Barycentric discrete conformal maps between piecewise flat approximations of Riemannian surfaces converge, under fullness and local rigidity assumptions, to conformal maps, generalizing Rodin-Sullivan circle packing convergence.
The convergence of inversive distance circle packings to the Riemann mapping
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abstract
Bowers and Stephenson introduced the notion of inversive distance circle packings as a natural generalization of Thurston's circle packings. They further conjectured that discrete conformal maps induced by inversive distance circle packings converge to the Riemann mapping. In this paper, we prove Bowers-Stephenson's conjecture for Jordan domains by establishing a solvability theorem of certain prescribing combinatorial curvature problems for inversive distance circle packings.
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Convergence of discrete conformal mappings on surfaces
Barycentric discrete conformal maps between piecewise flat approximations of Riemannian surfaces converge, under fullness and local rigidity assumptions, to conformal maps, generalizing Rodin-Sullivan circle packing convergence.