Cross-sections of line configurations in $R^3$ and $(d-2)$-flat configurations in $R^d$

We consider sets L={ℓ ₁,…,ℓ _n} of n labeled lines in general position in R ³, and study the order types of point sets {p ₁,…,p _n} 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(n ⁹) when considering all possible planes Π and O(n ³) 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 R ² moving with constant (but possibly different) speeds along straight lines forms at most O(n ³) different order types over time. We further generalize the setting from R ³ to R ^d 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 R ^d with planes is O(((n3)+nd(d−2))).