Representing geometric structures ind dimensions: Topology and order
- 141 Downloads
This work investigates data structures for representing and manipulatingd-dimensional geometric objects for arbitraryd ≥ 1. A class of geometric objects is defined, the “subdividedd-manifolds,” which is large enough to encompass many applications. A new representation is given for such objects, the “cell-tuple structure,” which provides direct access to topological structure, ordering information among cells, the topological dual, and boundaries.
The cell-tuple structure gives a simple, uniform representation of subdivided manifolds which unifies the existing work in the field and provides intuitive clarity in all dimensions. The dual subdivision, and boundaries, are represented consistently.
This work has direct applications in solid modeling, computer graphics, and computational geometry.
Unable to display preview. Download preview PDF.
- 2.B. G. Baumgart. A polyhedron representation for computer vision.AFIPS National Computer Conference Proceedings, 44:589–596, 1975.Google Scholar
- 3.I. C. Braid, R. C. Hillyard, and I. A. Stroud. Stepwise construction of polyhedra in geometric modelling. In K. W. Brodlie, editor,Mathematical methods in Computer Graphics and Design, pages 123–141. Academic Press, New York, 1980.Google Scholar
- 4.E. Brisson. Representation ofd-Dimensional Geometric Objects. Ph.D. thesis, Department of Computer Science and Engineering, University of Washington, 1990.Google Scholar
- 5.C. E. Buckley. A divide-and-conquer algorithm for computing 4-dimensional convex hulls. In H. Noltemeier, editor,Computational Geometry and Its Applications: Proceedings of the International Workshop on Computational Geometry CG '88, pages 113–135. Springer-Verlag, New York, 1988.CrossRefGoogle Scholar
- 6.B. Chazelle, L. J. Guibas, and D. T. Lee. The power of geometric duality. InProceedings of the 24th Symposium on Foundations of Computer Science, pages 217–225, 1983.Google Scholar
- 8.M. Dehn and P. Heegaard. Analysis situs.Encyklopadie der Mathematischen Wissenschaften, mit Einschluss ihrer Anwendungen, III AB3, pages 153–220. Teubner, Leipzig, 1907.Google Scholar
- 9.D. P. Dobkin and M. J. Laszlo. Primitives for the manipulation of three-dimensional subdivisions. InProceedings of the 3rd ACM Symposium on Computational Geometry, pages 86–99, 1987.Google Scholar
- 11.C. Eastman, J. Lividini, and D. Stoker. A database for designing large physical systems.AFIPS National Computer Conference Proceedings, 44:603–611, 1975.Google Scholar
- 14.R. Franz and D. Huson. The classification of quasi-regular polyhedra of genus 2. Unpublished manuscript, 1989.Google Scholar
- 17.E. L. Gursoz, Y. Choi, and F. B. Prinz. Vertex-based representation of non-manifold boundaries. In M. J. Wozny, J. U. Turner, and K. Preiss, editors,Geometric Modeling for Product Engineering, pages 107–130. North-Holland, Amsterdam, 1990.Google Scholar
- 18.S. Kawabe, K. Shimada, and H. Masuda. A framework for 3D modeling: constraint-based description and non-manifold geometric modeling. In T. Sata, editor,Organization of Engineering Knowledge for Product Modelling in Computer Integrated Manufacturing, pages 325–357. Elsevier, Amsterdam, 1989.Google Scholar
- 20.P. Lienhardt. Subdivisions of surfaces and generalized maps. Technical Report, Départment D'Informatique, Universite Louis Pasteur, 1988.Google Scholar
- 23.M. Mäntylä.An Introduction to Solid Modeling. Computer Science Press, Rockville, MD, 1988.Google Scholar
- 24.A. A. Markov. The problem of homeomorphy. InProceedings of the International Congress of Mathematicians, pages 300–306, 1958.Google Scholar
- 30.J. R. Rossignac and M. A. O'Connor. SGC: a dimension-independent model for point sets with internal structures and incomplete boundaries. In M. J. Wozny, J. U. Turner, and K. Preiss, editors,Geometric Modeling for Product Engineering, pages 145–180. North-Holland, Amsterdam, 1990.Google Scholar
- 32.G. T. Sallee,Incidence Graphs of Convex Polytopes. Ph.D. thesis, Department of Mathematics, University of Washington, 1966.Google Scholar
- 33.M. I. Shamos and D. Hoey. Closest-point problems.Proceedings of the 16th Symposium on Foundations of Computer Science, pages 151–162, 1975.Google Scholar
- 34.J.-C. Spehner. Merging in maps and in pavings. Technical Report, Laboratoire de Mathématiques et Informatique, Universite Haute Alsace, 1988.Google Scholar
- 35.J. Stillwell.Classical Topology and Combinatorial Group Theory. Graduate Texts in Mathematics, Vol. 72. Springer-Verlag, New York, 1984.Google Scholar
- 39.K. J. Weiler.Topological Structures for Geometric Modeling. Ph.D. thesis, Department of Computer and Systems Engineering, Rensselaer Polytechnic Institute, 1986.Google Scholar
- 40.K. Weiler and D. McLachlan. Generalized sweep operations in the nonmanifold environment. In M. J. Wozny, J. U. Turner, and K. Preiss, editors,Geometric Modeling for Product Engineering, pages 87–106. North-Holland, Amsterdam, 1990.Google Scholar
- 41.Y. Yamaguchi and F. Kimura. Boundary neighborhood representation for non-manifold topology. Unpublished manuscript, 1990.Google Scholar