Minimal surfaces from circle patterns: Geometry from combinatorics

Bobenko, Alexander I.; Hoffmann, Tim; Springborn, Boris

doi:10.4007/annals.2006.164.231

Cited by 104 publications

(146 citation statements)

References 29 publications

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“…In this case the four spheres for each quadrilateral touch cyclically [1,6]. The orthogonal circle then must pass through the four touching points.…”

Section: S-conical Nets As S-isothermic Netsmentioning

confidence: 99%

S-Conical CMC Surfaces. Towards a Unified Theory of Discrete Surfaces with Constant Mean Curvature

Bobenko

Hoffmann

2016

Advances in Discrete Differential Geometry

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“…In this case the four spheres for each quadrilateral touch cyclically [1,6]. The orthogonal circle then must pass through the four touching points.…”

Section: S-conical Nets As S-isothermic Netsmentioning

confidence: 99%

S-Conical CMC Surfaces. Towards a Unified Theory of Discrete Surfaces with Constant Mean Curvature

Bobenko

Hoffmann

2016

Advances in Discrete Differential Geometry

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“…We would like to note that there exists also a non-linear theory of discrete minimal surfaces based on the theory of circle patterns [3].…”

Section: Discrete Mean Curvature and Minimal Surfacesmentioning

confidence: 99%

A Discrete Laplace–Beltrami Operator for Simplicial Surfaces

Bobenko

Springborn

2007

Discrete Comput Geom

180

189

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We define a discrete Laplace-Beltrami operator for simplicial surfaces (Definition 16). It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the well known finiteelements Laplacian (the so called "cotan formula") except that it is based on the intrinsic Delaunay triangulation of the simplicial surface. This leads to new definitions of discrete harmonic functions, discrete mean curvature, and discrete minimal surfaces. The definition of the discrete Laplace-Beltrami operator depends on the existence and uniqueness of Delaunay tessellations in piecewise flat surfaces. While the existence is known, we prove the uniqueness. Using Rippa's Theorem we show that, as claimed, Musin's harmonic index provides an optimality criterion for Delaunay triangulations, and this can be used to prove that the edge flipping algorithm terminates also in the setting of piecewise flat surfaces.Keywords Laplace operator · Delaunay triangulation · Dirichlet energy · Simplicial surfaces · Discrete differential geometry Dirichlet Energy of Piecewise Linear FunctionsLet S be a simplicial surface in 3-dimensional Euclidean space, i.e. a geometric simplicial complex in R 3 whose carrier S is a 2-dimensional submanifold, possibly with Research for this article was supported by the DFG Research Unit 565 "Polyhedral Surfaces" and the DFG Research Center MATHEON "Mathematics for key technologies" in Berlin.A.I. Bobenko ( ) · B.A. Springborn

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“…New isothermic surfaces can be obtained from a given isothermic surface via the Christoffel duality and Darboux transformations [20]. Special discrete surfaces related to isothermic quadrilateral meshes were studied in [1,3].…”

Section: Example: Isothermic Quadrilateral Surfacesmentioning

confidence: 99%

Isothermic triangulated surfaces

Lam

Pinkall

2016

Math. Ann.

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Abstract. We found a class of triangulated surfaces in Euclidean space which have similar properties as isothermic surfaces in Differential Geometry. We call a surface isothermic if it admits an infinitesimal isometric deformation preserving the mean curvature integrand locally. We show that this class is Möbius invariant. Isothermic triangulated surfaces can be characterized either in terms of circle patterns or based on conformal equivalence of triangle meshes. This definition generalizes isothermic quadrilateral meshes.A consequence is a discrete analog of minimal surfaces. Here the Weierstrass data needed to construct a discrete minimal surface consist of a triangulated plane domain and a discrete harmonic function.

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Minimal surfaces from circle patterns: Geometry from combinatorics

Cited by 104 publications

References 29 publications

S-Conical CMC Surfaces. Towards a Unified Theory of Discrete Surfaces with Constant Mean Curvature

S-Conical CMC Surfaces. Towards a Unified Theory of Discrete Surfaces with Constant Mean Curvature

A Discrete Laplace–Beltrami Operator for Simplicial Surfaces

Isothermic triangulated surfaces

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