Special Issue on The First Korea-UK Workshop on Geometric Modeling and Computer Graphics; Guest Editors: M.-S. Kim & R. MartinNo Access
POINT-SETS AND CELL STRUCTURES RELEVANT TO COMPUTER AIDED DESIGN
Abstract
Several fields of mathematics are relevant to computer aided design and other software systems involving solid object geometry, topology, differential and algebraic geometry being particularly important. This paper discusses some of this mathematics in order to provide a theoretical foundation for geometric modelling kernels that support non-manifold objects with an internal cellular structure and subsets of different dimensions.
The paper shows relationships between relevant concepts from topology, differential geometry and computer aided geometric design that are not widely known in the CAD community. It also discusses semialgebraic, semianalytic and subanalytic sets as candidates for object representation. Stratifications of such sets are proposed for an object's cellular structure and new stratification concepts are introduced to support candidate applications.
Keywords:
References
- , Organisation of Engineering Knowledge for Product Modelling in Computer-Integrated Manufacturing, ed.
T. Sata (Elsevier, 1989) pp. 275–299. Google Scholar - EDS. Overview of parasolid. Parasolid Business Unit, March 1995 . Google Scholar
-
C. Armstrong , Djinn: a geometric interface for solid modelling ( Information Geometers Ltd. , Winchester, UK , 2000 ) . Google Scholar -
M. Armstrong , Basic topology ( McGraw-Hill Book Company (UK) Limited , 1979 ) . Google Scholar -
R. Brown , Topology: A Geometric Account of General Topology, Homotopy Types and the Fundamental Groupoid ,Mathematics and its Applications ( Ellis Horwood , 1988 ) . Google Scholar - , Differential and Combinatorial Topology, ed.
S. Cairns (Princeton University Press, Princeton, USA) pp. 205–244. Google Scholar -
W. Boothby , An introduction to Differentiate Manifolds and Riemannian Geometry ( Academic Press , 1975 ) . Google Scholar -
W. Fleming , Functions of several variables ( Addison-Wesley , 1965 ) . Google Scholar - , Dynamical Systems,
Lecture Notes in Mathematics , eds.A. Dold and B. Eckman (Springer-Verlag, Berlin, 1975) pp. 332–344. Crossref, Google Scholar -
M. Shiota , Geometry of Subanalytic and Semialgebraic Sets ,Progress in Mathematics 150 ( Birkhauser , Boston, USA , 1997 ) . Crossref, Google Scholar -
J. Mather , Notes on topological stability ,Lecture Notes ( Harvard University , 1970 ) . Google Scholar -
J. Rotman , An introdution to algebraic topology ,Graduate Texts in Mathematics ( Springer-Verlag , 1988 ) . Crossref, Google Scholar -
A. Lundell and S. Weingram , The Topology od CW Complexes ,The University Series in Higher Mathematics ( Van Nostrand Reinhold Company , 1969 ) . Crossref, Google Scholar - Bulletin American Mathematical Society 55, 213 (1949). Crossref, Google Scholar
-
A. Kosinski , Differential Manifolds ( Academic Press , 1993 ) . Google Scholar - Annali Delia Scuola Normale de Pisa 18, 449 (1964). Google Scholar
- S. Lojasiewicz. Ensembles semi-analytiques. Technical Report Cours Faculté des Sciences d'Orsay, Bures-sur-Yvette, Inst. Hautes Études Sci., 1965 . Google Scholar
- Transactions of the American Mathematical Society 211, (1975). Google Scholar
- Inventiones math. 28, 193 (1975). Crossref, Google Scholar
- , Number Theory, Algebraic Geometry and Commutative Algebra, eds.
Y. Kusunoki (Kinokuniya Book-Store Co., Ltd., Tokyo, Japan, 1973) pp. 453–493. Google Scholar - 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 - , Mathematical Methods in Computer Aided Geometric Design, eds.
T. Lyche and L. Schumaker (Academic Press, 1989) pp. 353–371. Google Scholar - Computer Graphics forum 2 11(2), 93 (1992). Crossref, Google Scholar
-
M. Hirsch , Differential Topology ,Graduate Texts in Mathematics ( Springer-Verlag , 1976 ) . Crossref, Google Scholar - Computer Aided Geometric Design 8, 51 (1991). Crossref, Google Scholar
-
M. Golubitsky and V. Guillemin , Stable mappings and their singularities ,Graduate Texts in Mathematics ( Springer-Verlag , 1986 ) . Google Scholar - Josef Hoschek and Dieter Lasser. Fundamentals of computer aided geometric design. Technical report, A K Peters, Wellesley, Massachusetts, 1993 . Google Scholar
- , Mathematical methods in computer aided geometric design , eds.
T. Lyche and L. Schumaker ( Academic Press, Inc. , 1989 ) . Google Scholar -
C. Rourke and B. Sanderson , Introduction to Piecewise-Linear Topology ,Ergebniesse der Mathematik und ihren Greuzgebiese ( Springer-Verlag , 1972 ) . Crossref, Google Scholar -
B. O'Neill , Elementary differential geometry ( Academic Press , 1966 ) . Google Scholar -
D. Hestenes , New Foundations for Classical Mechanics ( Reidel , Dordrecht , 1986 ) . Crossref, Google Scholar - A. Requicha. Mathematical models of rigid solid objects. Production Automation Project Tech. Memo 28, University of Rochester, 1977 . Google Scholar
- ACM Computing Surveys 12(4), 437 (1980). Crossref, Google Scholar
- K. Weiler. Topological Structures for Geometric Modeling. PhD thesis, Rensselaer Polytechnic Institute, Troy, New York, 1986 . Google Scholar
- , Geometric Modeling for CAD Applications, eds.
M. Wozny , H. McLaughlin and J. Encarnação (Elsevier Science Publishers B.V. (North-Holland), 1988) pp. 3–36. Google Scholar - , 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 - IEEE Computer Graphics & Applications 10(11), 76 (1990). Crossref, Google Scholar
- Computer Aided Design 23(2), 21 (1991). Crossref, Google Scholar
- A. Middleditch. Cellular models of mixed dimension. Technical Report BRU/CAE/92:3, Department of Computer Science, Brunel University, West London, 1992 . Google Scholar
- V. Shapiro. Representations of semi-algebraic sets in finite algebras generated by space decompositions. Cornell Programmable Automation CPA91-1, Cornell University, Ithaca, NY, February 1991 . Google Scholar
- International Journal of Computational Geometry and Applications 7(4), 383 (1997). Link, Google Scholar
