Thurston's circle packing approximation of the Riemann Mapping (proven togive the Riemann Mapping in the limit by Rodin-Sullivan) is largely based onthe theorem that any topological disk with a circle packing metric can bedeformed into a circle packing metric in the disk with boundary circlesinternally tangent to the circle. The main proofs of the uniformization usehyperbolic volumes (Andreev) or hyperbolic circle packings (by Beardon andStephenson). We reformulate these problems into a Euclidean context, whichallows more general discrete conformal structures and boundary conditions. Themain idea is to replace the disk with a double covered disk with one sideforced to be a circle and the other forced to have interior curvature zero. Theentire problem is reduced to finding a zero curvature structure. We also showthat these curvatures arise naturally as curvature measures on generalizedmanifolds (manifolds with multiplicity) that extend the usual discreteLipschitz-Killing curvatures on surfaces.
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