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Aperture-angle and Hausdorff-approximation of convex figures

Aperture-angle and Hausdorff-approximation of convex figures Aperture-Angle and Hausdorff-Approximation — of Convex Figures Hee-Kap Ahn Sejong University, Seoul, Korea. Sang Won Bae KAIST, Daejeon, Korea. heekap@gmail.com. Otfried Cheong KAIST, Daejeon, Korea. swbae@tclab.kaist.ac.kr Joachim Gudmundsson NICTA, Sydney, Australia. otfried@tclab.kaist.ac.kr ABSTRACT The aperture angle (x, Q) of a point x ˆ Q in the plane with respect to a convex polygon Q is the angle of the smallest cone with apex x that contains Q. The aperture angle approximation error of a compact convex set C in the plane with respect to an inscribed convex polygon Q ‚ C is the minimum aperture angle of any x ˆ C \Q with respect to Q. We show that for any compact convex set C in the plane and any k > 2, there is an inscribed convex k-gon Q ‚ C 2 with aperture angle approximation error 1 ’ k+1 . This bound is optimal, and settles a conjecture by Fekete from the early 1990s. The same proof technique can be used to prove a conjecture by Brass: If a polygon P admits no approximation by a sub-k-gon (the convex hull of k vertices of P ) with Hausdor € distance , but all subpolygons of P http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

Aperture-angle and Hausdorff-approximation of convex figures

Ahn, Hee-Kap; Bae, Sang Won; Cheong, Otfried; Gudmundsson, Joachim
Association for Computing Machinery — Jun 6, 2007
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References (11)

Datasource
Association for Computing Machinery
Copyright
Copyright © 2007 by ACM Inc.
ISBN
978-1-59593-705-6
doi
10.1145/1247069.1247076
Publisher site
See Article on Publisher Site

Abstract

Aperture-Angle and Hausdorff-Approximation — of Convex Figures Hee-Kap Ahn Sejong University, Seoul, Korea. Sang Won Bae KAIST, Daejeon, Korea. heekap@gmail.com. Otfried Cheong KAIST, Daejeon, Korea. swbae@tclab.kaist.ac.kr Joachim Gudmundsson NICTA, Sydney, Australia. otfried@tclab.kaist.ac.kr ABSTRACT The aperture angle (x, Q) of a point x ˆ Q in the plane with respect to a convex polygon Q is the angle of the smallest cone with apex x that contains Q. The aperture angle approximation error of a compact convex set C in the plane with respect to an inscribed convex polygon Q ‚ C is the minimum aperture angle of any x ˆ C \Q with respect to Q. We show that for any compact convex set C in the plane and any k > 2, there is an inscribed convex k-gon Q ‚ C 2 with aperture angle approximation error 1 ’ k+1 . This bound is optimal, and settles a conjecture by Fekete from the early 1990s. The same proof technique can be used to prove a conjecture by Brass: If a polygon P admits no approximation by a sub-k-gon (the convex hull of k vertices of P ) with Hausdor € distance , but all subpolygons of P

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