Relative Convex Hull Determination from Convex Hulls in the Plane

TL;DR

This paper introduces a novel algorithm for computing the relative convex hull of a polygon within another polygon in the plane, avoiding traditional triangulation methods by leveraging convex hull calculations of the involved polygons.

Contribution

The proposed algorithm computes the relative convex hull directly from vertex sequences, offering a new approach that simplifies the process without decomposition.

Findings

Efficient computation of the relative convex hull using convex hulls.

Algorithm does not require triangulation or decomposition.

Produces vertex sequence of the relative convex hull.

Abstract

A new algorithm for the determination of the relative convex hull in the plane of a simple polygon A with respect to another simple polygon B which contains A, is proposed. The relative convex hull is also known as geodesic convex hull, and the problem of its determination in the plane is equivalent to find the shortest curve among all Jordan curves lying in the difference set of B and A and encircling A. Algorithms solving this problem known from Computational Geometry are based on the triangulation or similar decomposition of that difference set. The algorithm presented here does not use such decomposition, but it supposes that A and B are given as ordered sequences of vertices. The algorithm is based on convex hull calculations of A and B and of smaller polygons and polylines, it produces the output list of vertices of the relative convex hull from the sequence of vertices of the convex hull of A.