Angles of Arc-Polygons and Lombardi Drawings of Cacti

TL;DR

This paper characterizes possible interior angles in non-self-crossing circular-arc triangles and polygons, and proves that every cactus graph admits a planar Lombardi drawing in its natural embedding.

Contribution

It provides a characterization of angle triples in circular-arc polygons and proves existence of planar Lombardi drawings for cacti in natural embeddings.

Findings

Characterization of interior angles in circular-arc triangles.

Proof that all cacti have planar Lombardi drawings in natural embeddings.

Existence of planar embeddings of cacti without Lombardi drawings.

Abstract

We characterize the triples of interior angles that are possible in non-self-crossing triangles with circular-arc sides, and we prove that a given cyclic sequence of angles can be realized by a non-self-crossing polygon with circular-arc sides whenever all angles are at most pi. As a consequence of these results, we prove that every cactus has a planar Lombardi drawing (a drawing with edges depicted as circular arcs, meeting at equal angles at each vertex) for its natural embedding in which every cycle of the cactus is a face of the drawing. However, there exist planar embeddings of cacti that do not have planar Lombardi drawings.