S-Conical CMC Surfaces. Towards a Unified Theory of Discrete Surfaces with Constant Mean Curvature

Bobenko, Alexander I.; Hoffmann, Tim

doi:10.1007/978-3-662-50447-5_9

Cited by 7 publications

(4 citation statements)

References 18 publications

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“…Equation (5) shows that Re(F) is A-minimal. Furthermore, Re(iF) can be written in the form of Equation (6). Exploiting Theorem 4.3.1, we conclude Re(iF) is C-minimal.…”

Section: Weierstrass Representationmentioning

confidence: 60%

“…Conical surfaces with vanishing mean curvature are called conical minimal surfaces. Though special conical minimal surfaces are recently studied [6], it is yet unknown if conical minimal surfaces in general admit conjugate minimal surfaces and an analogue of the Weierstrass representation. Furthermore, their relation to the variational approach is not clear.…”

Section: Introductionmentioning

confidence: 99%

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Discrete Minimal Surfaces: Critical Points of the Area Functional from Integrable Systems

Lam

2016

International Mathematics Research Notices

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Abstract. We obtain a unified theory of discrete minimal surfaces based on discrete holomorphic quadratic differentials via a Weierstrass representation. Our discrete holomorphic quadratic differential are invariant under Möbius transformations. They can be obtained from discrete harmonic functions in the sense of the cotangent Laplacian and Schramm's orthogonal circle patterns.We show that the corresponding discrete minimal surfaces unify the earlier notions of discrete minimal surfaces: circular minimal surfaces via the integrable systems approach and conical minimal surfaces via the curvature approach. In fact they form conjugate pairs of minimal surfaces.Furthermore, discrete holomorphic quadratic differentials obtained from discrete integrable systems yield discrete minimal surfaces which are critical points of the total area.

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“…Equation (5) shows that Re(F) is A-minimal. Furthermore, Re(iF) can be written in the form of Equation (6). Exploiting Theorem 4.3.1, we conclude Re(iF) is C-minimal.…”

Section: Weierstrass Representationmentioning

confidence: 60%

Section: Introductionmentioning

confidence: 99%

Discrete Minimal Surfaces: Critical Points of the Area Functional from Integrable Systems

Lam

2016

International Mathematics Research Notices

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“…We introduce orthogonal ring patterns that are natural generalizations of circle patterns. Our theory of orthogonal ring patterns has its origin in discrete differential geometry of S-isothermic cmc surfaces [3]. Recently, orthogonal double circle patterns (ring patterns) on the sphere have been used to construct discrete surfaces S-cmc by Tellier et al [7].…”

Section: Introductionmentioning

confidence: 99%

Orthogonal ring patterns in the plane

Bobenko,

Hoffmann,

Rörig

2023

Geom Dedicata

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We introduce orthogonal ring patterns consisting of pairs of concentric circles generalizing circle patterns. We show that orthogonal ring patterns are governed by the same equation as circle patterns. For every ring pattern there exists a one parameter family of patterns that interpolates between a circle pattern and its dual. We construct ring patterns analogues of the Doyle spiral, Erf and $$z^\alpha $$ z α functions. We also derive a variational principle and compute ring patterns based on Dirichlet and Neumann boundary conditions.

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Section: Introductionmentioning

confidence: 99%

Orthogonal ring patterns

Bobenko,

Hoffmann,

Rörig

2019

Preprint

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We introduce orthogonal ring patterns consisting of pairs of concentric circles generalizing circle patterns. We show that orthogonal ring patterns are governed by the same equation as circle patterns. For every ring pattern there exists a one parameter family of patterns that interpolates between a circle pattern and its dual. We construct ring pattern analogues of the Doyle spiral, Erf and z α functions. We also derive a variational principle and compute ring patterns based on Dirichlet and Neumann boundary conditions.

show abstract

S-Conical CMC Surfaces. Towards a Unified Theory of Discrete Surfaces with Constant Mean Curvature

Cited by 7 publications

References 18 publications

Discrete Minimal Surfaces: Critical Points of the Area Functional from Integrable Systems

Discrete Minimal Surfaces: Critical Points of the Area Functional from Integrable Systems

Orthogonal ring patterns in the plane

Orthogonal ring patterns

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