Optimal detection of intersections between convex polyhedra

For a polyhedron $P$ in $\mathbb{R}^d$, denote by $|P|$ its combinatorial complexity, i.e., the number of faces of all dimensions of the polyhedra. In this paper, we revisit the classic problem of preprocessing polyhedra independently so that given two preprocessed polyhedra $P$ and $Q$ in $\mathbb{R}^d$, each translated and rotated, their intersection can be tested rapidly. For $d=3$ we show how to perform such a test in $O(\log |P| + \log |Q|)$ time after linear preprocessing time and space. This running time is the best possible and improves upon the last best known query time of $O(\log|P| \log|Q|)$ by Dobkin and Kirkpatrick (1990). We then generalize our method to any constant dimension $d$, achieving the same optimal $O(\log |P| + \log |Q|)$ query time using a representation of size $O(|P|^{\lfloor d/2\rfloor + \varepsilon})$ for any $\varepsilon>0$ arbitrarily small. This answers an even older question posed by Dobkin and Kirkpatrick 30 years ago. In addition, we provide an alternative $O(\log |P| + \log |Q|)$ algorithm to test the intersection of two convex polygons $P$ and $Q$ in the plane.