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The distance trisector curve

Published: 21 May 2006 Publication History

Abstract

Given points P and Q in the plane, we are interested in separating them by two curves C1 and C2 such that every point of C1 has equal distance to P and to C2, and every point of C2 has equal distance to C1 and to Q. We show by elementary geometric means that such C1 and C2 exist and are unique. Moreover, for P = (0,1) and Q = (0,-1), C1 is the graph of a function ƒ: R → R, C2 is the graph of -f, and f is convex and analytic (i.e., given by a convergent power series at a neighborhood of every point). We conjecture that f is not expressible by elementary functions and, in particular, not algebraic. We provide an algorithm that, given x ∈ R and ε > 0, computes an approximation to f(x) with error at most ε in time polynomial in log 1+|x|/ε.The separation of two points by two "trisector" curves considered here is a special (two-point) case of a new kind of Voronoi diagram, which we call the Voronoi diagram with neutral zone and which we investigate in a companion paper.

References

[1]
T. Asano, J. Matoušek, T. Tokuyama. Voronoi Diagrams with Neutral Zone, in preparation.
[2]
T. Asano and T. Tokuyama. Drawing Equally-Spaced Curves between Two Points, Proc. Fall Conference on Computational Geometry, Boston, Massachusetts, November 2004, pages 24--25.
[3]
F. Aurenhammer. Voronoi Diagrams -- A Survey of a Fundamental Geometric Data Structure, ACM Computing Surveys 23,3(1991) 345--405.
[4]
Famous Curves Index, http://www-history.mcs.st-andrews.ac.uk/history/Curves/Curves.html, as of June 2005.
[5]
J. Milnor. Dynamics in one complex variable. Introductory lectures. Vieweg, Wiesbaden 1999.
[6]
M. Rosenlicht. Integration in finite terms. American Mathematical Monthly 79(1972), 963--972.
[7]
A. Okabe, B. Boots, K. Sugihara. Spatial Tessellations, Concepts and Applications of Voronoi Diagrams, John Wiley & Sons, New York, NY 1992.

Cited By

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  • (2018)On the Computation of Zone and Double Zone DiagramsDiscrete & Computational Geometry10.1007/s00454-017-9958-859:2(253-292)Online publication date: 1-Mar-2018
  • (2007)Zone diagramsProceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms10.5555/1283383.1283464(756-765)Online publication date: 7-Jan-2007
  • (2007)Distance Trisector of Segments and Zone Diagram of Segments in a PlaneProceedings of the 4th International Symposium on Voronoi Diagrams in Science and Engineering10.1109/ISVD.2007.19(66-73)Online publication date: 9-Jul-2007
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cover image ACM Conferences
STOC '06: Proceedings of the thirty-eighth annual ACM symposium on Theory of Computing
May 2006
786 pages
ISBN:1595931341
DOI:10.1145/1132516
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 21 May 2006

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STOC06: Symposium on Theory of Computing
May 21 - 23, 2006
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Cited By

View all
  • (2018)On the Computation of Zone and Double Zone DiagramsDiscrete & Computational Geometry10.1007/s00454-017-9958-859:2(253-292)Online publication date: 1-Mar-2018
  • (2007)Zone diagramsProceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms10.5555/1283383.1283464(756-765)Online publication date: 7-Jan-2007
  • (2007)Distance Trisector of Segments and Zone Diagram of Segments in a PlaneProceedings of the 4th International Symposium on Voronoi Diagrams in Science and Engineering10.1109/ISVD.2007.19(66-73)Online publication date: 9-Jul-2007
  • (2007)Curved Voronoi Diagrams Consisting of Influence Areas with Differentiable BoundariesProceedings of the 4th International Symposium on Voronoi Diagrams in Science and Engineering10.1109/ISVD.2007.12(270-275)Online publication date: 9-Jul-2007
  • (2006)Distance Trisector Curves in Regular Convex Distance MeProceedings of the 3rd International Symposium on Voronoi Diagrams in Science and Engineering10.1109/ISVD.2006.21(8-17)Online publication date: 2-Jul-2006

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