Geometric Modeling of Complex Shapes and Engineering Artifacts
Authors: Wolter F.‐E. , Peinecke N. , Reuter M. 
Published: January 2005

A geometric model of an object – in most cases being a subset of the 3‐D space – can be used to better understand the object's structure or behavior. 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, for example, the unit cube to 3‐D 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, for example, surface patches. These come in a variety of forms: planar polygons, parametric surfaces defined by a map from 2‐D space to 3‐D space, especially spline surfaces and trimmed surfaces, multiresolutionally represented surfaces, for example, wavelet‐based surfaces and surfaces obtained by subdivision schemes. 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. One of the (topologically) most complete schemes is the half‐edge data structure as described by Mäntylä. 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, for example, 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, that is, for a 3‐D 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 as a by‐product a natural meshing of the solid for FEM computations. Additional attributes can be attached to an object, that is, 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, which are functions that relate surface points to certain quantities of the attribute. The most common use for these are photometric textures, although they can also be used for roughness and so on. Logical attributes relate the object to its (data‐)environment. They can, for example, group objects which are somehow related or they can associate scripts to the object, such as callbacks for user interactions.