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

TL;DR

This paper introduces efficient algorithms for maximizing the overlap area between convex polyhedra and polygons under translation, with applications to multiple convex shape overlap problems in computational geometry.

Contribution

It presents a deterministic $O(n \, \log^2 n)$ algorithm for maximum overlap of a convex polyhedron and polygon, and extends to related problems involving multiple polygons and shape differences.

Findings

Algorithm achieves $O(n \log^2 n)$ time complexity for polyhedron-polygon overlap maximization.

Extended algorithms solve maximum overlap of three polygons and minimize symmetric difference under transformations.

Methods improve computational efficiency for shape overlap problems in 3D and 2D geometry.

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.