We consider sets L={ℓ1,…,ℓn} of n labeled lines in general position in R3, and study the order types of point sets {p1,…,pn} that stem from the intersections of the lines in L with (directed) planes Π not parallel to any line of L, that is, the proper cross-sections of L. As two main results, we show that the number of different order types that can be obtained as cross-sections of L is O(n9) when considering all possible planes Π and O(n3) when restricting considerations to sets of pairwise parallel planes, where both bounds are tight. The result for parallel planes implies that any set of n points in R2 moving with constant (but possibly different) speeds along straight lines forms at most O(n3) different order types over time. We further generalize the setting from R3 to Rd with d>3, showing that the number of order types that can be obtained as cross-sections of a set of n labeled (d−2)-flats in Rd with planes is O(((n3)+nd(d−2))).
