Rectilinear Convex Hull of Points in 3D

TL;DR

This paper presents optimal algorithms for computing the rectilinear convex hull of points in 3D and maintaining its vertices under rotation, along with a study of its properties.

Contribution

It introduces new efficient algorithms for 3D rectilinear convex hull computation and vertex maintenance during rotation, advancing computational geometry methods.

Findings

Optimal $O(n ext{log} n)$ algorithm for convex hull computation.

Efficient $O(n ext{log}^2 n)$ algorithm for vertex maintenance during rotation.

Study of properties of rectilinear convex hulls in 3D.

Abstract

Let $P$ be a set of $n$ points in $\mathbb{R}^3$ in general position, and let $RCH(P)$ be the rectilinear convex hull of $P$. In this paper we obtain an optimal $O(n\log n)$-time and $O(n)$-space algorithm to compute $RCH(P)$. We also obtain an efficient $O(n\log^2 n)$-time and $O(n\log n)$-space algorithm to compute and maintain the set of vertices of the rectilinear convex hull of $P$ as we rotate $\mathbb R^3$ around the $z$-axis. Finally we study some properties of the rectilinear convex hulls of point sets in $\mathbb{R}^3$.