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

Let P be a set of n points in R 3 , 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 2 n - 3 different directions if n is even and at least 2 n - 2 if n is odd. These bounds are sharp. The proof is based on a far-reaching generalization of Ungarʹs theorem concerning the analogous problem in the plane.