Abstract
This note addresses some fundamental questions concerning perturbations as they are used in computational geometry: How does one define them? What does it mean to compute with them? How can one compute with them? Is it sensible to use them?
We define perturbations to be curves, point out that computing with them amounts to computing with limits. and (re)derive some methods of computing with such limits automatically. In principle a line can always be used as a perturbation curve. We discuss a generic method for choosing such a line that is applicable in many situations.
Supported by NSF Presidential Young Investigator award CCR-9058440
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References
C. Burnikel, K. Mehlhorn, and S. Schirra, On Degeneracy in Geometric Computations. Proc. 5th Annual ACM-SIAM Symp. on Discrete Algorithms (1994).
J. Canny, Private Communication.
G.B. Dantzig, Linear Programming and Extensions. Princeton Univ. Press, Princeton, 1963.
K. Dobrindt, Algorithmen für Polyeder. Diplomarbeit, FB 14, Informatik, Univ. des Saarlandes, Saarbrücken (1990).
H. Edelsbrunner and E.P. Mücke, Simulation of Simplicity: A technique to Cope with Degenerate Cases in Geometric Algorithms. ACM Trans. Graphics, 9(1), (1990), 67–104.
I. Emiris and J. Canny, A General Approach to Removing Degeneracies. Proc. 32nd Annual IEEE Symp. FOCS (1991), 405–413.
I. Emiris and J. Canny, An Efficient Approach to Removing Geometric Degeneracies. Proc. 8th Annual ACM Symp. on Comp. Geom. (1991), 74–82.
A. Griewank and G.F. Corliss, Automatic Differentiation of Algorithms: Theory, Implementation, and Applications. SIAM (1991).
L.J. Guibas and J. Stolfi, Primitives for Manipulation of General Subdivisions and Computation of Voronoi Diagrams. ACM Trans. Graphics, 4(2), (1985), 74–123.
S.G. Krantz and H.R. Parks, A Primer of Real Analytic Functions. Birkhäuser Verlag (1992).
C. Monma, M. Paterson, S. Suri, and F. Yao, Computing Euclidean Maximum Spanning Trees. Proc. 4th Annual ACM Symp. on Comp. Geom. (1988), 241–251.
F.P. Preparata and M.I. Shamos, Computational Geometry, An Introduction. Springer Verlag (1985).
J.T. Schwartz, Fast Probabilistic Algorithms for Verification of Polynomial Identities. JACM 27(4), (1980), 701–717.
R. Seidel, Output-Size Sensitive Algorithms for Constructive Problems in Computational Geometry. PhD thesis, Computer Science Dept., Cornell Univ., (1986).
T. Thiele, Private Communication.
C.-K. Yap, Symbolic Treatment of Geometric Degeneracies, J. Symbolic Computation 10 (1990), 349–370.
C.-K. Yap, A Geometric Consistency Theorem for a Symbolic Perturbation Scheme. J. Computer and Systems Science 40 (1990), 2–18.
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© 1994 Springer-Verlag Berlin Heidelberg
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Seidel, R. (1994). The nature and meaning of perturbations in geometric computing. In: Enjalbert, P., Mayr, E.W., Wagner, K.W. (eds) STACS 94. STACS 1994. Lecture Notes in Computer Science, vol 775. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57785-8_127
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DOI: https://doi.org/10.1007/3-540-57785-8_127
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