Topological and Geometric Properties of Interval Solid Models

doi:10.1006/gmod.2001.0539

Graphical Models

Volume 63, Issue 3, May 2001, Pages 163-175

https://doi.org/10.1006/gmod.2001.0539 Get rights and content

Abstract

A solid is a connected orientable compact subset of R³ which is a 3-manifold with boundary. Moreover, its boundary consists of finitely many components, each of which is a subset of the union of finitely many almost smooth surfaces. Motivated by numerical robustness issues, we consider a finite collection of boxes, with faces parallel to the coordinate planes, which covers the boundary of the solid itself. An interval solid is the union of this collection and the solid. In this paper we develop sufficient conditions on the collection of the boxes and a 3-manifold, so that the union of the collection and the manifold is homeomorphic to the manifold itself. Finally, we outline an approach for constructing an interval solid, using interval arithmetic, homeomorphic to the solid.

References (26)

V. Milenkovic
Robust polygon modeling
Comput. Aided Design
(1993)
S. Fortune
Polyhedral modeling with multiprecision integer arithmet
Comput. Aided Design
(1997)
J. Keyser et al.
Efficient and accurate B-rep generation of low degree sculptured solids using exact arithmetic: I–representations
Comput. Aided Geom. Design
(1999)
J. Keyser et al.
Efficient and accurate B-rep generation of low degree sculptured solids using exact arithmetic: II–computation
Comput. Aided Geom. Design
(1999)
C.Y. Hu et al.
Robust interval solid modeling: Part I, representations
Comput. Aided Design
(1996)
C.Y. Hu et al.
Robust interval solid modeling: Part II, boundary evaluation
Comput. Aided Design
(1996)
M.O. Benouamer et al.
Error-free boundary evaluation based on a lazy rational arithmetic: A detailed implementation
Comput. Aided Design
(1994)
C.Y. Hu et al.
Robust interval algorithm for curve intersections
Comput. Aided Design
(1996)
C.Y. Hu et al.
Robust interval algorithm for surface intersections
Comput. Aided Design
(1997)
S.L. Abrams et al.
Efficient and reliable methods for rounded interval arithmetic
Comput. Aided Design
(1998)

T. Sakkalis et al.

Representational validity of boundary representation models

Comput. Aided Design

(2000)

T. J. Peters, N. F. Stewart, D. R. Ferguson, and P. S. Fussel, Algorithmic tolerances and semantics in data exchange,...

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This chapter provides an overview of computational topology. The first usage of the term “computational topology” appears to have occurred in the dissertation of M. Mantyla. The focus there was upon the connective topology joining vertices, edges, and faces in geometric models, frequently informally described as the symbolic information of a solid model. These vertices, edges, and faces are discussed as the operands for the classical Euler operations. One of the basic goals in computational topology is to create computer generated procedures for obtaining representations of objects having the same shape—at least in some acceptable approximate sense—as a given geometric object. Although computational topology is a relatively new discipline, it has grown and matured rapidly partially because of its increasing importance to many vital contemporary applications areas such as computer-aided design and manufacturing, (CAD/CAM), the life sciences, image processing, and virtual reality. It is leading to new techniques in algorithm and representation theory. These applications are evoking new connections between mathematical subdisciplines such as algebraic geometry, algebraic topology, differential geometry, differential topology, dynamical systems theory, general topology, and singularity, and stratification theory.
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Citation Excerpt :
Other approaches to assigning a meaning to the inconsistent representation are possible, but they will not be discussed in detail here. One possibility is to introduce the concepts of inner-interval and outer-interval solids [25], obtained by enclosing a solid’s boundary in boxes computed using interval arithmetic. For example, the larger set obtained in this way contains the original set, in the same manner as a maximum material condition (MMC) of geometric tolerancing [26].
This paper describes two important questions of semantics, in the context of accuracy in shape interrogation. The first of these questions is how to give meaning to an internally inconsistent solid description based on the widely used trimmed-surface boundary representation. The second question is the meaning of a request, made to a numerical method, to find a solution to a problem whose parameter values are uncertain. Answers to these questions are given, inspired in the case of the second question by a now-standard approach in numerical analysis.
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Citation Excerpt :
In this application, we will make use of this idea and approximate F by another surface which is generated by local perturbations of F. This construction yields a PL approximation and is motivated by the recent work of Sakkalis et al. [28], in which the concept of an interval solid is defined and some of its fundamental topological and geometric properties are proved. The following result summarizes several previously appearing results.
Given a nonsingular compact two-manifold F without boundary, we present methods for establishing a family of surfaces which can approximate F so that each approximant is ambient isotopic to F. The methods presented here offer broad theoretical guidance for a rich class of ambient isotopic approximations, for applications in graphics, animation and surface reconstruction. They are also used to establish sufficient conditions for an interval solid to be ambient isotopic to the solid it is approximating. Furthermore, the normals of the approximant are compared to the normals of the original surface, as these approximating normals play prominent roles in many graphics algorithms.
The methods are based on global theoretical considerations and are compared to existing local methods. Practical implications of these methods are also presented. For the global case, a differential surface analysis is performed to find a positive number ρ so that the offsets F_o(±ρ) of F at distances ±ρ are nonsingular. In doing so, a normal tubular neighborhood, F(ρ), of F is constructed. Then, each approximant of F lies inside F(ρ). Comparisons between these global and local constraints are given.

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Regular ArticleTopological and Geometric Properties of Interval Solid Models

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Regular Article
Topological and Geometric Properties of Interval Solid Models