Shortcut Hulls: Vertex-restricted Outer Simplifications of Polygons

TL;DR

This paper introduces a polynomial-time algorithm for constructing vertex-restricted shortcut hulls of polygons, balancing area and perimeter, with applications in geo-data simplification and polygon schematization.

Contribution

It presents the first polynomial-time algorithm for creating shortcut hulls with holes, extending previous shortest path solutions, and explores variants with edge and bend restrictions.

Findings

Polynomial-time algorithm for shortcut hulls with holes

Effective polygon simplification and schematization methods

Demonstrated applications in geo-data and point set visualization

Abstract

Let $P$ be a crossing-free polygon and $\mathcal C$ a set of shortcuts, where each shortcut is a directed straight-line segment connecting two vertices of $P$. A shortcut hull of $P$ is another crossing-free polygon that encloses $P$ and whose oriented boundary is composed of elements from $\mathcal C$. Shortcut hulls find their application in geo-related problems such as the simplification of contour lines. We aim at a shortcut hull that linearly balances the enclosed area and perimeter. If no holes in the shortcut hull are allowed, the problem admits a straight-forward solution via shortest paths. For the more challenging case that the shortcut hull may contain holes, we present a polynomial-time algorithm that is based on computing a constrained, weighted triangulation of the input polygon's exterior. We use this problem as a starting point for investigating further variants, e.g., restricting the number of edges or bends. We demonstrate that shortcut hulls can be used for drawing the rough extent of point sets as well as for the schematization of polygons.