Discrete conformal maps and ideal hyperbolic polyhedra

Details Discrete conformal maps and ideal hyperbolic polyhedra Discrete conformal maps and ideal hyperbolic polyhedra

2010-05-15

arxiv.org/abs/1005.2698v1

Mathematics

Geometric Topology

49 pages, 24 figures

Scientific paper

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

Affiliated with

Also associated with

No associations

Rating

Discrete conformal maps and ideal hyperbolic polyhedra does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Discrete conformal maps and ideal hyperbolic polyhedra, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Discrete conformal maps and ideal hyperbolic polyhedra will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-241233