Geometric Modeling: a New Fundamental Framework and its Practical Implications

Ari Rappoport

Geometric Modeling deals with the representation and manipulation of geometric objects in a computer. A fundamental framework for the field determines the types of questions being asked, hence in a way is more important than specific answers to specific questions.

In this paper we introduce definitions for the concepts modeling space, model, and modeling scheme. The new definitions generalize and enhance Requicha's seminal definitions of a representation and a representation scheme, definitions which have been continuously guiding both research work and commercial developments done in the field.

The new framework is based on two main observations: (1) the operations to be performed on a model are an inherent part of the modeled object, since a model exists for the sole purpose of performing operations on it; (2) a model results from a sequence of operations.

The new definitions enable formalization of important properties such as functionality enhancement, functionality diminution, functional dominance and expressive dominance in modeling spaces and valid, lossy, explicit and potentially lossless modeling schemes.

The importance of the new framework goes beyond theory alone. We discuss several major problems of the field and show how the difficulties towards their solutions can be better understood using the new framework. The deep practical implications of our theory are demonstrated by commercial developments in a related field. Finally, we present a comprehensive list of future research topics which are of importance in light of the new framework.

Proceedings, Third ACM/Siggraph Symposium on Solid Modeling and Applications (Solid Modeling '95), May 1995, Salt Lake City, ACM Press, pp. 31-42.