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

Authors:
János Pach;Rom Pinchasi;Micha Sharir
Affiliations:
City College, CUNY and Courant Institute of Mathematical Sciences, NYU, 251 Mercer Street, New York, NY;Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA;School of Computer Science, Tel Aviv University, Tel Aviv 69 978, Israel and Courant Institute of Mathematical Sciences, New York University, NY
Venue:
Journal of Combinatorial Theory Series A
Year:
2004

Citing 1
Cited 3

Proofs from THE BOOK

Proofs from THE BOOK

Graph drawings with few slopes

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On the number of tetrahedra with minimum, unit, and distinct volumes in three-space

Combinatorics, Probability and Computing

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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 (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.