On the $O_\beta$-hull of a planar point set

TL;DR

This paper introduces an efficient method to maintain and optimize the $O_eta$-hull of a planar point set as the angle $eta$ varies, enabling analysis of shape and fit for different orientations.

Contribution

It presents a novel approach to dynamically maintain the $O_eta$-hull and determine optimal angles for area, perimeter, and fit, generalizing the orthogonal convex hull concept.

Findings

Maintains the $O_eta$-hull in $O(n ext{ log } n)$ time for all $eta$ in [0, π].

Identifies $eta$ values that maximize area and perimeter of the $O_eta$-hull.

Finds the optimal $eta$ for best fitting with a two-joint chain.

Abstract

We study the $O_\beta$-hull of a planar point set, a generalization of the Orthogonal Convex Hull where the coordinate axes form an angle $\beta$. Given a set $P$ of $n$ points in the plane, we show how to maintain the $O_\beta$-hull of $P$ while $\beta$ runs from $0$ to $\pi$ in $O(n \log n)$ time and $O(n)$ space. With the same complexity, we also find the values of $\beta$ that maximize the area and the perimeter of the $O_\beta$-hull and, furthermore, we find the value of $\beta$ achieving the best fitting of the point set $P$ with a two-joint chain of alternate interior angle $\beta$.