A variational principle for weighted Delaunay triangulations and hyperideal polyhedra

By Boris A. Springborn. 2006

63 citations recorded by the source corpus.

We use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with non-intersecting site-circles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay triangulations may be interpreted as images of hyperbolic polyhedra with one vertex on and the remaining vertices beyond the infinite boundary of hyperbolic space. Thus, the main theorem states necessary and sufficient conditions for the existence and uniqueness of such polyhedra with prescribed combinatorial type and dihedral angles. More generally, we consider weighted Delaunay triangulations in piecewise flat surfaces, allowing cone singularities with prescribed cone angles in the vertices. The material presented here extends work by Rivin on Delaunay triangulations and ideal polyhedra.

Subfields: Computer Graphics and Computer-Aided Design, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7311b59b63e24ab5cac6add9831b12c47625ca8e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computer Graphics and Computer-Aided Design

1. David Glickenstein
2. Ren Guo
3. Ivan Izmestiev
4. Alexander I. Bobenko
5. Feng Luo
6. Xu Xu

In Discrete Mathematics and Combinatorics

1. Ren Guo
2. Ivan Izmestiev
3. Alexander I. Bobenko
4. David Glickenstein

In Geometry and Topology

1. David Glickenstein
2. Alexander I. Bobenko
3. Ivan Izmestiev
4. Ren Guo
5. Feng Luo
6. Xu Xu
7. Huabin Ge
8. Boris A. Springborn

Citations in the corpus

• Discrete conformal variations and scalar curvature on piecewise flat two and three dimensional manifolds (2009, background): “See, for instance, [13] [36] [11] [9] [5] [41] [21].”
• ALEXANDROV'S THEOREM, WEIGHTED DELAUNAY TRIANGULATIONS, AND MIXED VOLUMES (2006, methodology): “By the inverse function theorem, this allows us to realize any infinitesimal deformation of curvatures by an infinitesimal deformation of radii.”
• Local rigidity of inversive distance circle packing (2009, background): “For example, see Brägger [2], Rivin [15], Leibon [12], Chow-Luo [8], BobenkoSpringborn [5], Guo [10], Springborn [16], Luo [13], Guo-Luo [11] and others.”