Voronoi diagrams on piecewise flat surfaces and an application to biological growth

Indermitte, Claude; Liebling, Th. M.; Troyanov, Marc; Clémençon, H.

doi:10.1016/s0304-3975(00)00248-6

Cited by 43 publications

(38 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…If there are only finitely many geodesic triangulations of M , this suffices. If not, we use the fact that for any C there are only finitely many triangulations of M with all geodesics of length less than C, [17]. Thus it suffices to show that no long edges can appear when performing the algorithm.…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes

2008

View full text Add to dashboard Cite

We present a constructive proof of Alexandrov's theorem regarding the existence of a convex polytope with a given metric on the boundary. The polytope is obtained as a result of a certain deformation in the class of generalized convex polytopes with the given boundary. We study the space of generalized convex polytopes and discover a relation with the weighted Delaunay triangulations of polyhedral surfaces. The existence of the deformation follows from the non-degeneracy of the Hessian of the total scalar curvature of a positively curved generalized convex polytope. The latter is shown to be equal to the Hessian of the volume of the dual generalized polyhedron. We prove the non-degeneracy by generalizing the Alexandrov-Fenchel inequality. Our construction of a convex polytope from a given metric is implemented in a computer program.

show abstract

Section: Annales De L'institut Fouriermentioning

confidence: 99%

Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes

2008

View full text Add to dashboard Cite

show abstract

“…The maximum value for M is 1/4, realized when all the weights are zero (or equal) since in this case C({1, 2, 3}) is the circumcenter. Delaunay triangulations of closed surfaces have also been studied, for instance in [23] and later [15]. Bobenko and Springborn [2] note that Rivin's result that any Delaunay triangulation of a surface can be gotten by a sequence of flips (see [23]) implies that the full statement of Rippa's theorem applies to closed surfaces as well.…”

Section: Further Remarksmentioning

confidence: 99%

A Monotonicity Property for Weighted Delaunay Triangulations

Glickenstein

2007

Discrete Comput Geom

View full text Add to dashboard Cite

Given a triangulation of points in the plane and a function on the points, one may consider the Dirichlet energy, which is related to the Dirichlet energy of a smooth function. In fact, the Dirichlet energy can be derived from a finite element approximation. S. Rippa showed that the Dirichlet energy (which he refers to as the "roughness") is minimized by the Delaunay triangulation by showing that each edge flip which makes an edge Delaunay decreases the energy. In this paper, we introduce a Dirichlet energy on a weighted triangulation which is a generalization of the energy on unweighted triangulations and an analogue of the smooth Dirichlet energy on a domain. We show that this Dirichlet energy has the property that each edge flip which makes an edge weighted Delaunay decreases the energy. The proof is done by a direct calculation, and so gives an alternate proof of Rippa's result.

show abstract

“…His proof that the edge flipping algorithm terminates is flawed (see the discussion after Proposition 12 below). A correct proof was given by Indermitte et al [11]. (They seem to miss a small detail, a topological obstruction to edge-flipability.…”

Section: Delaunay Triangulations Of Piecewise Flat Surfacesmentioning

confidence: 95%

“…In their proof of Proposition 12, Indermitte et al [11] use the sum of squared circumcircle radii as proper function which decreases with every flip. (For a proof that it decreases with every flip, they refer to an unpublished PhD thesis.…”

Section: Definition 13mentioning

confidence: 99%

A Discrete Laplace–Beltrami Operator for Simplicial Surfaces

Bobenko

Springborn

2007

Discrete Comput Geom

180

189

View full text Add to dashboard Cite

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

show abstract

Voronoi diagrams on piecewise flat surfaces and an application to biological growth

Cited by 43 publications

References 7 publications

Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes

Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes

A Monotonicity Property for Weighted Delaunay Triangulations

A Discrete Laplace–Beltrami Operator for Simplicial Surfaces

Contact Info

Product

Resources

About