Representing Directed Trees as Straight Skeletons

TL;DR

This paper investigates whether any directed tree with a fixed edge order can be realized as the straight skeleton of a simple polygon, addressing a reverse problem in geometric graph representation.

Contribution

It provides a characterization and constructive method for realizing directed trees as straight skeletons of polygons, extending understanding of geometric graph representations.

Findings

Characterization of trees realizable as straight skeletons

Constructive algorithms for polygon reconstruction

Extension of straight skeleton theory to directed trees

Abstract

The straight skeleton of a polygon is the geometric graph obtained by tracing the vertices during a mitered offsetting process. It is known that the straight skeleton of a simple polygon is a tree, and one can naturally derive directions on the edges of the tree from the propagation of the shrinking process. In this paper, we ask the reverse question: Given a tree with directed edges, can it be the straight skeleton of a polygon? And if so, can we find a suitable simple polygon? We answer these questions for all directed trees where the order of edges around each node is fixed.