## A discrete Laplace-Beltrami operator for simplicial surfaces (2006)

Citations: | 37 - 3 self |

### BibTeX

@MISC{Bobenko06adiscrete,

author = {Alexander I. Bobenko and Boris A. Springborn},

title = {A discrete Laplace-Beltrami operator for simplicial surfaces},

year = {2006}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

369 | H.: Discrete differential-geometry operators for triangulated 2-manifolds
- MEYER, DESBRUN, et al.
- 2003
(Show Context)
Citation Context ...lace–Beltrami operator of Definition 16 and A(C(x)) is the area of the Voronoi cell C(x). This definition is similar to the definition of mean curvature suggested by Meyer, Desbrun, Schröder and Barr =-=[14]-=-. Again, the difference is that we contend that one should use the discrete Laplace–Beltrami operator. Definition 23 (Wide definition of simplicial minimal surfaces). A simplicial surface is called mi... |

275 | POLTHIER K.: Computing discrete minimal surfaces and their conjugates
- PINKALL
- 1993
(Show Context)
Citation Context ...e operator on simplicial surfaces, but without stating the cotan formula explicitly [9]. It seems to have been rediscovered by Pinkall and Polthier in their investigation of discrete minimal surfaces =-=[17]-=-, and turned out to be extremely important in geometry processing where it found numerous applications, e.g. [4, 6] to name but two. In particular, harmonic parameterizations u : V → R 2 are used in c... |

179 | Geometry and Topology for Mesh Generation - Edelsbrunner - 2001 |

174 | Intrinsic parameterizations of surface meshes
- Desbrun, Meyer, et al.
- 2002
(Show Context)
Citation Context ...discovered by Pinkall and Polthier in their investigation of discrete minimal surfaces [17], and turned out to be extremely important in geometry processing where it found numerous applications, e.g. =-=[4, 6]-=- to name but two. In particular, harmonic parameterizations u : V → R 2 are used in computer graphics for texture mapping. The cotan-formula also forms the basis for a theory of discrete holomorphic f... |

153 | On the peeper’s voronoi diagram
- Aurenhammer, Stckl
- 1991
(Show Context)
Citation Context ...ent [5] forDelaunay triangulations in R n . (See also Edelsbrunner [10, p. 8] for a more easily available modern exposition.) Alternatively, one could also adapt the argument of Aurenhammer and Klein =-=[2]-=- for Delaunay triangulations in the plane.748 Discrete Comput Geom (2007) 38: 740–756 Fig. 5 Left: The layed out triangles. Right: The power line of ci and ci+1 is gi,i+1. Since the edge ei,i+1 is as... |

99 |
Sur la sphere vide
- Delaunay
- 1934
(Show Context)
Citation Context ...faces. We decided to give a detailed exposition because not all necessary proofs can be found elsewhere. The concept of a Delaunay triangulation in n-dimensional Euclidean space goes back to Delaunay =-=[5]-=-. Piecewise flat surfaces (Definition 1) were studied by (his student) Alexandrov [1] and more recently by Troyanov [20]. The idea of a Delaunay triangulation of a piecewise flat surface was apparentl... |

69 |
Euclidean Structures on Simplicial Surfaces and Hyperbolic Volume
- Rivin
- 1994
(Show Context)
Citation Context ...aces, however, the set of triangulations on the marked points may be infinite. (The fact that the number of750 Discrete Comput Geom (2007) 38: 740–756 combinatorial types of triangulations is finite =-=[19]-=- is not sufficient to make the argument.) For example, the surface of a cube has infinitely many geodesic triangulations on the eight vertices. To prove Proposition 12 by means of a function which dec... |

61 |
Finite elements for the Beltrami operator on arbitrary surfaces
- Dziuk
- 1988
(Show Context)
Citation Context ...) = ∑ xj ∈V :(xi,xj )∈E wij (f (xi) − f(xj )). Dziuk was the first to treat a finite element approach for the Laplace operator on simplicial surfaces, but without stating the cotan formula explicitly =-=[9]-=-. It seems to have been rediscovered by Pinkall and Polthier in their investigation of discrete minimal surfaces [17], and turned out to be extremely important in geometry processing where it found nu... |

58 | Delaunay Triangulations and Voronoi Diagrams for Riemannian Manifolds
- Leibon, Letscher
- 2000
(Show Context)
Citation Context ...t of cone points of the piecewise flat surface, so that the surface is flat away from the vertices. (This is very different from considering Delaunay triangulations in surfaces with Riemannian metric =-=[13]-=-.) Rivin claims but does not prove an existence and uniqueness theorem for Delaunay triangulations in piecewise flat surfaces. His proof that the edge flipping algorithm terminates is flawed (see the ... |

57 | Discrete Riemann surfaces and the Ising model
- Mercat
- 2001
(Show Context)
Citation Context ...ic parameterizations u : V → R 2 are used in computer graphics for texture mapping. The cotan-formula also forms the basis for a theory of discrete holomorphic functions and discrete Riemann surfaces =-=[8, 15]-=-. Two important disadvantages of this definition of a discrete Laplace operator are: 1. The weights may be negative. The properties of the discrete Laplace operator with positive weights (wij > 0) are... |

53 |
Sur la sphère vide. Izvestia Akademia Nauk
- Delaunay
- 1934
(Show Context)
Citation Context ...faces. We decided to give a detailed exposition because not all necessary proofs can be found elsewhere. The concept of a Delaunay triangulation in n-dimensional Euclidean space goes back to Delaunay =-=[5]-=-. Piecewise flat surfaces (Definition 1) were studied by (his student) Alexandrov [1] and more recently by Troyanov [20]. The idea of a Delaunay triangulation of a piecewise flat surface was apparentl... |

49 | Minimal roughness property of the delaunay triangulation - Rippa - 1990 |

42 | Minimal surfaces from circle patterns: Geometry from combinatorics
- BOBENKO, HOFFMANN, et al.
- 2006
(Show Context)
Citation Context ...the area functional under variations of the vertex positions [17]. 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]-=-. The mean curvature flow for simplicial surfaces is given by the equation df (x) = H(x). dt This flow may change the Delaunay triangulation of the surface. At some moment two Delaunay circles coincid... |

37 |
Les surfaces euclidiennes a singularites coniques. l'Ens
- Troyanov
(Show Context)
Citation Context ... a Delaunay triangulation in n-dimensional Euclidean space goes back to Delaunay [5]. Piecewise flat surfaces (Definition 1) were studied by (his student) Alexandrov [1] and more recently by Troyanov =-=[20]-=-. The idea of a Delaunay triangulation of a piecewise flat surface was apparently first considered by Rivin [19, Sect. 10]. The vertex set of the Delaunay triangulation is assumed to contain the set o... |

23 |
Potential theory on a rhombic lattice
- Duffin
- 1968
(Show Context)
Citation Context ...ic parameterizations u : V → R 2 are used in computer graphics for texture mapping. The cotan-formula also forms the basis for a theory of discrete holomorphic functions and discrete Riemann surfaces =-=[8, 15]-=-. Two important disadvantages of this definition of a discrete Laplace operator are: 1. The weights may be negative. The properties of the discrete Laplace operator with positive weights (wij > 0) are... |

19 |
Convex Polyhedra, Springer Monographs in Mathematics
- Alexandrov
- 2005
(Show Context)
Citation Context ...be found elsewhere. The concept of a Delaunay triangulation in n-dimensional Euclidean space goes back to Delaunay [5]. Piecewise flat surfaces (Definition 1) were studied by (his student) Alexandrov =-=[1]-=- and more recently by Troyanov [20]. The idea of a Delaunay triangulation of a piecewise flat surface was apparently first considered by Rivin [19, Sect. 10]. The vertex set of the Delaunay triangulat... |

18 | Properties of the Delaunay triangulation
- Musin
- 1997
(Show Context)
Citation Context ...has to terminate. As a further consequence such a function attains its minimal (or maximal) value on the Delaunay triangulations. Several such functions are known, see for example Lambert [12], Musin =-=[16]-=-, Rivin [19, Sect. 10], and the survey article [2]. When we consider Delaunay triangulations of PF surfaces, however, the set of triangulations on the marked points may be infinite. (The fact that the... |

11 |
Voronoi diagrams on piecewise flat surfaces and an application to biological growth, Theor
- Liebling, Troyanov, et al.
(Show Context)
Citation Context ...angulations in piecewise flat surfaces. 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. See our proof of Proposition 11.) Furthermore, for our definition of the discrete Laplace–Beltrami operator we also ... |

9 |
Distributed and Lumped Networks
- DUFFIN
- 1959
(Show Context)
Citation Context ...interior edges, wij = 1 2 cot αij for boundary edges and αij , αji are the angle(s) opposite edge (xi,xj) in the adjacent triangle(s) (see Fig. 1). This formula was, it seems, first derived by Duffin =-=[7]-=-, who considers triangulated planar regions. It follows (by summation over the triangles) from the observation that the Dirichlet energy of a linear function on a triangle (x1,x2,x3) with ∑ i∈Z/3Z cot... |

8 | The delaunay triangulation maximizes the mean inradius
- Lambert
- 1994
(Show Context)
Citation Context ...e algorithm has to terminate. As a further consequence such a function attains its minimal (or maximal) value on the Delaunay triangulations. Several such functions are known, see for example Lambert =-=[12]-=-, Musin [16], Rivin [19, Sect. 10], and the survey article [2]. When we consider Delaunay triangulations of PF surfaces, however, the set of triangulations on the marked points may be infinite. (The f... |

2 |
An intuitive framework for real time freeform modeling
- Botsch, Kobbelt
(Show Context)
Citation Context ...discovered by Pinkall and Polthier in their investigation of discrete minimal surfaces [17], and turned out to be extremely important in geometry processing where it found numerous applications, e.g. =-=[4, 6]-=- to name but two. In particular, harmonic parameterizations u : V → R 2 are used in computer graphics for texture mapping. The cotan-formula also forms the basis for a theory of discrete holomorphic f... |