Introduction to Linear and Nonlinear Integrable Theories in Discrete Complex Analysis

Bücking, Ulrike

doi:10.1007/978-3-319-56666-5_4

Cited by 2 publications

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References 51 publications

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“…Then there exists a one parameter family of generalized Doyle spirals T(t) with cross-ratio function q(t) : E L → {q A (t), q B (t), q C (t)}, where q A (t) = exp(R(A, ϑ)e iϑ + ie iϑ (tIm[e −iϑ log(a)] + (1 − t)I(A, ϑ))) and q B (t) and q C (t) are defined analogously. The logarithmic derivatives q i j defined by (10) satisfy (11) and additionally v j :v j adjacent to triangle∆ q i j = 0 for all triangular faces ∆ of T L C . This last property means that this discrete holomorphic quadratic differential is integrable.…”

Section: E ϑ-mentioning

confidence: 99%

“…Furthermore, circle patterns (and circle packings) can be obtained from variational principles for the radii of the circles, see [17,9,23]. Moreover, two circle patterns with the same underlying combinatorics may be considered as discrete conformal map if all intersection angles of corresponding pairs of circles for incident vertices agree, see for example [25,23,11].…”

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Conformally symmetric triangular lattices and discrete $\vartheta$-conformal maps

Bücking

2018

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A. Two immersed triangulations in the plane with the same combinatorics are considered as preimage and image of a discrete immersion F. We compare the cross-ratios Q and q of corresponding pairs of adjacent triangles in the two triangulations. If for every pair the arguments of these cross-ratios (i.e. intersection angles of circumcircles) agree, F is a discrete conformal map based on circle patterns. Similarly, if for every pair the absolute values of the corresponding cross-ratios Q and q (i.e. length cross-ratios) agree, the two triangulations are discrete conformally equivalent. We introduce a new notion, discrete ϑ-conformal maps, which interpolates between these two known definitions of discrete conformality for planar triangulations. We prove that there exists an associated variational principle. In particular, discrete ϑ-conformal maps are unique minimizers of a locally defined convex functional F ϑ in suitable variables. Furthermore, we study conformally symmetric triangular lattices which contain examples of discrete ϑ-conformal maps.

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Section: E ϑ-mentioning

confidence: 99%

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confidence: 99%

Conformally symmetric triangular lattices and discrete $\vartheta$-conformal maps

Bücking

2018

Preprint

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Conformally Symmetric Triangular Lattices and Discrete ϑ-Conformal Maps

Bücking

2019

International Mathematics Research Notices

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Two immersed triangulations in the plane with the same combinatorics are considered as preimage and image of a discrete immersion $F$. We compare the cross-ratios $Q$ and $q$ of corresponding pairs of adjacent triangles in the two triangulations. If for every pair the arguments of these cross-ratios (i.e., intersection angles of circumcircles) agree, $F$ is a discrete conformal map based on circle patterns. Similarly, if for every pair the absolute values of the corresponding cross-ratios $Q$ and $q$ (i.e., length cross-ratios) agree, the two triangulations are discretely conformally equivalent. We introduce a new notion, discrete $\vartheta $-conformal maps, which interpolates between these two known notions of discrete conformality for planar triangulations. We prove that there exists an associated variational principle. In particular, discrete $\vartheta $-conformal maps are unique maximizers of a locally defined concave functional ${\mathcal{F}}_{\vartheta}$in suitable variables. Furthermore, we study conformally symmetric triangular lattices that contain examples of discrete $\vartheta $-conformal maps.

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Introduction to Linear and Nonlinear Integrable Theories in Discrete Complex Analysis

Cited by 2 publications

References 51 publications

Conformally symmetric triangular lattices and discrete $\vartheta$-conformal maps

Conformally symmetric triangular lattices and discrete $\vartheta$-conformal maps

Conformally Symmetric Triangular Lattices and Discrete ϑ-Conformal Maps

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