This paper presents a state of the art report of our project, the main objectives of which are: -- Categorization and summary of the geometric concepts required in a functionally based modeling environment; -- Elaboration of a rich system of geometric operations closed on functionally represented objects; -- Treatment of multidimensional and particularly space-time objects in a uniform manner; 6

Theory of R-functions [12] provides the methodology for constructing exact implicit functions for any semianalytic set. This paper systematically explores and compares the known constructions in terms of their differential properties and explains how such functions may be constructed automatically from CSG and boundary representations of solids. The constructed functions may be automatically differentiated and integrated and have many important applications in meshfree engineering analysis, motion planning, and scientific visualization.

Nearly all major commercial computer-aided design systems have adopted a feature-based design approach to solid modeling. Models are created via a sequence of operations which apply design features to incremental versions of a design model. Even surfacing, free-form surface shaping, and deformation operations are internally represented in modeling systems as features in a "history tree" that generates the final design. Much in the same manner that Constructive Solid Geometry (CSG) trees for an individual model can be non-unique, these design feature histories for solid models might be ordered in a number of ways and still result in the same final geometry and topology. We formulate this problem symbolically and present geometric reasoning techniques to generate a canonical form for certain classes of design feature histories. We define this representation as a Model Dependency Graph (MDG) and show how it can be used as a basis for developing techniques for managing databases of solid ...

One of the major unsolved problems in parametric solid modeling is a robust update (regeneration) of the solid's boundary representation, given a specified change in the solid's parameter values. The fundamental difficulty lies in determining the mapping between boundary representations for solids in the same parametric family. Several heuristic approaches have been proposed for dealing with this problem, but the formal properties of such mappings are not well understood. We propose a formal definition for Boundary Representation (BR-)deformation for solids in the same parametric family, based on the assumption of continuity: small changes in solid parameter values should result in small changes in the solid's boundary representation, which may include local collapses of cells in the boundary representation. The necessary conditions that must be satisfied by any BR-deforming mappings between boundary representations are powerful enough to identify invalid updates in many (but...

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

....................................... ix 1 INTRODUCTION . . .............................. 1 1.1 ProblemStatement ............................... 1 1.2 OverviewofApproach............................. 4 1.3 Outline of Thesis . . .............................. 5 2 BACKGROUND . . . .............................. 6 2.1 GraphMatching ................................ 6 2.1.1 Definitions and Background . . ....................... 8 2.1.2 CommonApproaches ............................ 9 2.1.3 Invariants................................... 11 2.1.4 ConventionalApproaches .......................... 13 2.1.5 OtherApproaches .............................. 17 2.2 SolidModelingandFeatureBasedDesign................... 22 2.2.1 Constructive Solid Geometry (CSG) . . . . . ................ 22 2.2.2 Boundary Representation (B-rep) . . . . . . ................ 22 2.2.3 Feature-based Modeling . . . . ....................... 23 2.2.4 Feature Recognition From Solid Models . . ...................

... this article are to review and summarize the development of research on machine recognition of features from CAD data, to discuss the advantages and potential problems of each approach, and to point out some of the promising directions future investigations may take. Since most work in this field has focused on machining features, the article primarily covers those features associated with the manufacturing domain. In order to better understand the state of the art, methods of automated feature recognition are divided into the following categories of methods based on their approach: graph-based, syntactic pattern recognition, rule-based, and volumetric. Within each category we have studied issues such as the definition of features, mechanisms developed for recognition of features, the application scope, and the assumptions made. In addition, the problem is addressed from the perspective of information input requirements and the advantages and disadvantages of boundary representation, constructive solid geometry (CSG), and 2D drawings with respect to machine recognition of features are examined. Emphasis is placed on the mechanisms for attacking problems associated with interacting features

Modeling of families of geometric objects is a major topic in modern geometric and solid modeling. Object families are central to many important solid modeling applications, including parametric modeling schemes based on features, constraints and design history. In this paper we introduce the Generic Geometric Complex (GGC), a modeling scheme for families of decomposed pointsets. Each member of the modeled family is modeled using an improved version of the selective geometric complex. Hence, the GGC can be viewed as a generalization of the boundary representation to a modeling scheme for families of objects. The GGC models a family in the classifying sense, supporting the object membership classification query. Association of corresponding boundary entities (e.g. vertices, edges and faces) in different members of the modeled family is supported by the entity-to-name (E2N) and name-to-entity (N2E) queries. We refer to generic naming mechanisms that possess knowledge only about the boundaries of the modeled objects as invariant naming schemes. We discuss several concrete ingredients of generic names, present a general algorithm for invariant naming of entities in selective geometric complexes in any dimension, and completely characterize invariant naming in the 2-D case.

Abstract. Approximate geometric models, e.g. as created by reverse engineering, describe the approximate shape of an object, but do not record the underlying design intent. Automatically inferring geometric aspects of the design intent, represented by feature trees and geometric constraints, enhances the utility of such models for downstream tasks. One approach to design intent detection in such models is to decompose them into regularity features. Geometric regularities such as symmetries may then be sought in each regularity feature, and subsequently be com-bined into a global, consistent description of the model’s geometric design intent. This paper describes a systematic approach for finding such regu-larity features based on recovering broken symmetries in the model. The output is a tree of regularity features for subsequent use in regularity detection and selection. Experimental results are given to demonstrate the operation and efficiency of the algorithm. 1

The ability to transform between distinct geometric representations is the key to success of multiple-representation modeling systems. But the existing theory of geometric modeling does not directly address or support construction, conversion, and comparison of geometric representations. A study of classical problems of CSG $ b-rep conversions, CSG optimization, and other representation conversions suggests a natural relationship between a representation scheme and an appropriate decomposition of space. We show that a hierarchy of space decompositions corresponding to different representation schemes can be used to enhance the theory and to develop a systematic approach to maintenance of geometric representations. 1. Motivation 1.1. Modern theory of representations The modern field of solid modeling owes much of its success to the theoretical foundations laid by members of the Production Automation Project at the University of Rochester in the 1970's. The history of these development...