Ortho-integral surfaces

Abstract

This paper introduces a natural combinatorial structure of orthogeodesics on hyperbolic surfaces and presents Ptolemy relations among them. As a primary application, we propose a recursive method for computing the trace (the hyperbolic cosine of the length) of orthogeodesics and establish the existence of surfaces where the trace of each orthogeodesic is an integer. These surfaces and their orthogeodesics are closely related to integral Apollonian circle packings. Notably, we found a new type of root-flipping that transitions between roots in different quadratic equations of a certain type, with Vieta root-flipping as a special case. Finally, we provide a combinatorial proof of Basmajian's identity for hyperbolic surfaces, akin to Bowditch's combinatorial proof of the McShane identity.