Maximum overlap area of a convex polyhedron and a convex polygon under translation

TL;DR

This paper delivers a deterministic algorithm to maximize the overlap between a convex polyhedron $P$ and a convex polygon $Q$ under translation in $\mathbb{R}^3$ in $O(n \log^2 n)$ time, using a generalized prune-and-search framework that operates on the configuration space cut by event polygons. It extends the approach to two related problems: (i) maximizing the intersection of three planar convex polygons in $O(n \log^3 n)$ time and (ii) minimizing the area of symmetric difference under a homothety in $O(n \log^2 n)$ time. The core technique blends unimodality properties of the overlap function with innovative $ frac{1}{2}$-cutting recursions and a generalized prune-and-search for unions of parallel-line families, enabling efficient pruning of large combinatorial structures. The work advances deterministic, dimension-mixed shape-matching methods and introduces tools (generalized prune-and-search, event-polynomial/hyperplane handling, and cone-based reformulations) that could impact broader computational-geometry problems involving polytope-polytope interactions under motion.

Abstract

Let $P$ be a convex polyhedron and $Q$ be a convex polygon with $n$ vertices in total in three-dimensional space. We present a deterministic algorithm that finds a translation vector $v \in \mathbb{R}^3$ maximizing the overlap area $|P \cap (Q + v)|$ in $O(n \log^2 n)$ time. We then apply our algorithm to solve two related problems. We give an $O(n \log^3 n)$ time algorithm that finds the maximum overlap area of three convex polygons with $n$ vertices in total. We also give an $O(n \log^2 n)$ time algorithm that minimizes the symmetric difference of two convex polygons under scaling and translation.

Figures (7)

• Figure 1: $P_1$, $P_2$ are $P_3$ are represented by colors.
• Figure 2: The slice of $P$ with $z \in [z_0, z_1]$.
• Figure 3: The convex polyhedron $I$ is formed by interesecting $P$ and $(Q+l)$.
• Figure 4: Projecting the configuration space onto the $xz$-plane. The projection of $S$ is the magenta line segment, and the projection of the strip $R$ obtained form \ref{['lemma_plane_step_2.1']} is the cyan line segment.
• Figure 5: Projecting onto the xz-plane.

Theorems & Definitions (48)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1
• Corollary 2.2: avis1996
• Lemma 2.3: frederickson1984
• Lemma 2.4: chazelle1993
• Theorem 3.1: megiddo1984
• Lemma 3.2: cormen2009