No Access

LEAST STRATIFICATIONS AND CELL-STRUCTURED OBJECTS IN GEOMETRIC MODELLING

C. M. P. READE

School of Business Information Management, Kingston University, Kingston Hill, Kingston, Surrey KT2 7LB, UK

Search for more papers by this author

A. E. MIDDLEDITCH

Centre for Geometric Modelling and Design, Brunei University, Kingston Lane, Uxbridge, Middlesex, UB10 3PH, UK

Search for more papers by this author

, and

A. J. GOMES

Department Informatica, Universidade de Bern Interior, Rua Marques d'Avila e Bolama, 6200-048, Covilha, Portugal

Search for more papers by this author

https://doi.org/10.1142/S0218654302000054Cited by:1 (Source: Crossref)

Abstract

Results from the study of O-minimal structures are used to formalise fundamental objects and operations for geometric modelling kernels. O-minimal structures provide a more general setting than the traditional semialgebraic sets; they allow semianalytic sets such as screw-threads to be modelled accurately. O-minimality constrains the class of sets to ensure the retention of the key finitary properties necessary for many geometric operations. The formalism is independent of internal representations and implementations, in the spirit of the mixed-dimension cellular objects of the Djinn API (Application Programming Interface) [2].

Topological stratification is used to provide an object with a cellular structure consisting of a finite set of piecewise-smooth manifold cells. The main contribution of the paper is an existence proof for least stratifications (of sets in O-minimal structures). Set-like combination of two cell-structured objects requires a common partition of space from which a cell-structured result can be obtained. The existence of least stratifications facilitates the definition of combination operators that provide unique results.

Keywords:

References

F. Arbab, IEEE Computer Graphics & Applications 10(11), 76 (1990). Crossref, Google Scholar
C. Armstrong et al. , Djinn: a geometric interface for solid modelling ( Information Geometers Ltd. , Winchester, UK , 2000 ) . Google Scholar
A. Gabrielov, Funct. Anal. Appl. 2, 282 (1968). Crossref, Google Scholar
H. Hironaka, Number Theory, Algebraic Geometry and Commutative Algebra, eds. Y. Kusunokiet al. (Kinokuniya Book-Store Co., Ltd., Tokyo, Japan, 1973) pp. 453–493. Google Scholar
S. Lojasiewicz, Annali Della Scuola Normale de Pisa 18, 449 (1964). Google Scholar
J. Mather , Lecture Notes ( Harvard University , 1970 ) . Google Scholar
A. Middleditch. Cellular models of mixed dimension. Technical Report BRU/CAE/92:3, Department of Computer Science, Brunei University, West London, 1992 . Google Scholar
A. Middleditch and C. Reade, International Journal of Shape Modeling 6(1), 79 (2000). Link, Google Scholar
A. Middleditch, C. Reade, and A. Gomes. Rationale and formal definitions for the objects of the Djinn API to a geometric modelling kernel. Technical Report BRU/CAE/98:2, Brunel University, September 1998 . Google Scholar
A. Middleditch, C. Reade and A. Gomes, Computer Aided Design 31, 683 (1999). Crossref, Google Scholar
A. Middleditch, C. Reade and A. J. Gomes, International Journal of Shape Modeling 6(2), 175 (2000). Link, Google Scholar
C. Reade and A. Middleditch. Partitioning and departitioning mixed dimension cellular objects of the djinn api. (Submitted for publication), 2001 . Google Scholar
A. Requicha. Mathematical models of rigid solid objects. Production Automation Project Tech. Memo 28, University of Rochester, 1977 . Google Scholar
A. Requicha, ACM Computing Surveys 12(4), 437 (1980). Crossref, Google Scholar
J. Rossignac and M. O'Connor, Geometric Modeling for Product Engineering, eds. M. Wozny, J. Turner and K. Preiss (IFIP, Elsevier Science Publishers B.V. (North-Holland), 1990) pp. 145–180. Google Scholar
J. Rossignac and A. Requicha, Computer Aided Design 23(2), 21 (1991). Crossref, Google Scholar
V. Shapiro, Computer Aided Geometric Design 11(2), 153 (1994). Crossref, Google Scholar
V. Shapiro, International Journal of Computational Geometry and Applications 7(4), 383 (1997). Link, Google Scholar
L. van den Dries, Bulletin of the AMS 15(2), 189 (1986). Crossref, Google Scholar
L. van den Dries , Tame Topology and O-minimal Structures , LMS Lecture Note Series 248 ( Cambridge University Press , 1998 ) . Crossref, Google Scholar
L. van den Dries and C. Miller, Duke Math 84, 497 (1996). Crossref, Google Scholar
C. Wall, Dynamical Systems, Lecture Notes in Mathematics, eds. A. Dold and B. Eckman (Springer-Verlag, Berlin, 1975) pp. 332–344. Crossref, Google Scholar
H. Whitney, Annals of Mathematics 66(3), 545 (1957). Crossref, Google Scholar
A. Wilkie, Journal of the AMS 9, 1051 (1996). Google Scholar