article

Counting triangle crossings and halving planes

Authors:

T. K. Dey,

H. EdelsbrunnerAuthors Info & Claims

Discrete & Computational Geometry, Volume 12, Issue 3

Pages 281 - 289

https://doi.org/10.1007/BF02574381

Published: 01 December 1994 Publication History

Abstract

Every collection oft 2n2 triangles with a total ofn vertices in 3 has Ω(t4/n6) crossing pairs. This implies that one of their edges meets Ω(t3/n6) of the triangles. From this it follows thatn points in 3 have onlyO(n8/3) halving planes.

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Every collection of t ≥ 2n² triangles with a total of n vertices in R³ has Ω(t⁴/n⁶) crossing pairs. This implies that one of their edges meets Ω(t³/n⁶) of the triangles. From this it follows that n points in R³ have only O(^8/3) halving planes.
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cover image Discrete & Computational Geometry

Discrete & Computational Geometry Volume 12, Issue 3

September 1994

142 pages

ISSN:0179-5376

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 December 1994

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https://dl.acm.org/doi/10.1007/978-3-030-35802-0_2
Parsa S(2018)On the Links of Vertices in Simplicial d-Complexes Embeddable in the Euclidean 2d-SpaceDiscrete & Computational Geometry10.1007/s00454-017-9936-159:3(663-679)Online publication date: 1-Apr-2018
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Bukh BMatoušek JNivasch G(2010)Stabbing Simplices by Points and FlatsDiscrete & Computational Geometry10.5555/3116275.311654143:2(321-338)Online publication date: 1-Mar-2010
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Sharir M(2003)The Clarkson–Shor Technique Revisited and ExtendedCombinatorics, Probability and Computing10.1017/S096354830200552712:2(191-201)Online publication date: 1-Mar-2003
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