Abstract
A polyhedron is any set that can be obtained from the open halfspaces by a finite number of set complement and set intersection operations. We give an efficient and complete algorithm for intersecting two three-dimensional polyhedra, one of which is convex. The algorithm is efficient in the sense that its running time is bounded by the size of the inputs plus the size of the output times a logarithmic factor. The algorithm is complete in the sense that it can handle all inputs and requires no general position assumption. We also describe a novel data structure that can represent all three-dimensional polyhedra (the set of polyhedra representable by all previous data structures is not closed under the basic boolean operations).
The research of all three authors was partly supported by the ESPRIT Basic Research Actions Program, under contract No. 7141 (project ALCOM II). The research of the second author was also partially supported by the BMFT (Förderungskennzeichen ITS 9103). The paper is based on the first author's master's thesis [Dob90].
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
H. Bieri and W. Nef. Elementary set operations with d-dimensional polyhedra. In Computational Geometry and its Applications, volume 333 of Lecture Notes in Computer Science, pages 97–112. Springer-Verlag, 1988.
D. P. Dobkin and D. G. Kirkpatrick. Fast detection of polyhedral intersection. In Proc. 9th Internat. Colloq. Automata Lang. Program., volume 140 of Lecture Notes in Computer Science, pages 154–165. Springer-Verlag, 1982.
K. Dobrindt. Algorithmen für Polyeder. Master's thesis, Fachbereich Informatik, Universität des Saarlandes, June 1990.
H. Edelsbrunner and H. Maurer. Finding extreme points in three dimensions and solving the post office problem in the plane. Information Processing Letters, 21:39–47, 1985.
H. Edelsbrunner and E.P. Mücke. Simulation of simplicity: A technique to cope with degenerate cases in geometric algorithms. ACM Transactions on Graphics, 9:66–104, 1990.
C.H. Hoffmann. Geometric and Solid Modeling. Morgan Kaufmann, San Mateo, Calif., 1989.
M. Mäntylä. An Introduction to Solid Modeling. Computer Science Press, Rockville, Md., 1988.
K. Mehlhorn. Data Structures and Efficient Algorithms. Springer Verlag, 1984.
K. Mehlhorn and K. Simon. Intersecting two polyhedra one of which is convex. In Proc. Found. Gomput. Theory, volume 199 of Lecture Notes in Computer Science, pages 534–542. Springer-Verlag, 1985.
W. Nef. Beiträge zur Theorie der Polyeder. Herbert Lang Bern, 1978.
F. Preparata and M.I. Shamos. Computational Geometry: An Introduction. Springer, New York Berlin Heidelberg Tokyo, 1985.
A.A.G. Requicha. Representations for rigid solids: Theory, methods, and systems. ACM Computing Surveys, 12:437–464, 1980.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1993 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Dobrindt, K., Mehlhorn, K., Yvinec, M. (1993). A complete and efficient algorithm for the intersection of a general and a convex polyhedron. In: Dehne, F., Sack, JR., Santoro, N., Whitesides, S. (eds) Algorithms and Data Structures. WADS 1993. Lecture Notes in Computer Science, vol 709. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57155-8_258
Download citation
DOI: https://doi.org/10.1007/3-540-57155-8_258
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-57155-1
Online ISBN: 978-3-540-47918-5
eBook Packages: Springer Book Archive