Angles of Arc-Polygons and Lombardi Drawings of Cacti
Computational Geometry
2023-09-29 v1
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.
Keywords
Cite
@article{arxiv.2107.03615,
title = {Angles of Arc-Polygons and Lombardi Drawings of Cacti},
author = {David Eppstein and Daniel Frishberg and Martha C. Osegueda},
journal= {arXiv preprint arXiv:2107.03615},
year = {2023}
}
Comments
12 pages, 8 figures. To be published in Proc. 33rd Canadian Conference on Computational Geometry, 2021