Hyper-ideal circle patterns with cone singularities

TL;DR

This paper develops a new method combining topological and variational techniques to construct hyper-ideal circle patterns with cone singularities from given angle data, providing a convex optimization framework and a new proof of existing results.

Contribution

It introduces a hybrid approach for constructing hyper-ideal circle patterns, enhancing applicability and offering a new proof of key theoretical results.

Findings

Constructed hyper-ideal circle patterns from angle data using convex optimization.

Developed a hybrid topological and variational method for pattern construction.

Provided a new proof of Schlenker's main results on hyper-ideal circle patterns.

Abstract

The main objective of this study is to understand how geometric hyper-ideal circle patterns can be constructed from given combinatorial angle data. We design a hybrid method consisting of a topological/deformation approach augmented with a variational principle. In this way, together with the question of characterization of hyper-ideal patterns in terms of angle data, we address their constructability via convex optimization. We presents a new proof of the main results from Jean-Marc Schlenker's work on hyper-ideal circle patterns by developing an approach that is potentially more suitable for applications.