insertion of Bezier curves in a planar map are described. A prototype illustration tool using this method is also discussed. R esum e Considerons la figure plane formee par un ensemble de segments de droite et d'arcs de courbe intersectants. Une carte planaire est une ...

Combinatorial topology is a recent field of mathematics which promises to be of great benefit to geometric modeling and CAD. As such, this article shows how the notion of Generalized Map (GMap) can be used to implement a dimension-independent topological kernel for industrial scale modelers and partial derivative equation (PDE) solvers. Classic approaches to this issue either require a large number of entities and relations between them to be defined, or are limited to objects made of simplices. The G-Map representation relies on no more than a single type of element together with a single type of relation to define the topology of arbitrary dimensional objects (surfaces, solids, hyper-solids . . . ) containing primitives with an arbitrary number of edges and faces. The mathematical origin of G-Maps facilitates the characterization and the definition of validity checks for the objects, which can be important for industrial scale applications. The method might also have important implications for topology-intensive computations such as mesh compression, mesh optimization or multi-resolution editing. Teaching abstract mathematics, such as the notion of orientability and cellular partition, is another possible application of the method, since it provides a way to intuitively visualize some of these notions.

Abstract: This paper presents a description of the reorganization of a geometric modeler, MG, designed to support new capabilities of a topological module (CGC) that allows the detection of closed-off solid regions described by surface patches in non-manifold geometric models defined by NURBS. These patches are interactively created by the user by means of the modeler’s graphics interface, and may result from parametric-surface intersection in which existing surface meshes are used as a support for a discrete definition of intersection curves. The geometry of realistic engineering objects is intrinsically complex, usually composed by several materials and regions. Therefore, automatic and adaptive meshing algorithms have become quite useful to increase the reliability of the procedures of a FEM numerical analysis. The present approach is concerned with two aspects of 3D FEM simulation: geometric modeling, with automatic multi-region detection, and support to automatic finite-element mesh generation.

this paper.

Abstract: This paper presents an object-oriented approach for creating multi-region non-manifold models with NURBS. The main motivation is that the geometry and shape of realistic engineering objects are intrinsically complex, usually composed by several materials and regions. Therefore, automatic and/or adaptive meshing algorithms have become revealed themselves quite useful to increase the reliability of the procedures of a FEM numerical analysis. The present approach is concerned with two aspects of 3D FEM simulation: geometric modeling, with automatic multi-region detection, and support to automatic finite element mesh generation. The final objective is to use geometric models directly in numerical applications. 1

This paper deals with the problem of creating and maintaining a spa-tial subdivision, defined by a set of surface patches. The main goal is to create a set of functions which provides a layer of abstraction capable of hiding the geometric and topological problems which occur when one cre-ates and manipulates spatial subdivisions. The study of arbitrary spatial subdivisions extends and unifies the techniques used in nonmanifold solid modeling and allows the modeling of heterogeneous objects.