Geometric Modeling of Complex Shapes and Engineering Artifacts

Abstract

A geometric model of an object -in most cases being a subset of the three dimensional space -can be used to better understand the object's structure or behaviour. Therefore data such as the geometry, the topology and other application specific data have to be represented by the model. With the help of a computer it is possible to manipulate, process or display these data. We will discuss different approaches for representing such an object: Volume based representations describe the object in a direct way, whereas boundary representations describe the object indirectly by specifying its boundary. A variety of different surface patches can be used to model the object's boundary. For many applications it is sufficient to know only the boundary of an object. For special objects explicit or implicit mathematical representations can easily be given. An explicit representation is a map from a known parameter space e.g. the unit cube to 3D-space. Implicit representations are equations or relations such as the set of zeros of a functional with three unknowns. These can be very efficient in special cases. As an example of volume based representations we will give a brief overview of the voxel representation. We also show how the boundary of complex objects can be assembled by simpler parts e.g. surface patches. These come in a variety of forms: planar polygons, parametric surfaces, especially spline surfaces and trimmed surfaces, multiresolutionally represented surfaces, e.g. wavelet-based surfaces. In a boundary representation only the boundary of a solid is described. This is usually done by describing the boundary as a collection of surface patches attached to each other at outer edges. Simple objects constructed via any of the methods above can be joined to build more complex objects via Boolean operators (constructive solid geometry, CSG). Constructing an object one has to assure that the object is in agreement with the topological requirements of the modeling system. Notoriously difficult problems are caused by the fact that most modeling systems can compute surface intersections only with a limited precision. This yields numerical results that may finally cause major errors e.g. topologically contradictory conclusions. The rather new method of "Medial Modeling" is also presented. Here an object is described by its medial axis and an associated radius function. The medial axis itself is a collection of lower dimensional objects, i.e. for a 3D-solid a set of points, curves and surface patches. This medial modeling concept developed at the Welfenlab yields a very intuitive user interface useful for solid modeling, and also gives a natural meshing of the solid for FEM computations. Additional attributes can be attached to an object, i.e. attributes of physical origin or logical attributes. Physical attributes include photometric, haptical and other material properties, such as elasticity or roughness. Physical attributes are often specified by textures, i.e. functions that relate surface points to certain quantities of ...

Medial Representations

“…In addition, it has been used for surface meshing (Hoffmann, 1995;Amenta et al, 1998b), volume meshing and finite element analysis Armstrong, 1995, 1997;Patrikalakis and Maekawa, 2002;Wolter et al, 2004), dimension reduction and detail suppression (Sheehy et al, 1996;Leymarie et al, 2004b), as well as shape morphing (Blanding et al, 2000) and haptic exploration of surfaces (Okamura, 2000(Okamura, , 2003. The ability of the MA to implicitly represent the distance of a point to a (complex polygonal) shape is useful (Fig.…”

“…The MA or some equivalent skeletal representation can be used as a tool for shape interrogation (Patrikalakis and Maekawa, 2002), reconstruction, modification and design, and even as a basic element for building intuitive new interfaces for shape modelers (Armstrong et al, 1998;Blanding et al, 1999;Wolter and Friese, 2000;Wolter et al, 2004) (Fig. 11.8).…”