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

Authors:
Janos Pach;Rom Pinchasi;Micha Sharir
Affiliations:
City College, CUNY and Courant Institute of Mathematical Sciences, NYU, New York, NY;Massachusetts Institute of Technology, Cambridge, MA;Tel Aviv University, Tel Aviv, Israel and New York University, New York, NY
Venue:
Proceedings of the nineteenth annual symposium on Computational geometry
Year:
2003

Citing 1
Cited 4

Proofs from THE BOOK

Proofs from THE BOOK

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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.