Discrete conformal maps and ideal hyperbolic polyhedra

摘要: We establish a connection between two previously unrelated topics: particular discrete version of conformal geometry for triangulated surfaces, and the ideal polyhedra in hyperbolic three-space. Two surfaces are considered discretely conformally equivalent if edge lengths related by scale factors associated with vertices. This simple definition leads to surprisingly rich theory featuring M\"obius invariance, maps as circumcircle preserving piecewise projective maps, variational principles. show how literally same can be reinterpreted addresses problem constructing an polyhedron prescribed intrinsic metric. synthesis enables us derive companion triangulations. It also shows definitions conformality here closely established terms circle packings.