Angle-monotone Paths in Non-obtuse Triangulations

TL;DR

This paper revisits a known property of non-obtuse triangulations, proving that any two vertices can be connected by an angle-monotone path, and explores extensions to spanning forests with implications for polyhedral unfolding.

Contribution

It provides a new proof of angle-monotone paths in non-obtuse triangulations and extends the concept to boundary-rooted spanning forests, with applications to polyhedral unfolding.

Findings

Every two vertices in a non-obtuse triangulation are connected by an angle-monotone path.

The result cannot be extended to angle-monotone spanning trees.

The extension to boundary-rooted spanning forests suggests a conjectural unfolding method for convex caps.

Abstract

We reprove a result of Dehkordi, Frati, and Gudmundsson: every two vertices in a non-obtuse triangulation of a point set are connected by an angle-monotone path--an xy-monotone path in an appropriately rotated coordinate system. We show that this result cannot be extended to angle-monotone spanning trees, but can be extended to boundary-rooted spanning forests. The latter leads to a conjectural edge-unfolding of sufficiently shallow polyhedral convex caps.