1 Introduction and Main Results In the study of hyperbolic metrics on 3-dimensional manifolds,Thurston(see[9],Chapter 13)introduced Thurston’s circle packing on surfaces,which allows the circles to intersects with non-obtuse angles and is a natural generalization of the tangential circle packing studied by Andreev[1,2]and Koebe[6].Thurston further proved the Andreev-Thurston theorem.
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