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arXiv · 2107.03615

Angles of Arc-Polygons and Lombardi Drawings of Cacti

David Eppstein·Daniel Frishberg·Martha C. Osegueda
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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.

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David Eppstein, Daniel Frishberg, Martha C. Osegueda. 2021-07-08. Angles of Arc-Polygons and Lombardi Drawings of Cacti. https://doi.org/10.1016/j.comgeo.2023.101982

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