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### Citations

130 |
Gesammelte mathematische Abhandlungen. Band I, II.
- Schwarz
- 1972
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Citation Context .... Scherk’s tower unsymmetric version Figure 2.28. From the plane to Scherk’s tower 25Figure 2.29. Scherk tower, medial combinatorics of a singularity Figure 2.30. Gergonne’s surface by H. A. Schwarz =-=[Sch90]-=- (left) and discrete analog (right) correspond to points at infinity. Figure 2.28 shows an attempt to cope with this behavior. As the angles at the corners approach π the vertices go to infinity and f... |

125 |
Kontaktprobleme der konformen Abbildung,
- Koebe
- 1936
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Citation Context ...orrespond to vertices and two circles touch if and only if the corresponding vertices are adjacent. This circle pattern is unique up to Möbius transformations of the sphere. The theorem, published in =-=[Koe36]-=-, can be generalized to polytopal cellular decompositions of the sphere (Definition 1.1.2) and to circle patterns. In this setting there is a circle for every vertex and every face and these circles i... |

79 | Variational principles for circle patterns and koebe’s theorem
- Bobenko, Springborn
- 2004
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Citation Context ...age Java. viCHAPTER 1 Koebe Polyhedra 1.1. Theory In this chapter we describe how to construct Koebe polyhedra, these are convex polyhedra with edges tangent to the sphere. We follow the approach of =-=[BS04]-=- and add implementational details. The goal is to create an algorithm that computes Koebe polyhedra with the combinatorics of a given planar graph. To understand for which planar graphs we can compute... |

54 | Minimal surfaces from circle patterns: Geometry from combinatorics
- Bobenko, Hoffmann, et al.
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Citation Context ...mal surfaces have been studied quite comprehensively in the past century. The latest efforts on this subject now seek for discrete analogs for the well developed theory of smooth minimal surfaces. In =-=[BHS06]-=- discrete minimal surfaces are defined and these definitions directly give rise to algorithms for calculating the surfaces. In the smooth case minimal surfaces are a subclass of isothermic surfaces an... |

20 |
Geometry and Topology for Mesh Generation, Cambridge Monographs on Applied and Computational Mathematics,
- Edelsbrunner
- 2001
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Citation Context ...ly Delaunay then 4: Flip ab tocd 5: for xy ∈ ac, cb, bd, da do 6: if xy is not marked then 7: mark xy and push it on S 8: end if 9: end for 10: end if 11: end while There is an algorithm described in =-=[Ede01]-=- pp. 7-9 that transforms an arbitrary triangulation into a Delaunay triangulation (Algorithm 1). The algorithm uses the flip transformation. The length of the flipped edge is then l ′2 = l 2 a + l 2 d... |

15 | Variational principles for circle patterns - Springborn |

11 |
Bobenko and Ivan Izmestiev. Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes. Annales de l’Institut Fourier
- Alexander
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Citation Context ...alog of the circle (Figure 2.49). Here we use the combinatorics of the generalized Schoen I-6 surface. 33Figure 2.49. Non-regular catenoid approximation 34CHAPTER 3 Alexandrov’s Theorem The article =-=[BI06]-=- provides a new poof of Alexandrov’s theorem. We implemented an algorithm following the constructive proof of the article. In the end the user could enter some metric via editor or file and calculate ... |

5 | Convex Polytopes: Extremal Constructions and f-Vector Shapes, IAS/Park City Mathematics Series
- Ziegler
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Citation Context ...onally, so that the resulting circle pattern is bounded by a rectangle. Figures 1.5 and 1.6 show the circle patterns of the cube and the icosahedron respectively. This proceeding is also described in =-=[Zie04]-=-. 5Figure 1.5. The medial combinatorics and circle pattern of the cube in the Euclidean plane Figure 1.6. The circle pattern of the icosahedron Assuming circles of infinite radius at the boundary, we... |

2 |
Java Tools for Experimental Mathematics, http://www.jtem.de
- TU-Berlin
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Citation Context ... and β = 0.5. See [BV04] pp. 463-466 for a comprehensive description of this step-size controller. We utilize the numerical algorithms of the “Matrix Toolkits for Java API [Hei]” and the package jtem =-=[TBa]-=- for linear system solving and matrix factorization. Minimizing the Euclidean functional produces correct radii for the given combinatorics in the plane. We calculate the coordinates (centers and inte... |

1 |
Convex optimization, Camebridge
- Boyd, Vandenberghe
- 2004
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Citation Context ... 0 < β < 1 and for every step find n ∈ N such that SEuc(ρ 0 + β n ρ s ) ≤ SEuc(ρ 0 ) + α < S ′ Euc(ρ 0 ), β n ρ s > . Then set ρ n−1 := ρ n +β n ρ s for the next step. We use α = 0.2 and β = 0.5. See =-=[BV04]-=- pp. 463-466 for a comprehensive description of this step-size controller. We utilize the numerical algorithms of the “Matrix Toolkits for Java API [Hei]” and the package jtem [TBa] for linear system ... |

1 |
Matrix toolkits for java (mtj), http://rs.cipr.uib.no/ mtj
- Heimsund
(Show Context)
Citation Context ...he next step. We use α = 0.2 and β = 0.5. See [BV04] pp. 463-466 for a comprehensive description of this step-size controller. We utilize the numerical algorithms of the “Matrix Toolkits for Java API =-=[Hei]-=-” and the package jtem [TBa] for linear system solving and matrix factorization. Minimizing the Euclidean functional produces correct radii for the given combinatorics in the plane. We calculate the c... |