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

Lam, Wai Yeung

doi:10.1093/imrn/rnw267

Cited by 23 publications

(23 citation statements)

References 39 publications

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“…A similar formula for discrete minimal surfaces is elaborated in [28], where a notion of holomorphic quadratic differentials is introduced to link discrete harmonic functions and discrete minimal surfaces by considering infinitesimal conformal deformations. Furthermore, it turns out that our definition of discrete minimal surfaces bridges the gap between earlier notions of discrete minimal surfaces [27].…”

Section: Example: Inscribed Triangular Meshesmentioning

confidence: 83%

Isothermic triangulated surfaces

Lam

Pinkall

2016

Math. Ann.

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Abstract. We found a class of triangulated surfaces in Euclidean space which have similar properties as isothermic surfaces in Differential Geometry. We call a surface isothermic if it admits an infinitesimal isometric deformation preserving the mean curvature integrand locally. We show that this class is Möbius invariant. Isothermic triangulated surfaces can be characterized either in terms of circle patterns or based on conformal equivalence of triangle meshes. This definition generalizes isothermic quadrilateral meshes.A consequence is a discrete analog of minimal surfaces. Here the Weierstrass data needed to construct a discrete minimal surface consist of a triangulated plane domain and a discrete harmonic function.

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Section: Example: Inscribed Triangular Meshesmentioning

confidence: 83%

Isothermic triangulated surfaces

Lam

Pinkall

2016

Math. Ann.

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“…The following discretization of A-nets appears several times in discrete differential geometry (cf. [2,12,20]). Definition 4.1 A discrete asymptotic net or discrete A-net is a map f : Z 2 → R 3 , wherein each vertex star is planar, i.e., the five points f , f 1 , f 2 , f1, f2 lie in a plane, as depicted in Fig.…”

Section: Discretization With Asymptotic Nets (A-nets)mentioning

confidence: 99%

Discretizations of Surfaces with Constant Ratio of Principal Curvatures

Jimenez

Müller

Pottmann

2019

Discrete Comput Geom

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Motivated by applications in architecture, we study surfaces with a constant ratio of principal curvatures. These surfaces are a natural generalization of minimal surfaces, and can be constructed by applying a Christoffel-type transformation to appropriate spherical curvature line parametrizations, both in the smooth setting and in a discretization with principal nets. We link this Christoffel-type transformation to the discrete curvature theory for parallel meshes and characterize nets that admit these transformations. In the case of negative curvature, we also present a discretization of asymptotic nets. This case is suitable for design and computation, and forms the basis for a special type of architectural support structures, which can be built by bending flat rectangular strips of inextensible material, such as sheet metal.

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“…This family of discrete surfaces can be regarded as an associate family of minimal surfaces and is investigated in [7].…”

Section: Remark 64mentioning

confidence: 99%

Holomorphic Vector Fields and Quadratic Differentials on Planar Triangular Meshes

Lam

Pinkall

2016

Advances in Discrete Differential Geometry

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Given a triangulated region in the complex plane, a discrete vector field Y assigns a vector Y i ∈ C to every vertex. We call such a vector field holomorphic if it defines an infinitesimal deformation of the triangulation that preserves length cross ratios. We show that each holomorphic vector field can be constructed based on a discrete harmonic function in the sense of the cotan Laplacian. Moreover, to each holomorphic vector field we associate in a Möbius invariant fashion a certain holomorphic quadratic differential. Here a quadratic differential is defined as an object that assigns a purely imaginary number to each interior edge. Then we derive a Weierstrass representation formula, which shows how a holomorphic quadratic differential can be used to construct a discrete minimal surface with prescribed Gauß map and prescribed Hopf differential.

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

Cited by 23 publications

References 39 publications

Isothermic triangulated surfaces

Isothermic triangulated surfaces

Discretizations of Surfaces with Constant Ratio of Principal Curvatures

Holomorphic Vector Fields and Quadratic Differentials on Planar Triangular Meshes

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