Let P be a set of n points in the plane, each point moving along a given trajectory. A k-collinearity is a pair (L, t) of a line L and a time t such that L contains at least k points at time t, L is spanned by the points at time t (i.e., the points along L are not all coincident), and not all of the points are collinear at all times. We show that, if the points move with constant velocity, then the number of 3-collinearities is at most 2 ()

n Abstract. We show that given a collection of A lines in, n � 2, the maximum number of their joints (points incident to at least n lines whose directions form a linearly independent set) is O(An/(n−1)). An analogous result for smooth algebraic curves is also proven. 1.