TY - JOUR

T1 - On the number of directions determined by a three-dimensional points set

AU - Pach, János

AU - Pinchasi, Rom

AU - Sharir, Micha

N1 - Funding Information:
Work on this paper by János Pach and Micha Sharir has been supported by NSF Grants CCR-97-32101 and CCR-00-98246, and by a joint grant from the US–Israel Binational Science Foundation. Work by János Pach has also been supported by PSC-CUNY Research Award 63382-0032 and by Hungarian Science Foundation Grant OTKA T-043520. Work by Micha Sharir has also been supported by a grant from the Israeli Academy of Sciences for a Center of Excellence in Geometric Computing at Tel Aviv University, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University.

PY - 2004/10

Y1 - 2004/10

N2 - 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 (Amer. Math. Monthly 77 (1970) 502) 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.

AB - 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 (Amer. Math. Monthly 77 (1970) 502) 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.

UR - http://www.scopus.com/inward/record.url?scp=9244239177&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2004.04.010

DO - 10.1016/j.jcta.2004.04.010

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AN - SCOPUS:9244239177

SN - 0097-3165

VL - 108

SP - 1

EP - 16

JO - Journal of Combinatorial Theory. Series A

JF - Journal of Combinatorial Theory. Series A

IS - 1

ER -