Optimal Möbius Transformations for Information Visualization and Meshing

Bern, Marshall; Eppstein, David

doi:10.1007/3-540-44634-6_3

Cited by 25 publications

(23 citation statements)

References 29 publications

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“…The equation A similar interplay of geometries leads Bern and Eppstein, to another choice of a unique representative for each polyhedral type. Given n spheres in S d , Bern and Eppstein apply that Möbius transformation which makes the smallest sphere as large as possible [3]. It is not difficult to see that this Möbius transformation is unique up to post-composition with a rotation if n ≥ 3.…”

Section: Theoremmentioning

confidence: 99%

A unique representation of polyhedral types. Centering via M�bius transformations

Springborn

2004

Math. Z.

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For n ≥ 3 distinct points in the d-dimensional unit sphere S d ⊂ R d+1 , there exists a Möbius transformation such that the barycenter of the transformed points is the origin. This Möbius transformation is unique up to post-composition by a rotation. We prove this lemma and apply it to prove the uniqueness part of a representation theorem for 3-dimensional polytopes as claimed by Ziegler (1995): For each polyhedral type there is a unique representative (up to isometry) with edges tangent to the unit sphere such that the origin is the barycenter of the points where the edges touch the sphere. (2000): 52B10 In today's language, Steinitz' fundamental theorem of convex types [14], [15] (for a modern treatment see [7], [17]) states that the combinatorial types of convex 3-dimensional polyhedra correspond to the strongly regular cell decompositions of the 2-sphere. (A cell complex is regular if the closed cells are attached without identifications on the boundary. A regular cell complex is strongly regular if the intersection of two closed cells is a closed cell or empty.) Grünbaum and Shephard [8] posed the question whether for every combinatorial type there is a polyhedron with edges tangent to a sphere. This question has been answered affirmatively: Theorem 1 (Koebe [10], Andreev [1], [2], Thurston [16], Brightwell and Scheinerman [5], Schramm [12] Mathematics Subject Classification). For every combinatorial type of convex 3-dimensional polyhedra, there is a representative with edges tangent to the unit sphere. This representative is unique up to projective transformations which fix the sphere and do not make the polyhedron intersect the plane at infinity.

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Section: Theoremmentioning

confidence: 99%

A unique representation of polyhedral types. Centering via M�bius transformations

Springborn

2004

Math. Z.

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“…This problem of finding a radius-maximizing Möbius transformation may be expressed (using the connection between Möbius transformations and three-dimensional hyperbolic geometry) as a quasiconvex program, a problem of finding the minimum value of a pointwise maximum of quasiconvex functions. It may be solved either combinatorially in linear time using LP-type optimization algorithms or numerically using local improvement procedures; the theory of quasiconvex programs guarantees that there are no local optima in which the local improvement might get stuck [1,11]. Since we are already using a numerical method to find circle packings, our implementation also takes the numerical approach to find the best transformation.…”

Section: Circle Packingmentioning

confidence: 99%

Planar Lombardi Drawings for Subcubic Graphs

Eppstein

2013

Graph Drawing

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Abstract. We prove that every planar graph with maximum degree three has a planar drawing in which the edges are drawn as circular arcs that meet at equal angles around every vertex. Our construction is based on the Koebe-Thurston-Andreev circle packing theorem, and uses a novel type of Voronoi diagram for circle packings that is invariant under Möbius transformations, defined using three-dimensional hyperbolic geometry. We also use circle packing to construct planar Lombardi drawings of a special class of 4-regular planar graphs, the medial graphs of polyhedral graphs, and we show that not every 4-regular planar graph has a Lombardi drawing. We have implemented our algorithm for 3-connected planar cubic graphs.

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“…Quasiconvex programming is a general framework for LP-type computational geometry optimization problems, formulated by Amenta et al [4] in the context of mesh smoothing procedures, and later used by the authors for several applications in information visualization and mesh generation [5]. Define a nested convex family to be a function κ(t) mapping real values t to convex sets in some Euclidean space E d , such that for any t < t ′ , κ(t) ⊆ κ(t ′ ), and such that for all t, κ(t) = t ′ >t κ(t ′ ).…”

Section: Quasiconvex Programmingmentioning

confidence: 99%

Optimized color gamuts for tiled displays

Bern¹,

Eppstein

2003

Proceedings of the Nineteenth Conference on Computational Geometry - SCG '03

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Abstract. We consider the problem of finding a large color space that can be generated by all units in multiprojector tiled display systems. Viewing the problem geometrically as one of finding a large parallelepiped within the intersection of multiple parallelepipeds, and using colorimetric principles to define a volume-based objective function for comparing feasible solutions, we develop an algorithm for finding the optimal gamut in time O(n 3 ), where n denotes the number of projectors in the system. We also discuss more efficient quasiconvex programming algorithms for alternative objective functions based on maximizing the quality of the color space extrema.

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Optimal Möbius Transformations for Information Visualization and Meshing

Cited by 25 publications

References 29 publications

A unique representation of polyhedral types. Centering via M�bius transformations

A unique representation of polyhedral types. Centering via M�bius transformations

Planar Lombardi Drawings for Subcubic Graphs

Optimized color gamuts for tiled displays

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