Discrete conformal maps and ideal hyperbolic polyhedra
By Alexander I. Bobenko, Ulrich Pinkall, Boris A. Springborn. 2010
199 citations recorded by the source corpus.
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 Mobius invariance, the definition of discrete conformal maps as circumcirclepreserving piecewise projective maps, and two variational principles. We show how literally the same theory can be reinterpreted to address 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. 52C26; 52B10, 57M50
Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/73c15415e139628632f81f2281a199aa0260aa60
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Citations in the corpus
- On the deformation of inversive distance circle packings, III (2016, methodology): “vector (1,1,1)T and two negative eigenvalues. For a proof of Lemma 3.1, see Lemma 5 and Lemma 6 in Guo [11]. Now we see the notion of generalized Euclidean triangles and their angles. Definition 3.2. ([3, 16, 17]) A generalized Euclidean triangle △ is a (topological) triangle of vertices v1,v2,v3 so ”
- Rigidity of inversive distance circle packings revisited (2017, methodology): “The main tools are the variational principle established by Guo [19] for inversive distance circle packings and the extension of locally convex function introduced by Bobenko, Pinkall and Springborn [3] and systemly developed by Luo [22].”
- A discrete uniformization theorem for polyhedral surfaces (2013, methodology): “Let us begin with an important observation of Bobenko–Pinkall– Springborn [6] which relates Euclidean polyhedral metrics to decorated hyperbolic metrics.”