A tight bound for the number of different directions in three dimensions

János Pach; Rom Pinchasi; Micha Sharir

doi:10.1145/777806.777809

A tight bound for the number of different directions in three dimensions

János Pach^*, Rom Pinchasi, Micha Sharir

^*Corresponding author for this work

School of Computer Science

Research output: Contribution to conference › Paper › peer-review

Abstract

Let P be a set of n points in ℝ³, not all of which are in a plane and no three on a line. We partially answer a question of Scott (1970) by showing that the connecting lines of P assume at least 2n -3 different directions if n is even and at least 2n - 2 if n is odd. These bounds are sharp. The proof is based on a far-reaching generalization of Ungar's theorem concerning the analogous problem in the plane.

Original language	English
Pages	106-113
Number of pages	8
DOIs	https://doi.org/10.1145/777806.777809
State	Published - 2003
Event	Nineteenth Annual Symposium on Computational Geometry - san Diego, CA, United States Duration: 8 Jun 2003 → 10 Jun 2003

Conference

Conference	Nineteenth Annual Symposium on Computational Geometry
Country/Territory	United States
City	san Diego, CA
Period	8/06/03 → 10/06/03

Keywords

Directions
Slope Problem
Ungar's Theorem

Access to Document

10.1145/777806.777809

Cite this

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KW - Ungar's Theorem

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A tight bound for the number of different directions in three dimensions

Abstract

Conference

Keywords

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