Deformation of circle patterns and its applications

Given a circle pattern on the Riemann sphere $\overline{\mathbb C}$, in this talk we prove that its quasiconformal deformation space can be naturally identified with the product of the Teichmüller spaces of its interstices. By using the intersection number technique, together with Teichmüller theory of circle packings, we provides a rigidity result of the Midscribability Theorem. Furthermore, by using these methods, we shall investigate the stability of some inscribable graphs.