  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.
