Related papers: A Discrete Surface Theory

Finding appropriate notions of discrete holomorphic maps and, more generally, conformal immersions of discrete Riemann surfaces into 3-space is an important problem of discrete differential geometry and computer visualization. We propose an…

This paper is the second in a series where we attempt to give a complete description of the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3-manifold. The key for understanding such surfaces is to…

The combination of words ``discrete curvature'' is only an apparent contradiction. In this survey we describe curvature notions associated with polygons, polyhedral surfaces, and with abstract polyhedral manifolds. Several theorems about…

This article introduces a new approach to discrete curvature based on the concept of effective resistances. We propose a curvature on the nodes and links of a graph and present the evidence for their interpretation as a curvature. Notably,…

In this paper we study curvature types of immersed surfaces in three-dimensional (normed or) Minkowski spaces. By endowing the surface with a normal vector field, which is a transversal vector field given by the ambient Birkhoff…

Discrete forms of the mean and directed curvature are constructed on piecewise flat manifolds, providing local curvature approximations for smooth manifolds embedded in both Euclidean and non-Euclidean spaces. The resulting expressions take…

The aim of this paper is to verify that the study of generic conformally flat hypersurfaces in 4-dimensional space forms is reduced to a surface theory in the standard 3-sphere. The conformal structure of generic conformally flat…

We define discrete flat surfaces in hyperbolic 3-space from the perspective of discrete integrable systems and prove properties that justify the definition. We show how these surfaces correspond to previously defined discrete constant mean…

We study a new discretization of the Gaussian curvature for polyhedral surfaces. This discrete Gaussian curvature is defined on each conical singularity of a polyhedral surface as the quotient of the angle defect and the area of the Voronoi…

We present a discrete theory for modeling developable surfaces as quadrilateral meshes satisfying simple angle constraints. The basis of our model is a lesser known characterization of developable surfaces as manifolds that can be…

We study a generalization of constant Gauss curvature -1 surfaces in Euclidean 3-space, based on Lorentzian harmonic maps, that we call pseudospherical frontals. We analyze the singularities of these surfaces, dividing them into those of…

Using Green's theorem we reduce the variation of the total mean curvature of a smooth surface in the Euclidean 3-space to a line integral of a special vector field and obtain the following well-known theorem as an immediate consequence: the…

In this paper we define and analyze singularities of discrete linear Weingarten surfaces with Weierstrass-type representations in $3$-dimensional Riemannian and Lorentzian spaceforms. In particular, we discuss singularities of discrete…

We define and discuss the notion of pseudospherical surfaces in asymptotic coordinates on time scales. Two special cases, namely dicrete pseudospherical surfaces and smooth pseudosperical surfaces are consistent with this description. In…

A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete…

It is well-known that for a surface in a 3-dimensional real space form the constancy of the mean curvature is equivalent to the harmonicity of the Gauss map. However, this is not true in general for surfaces in an arbitrary 3-dimensional…

This is an unrefereed lecture note based on lectures in 'Introductory Workshop on Discrete Differential Geometry' at Korea University on January 21--24, 2019. In this note, we discuss topological crystallography, which is a mathematical…

A classification of discrete integrable systems on quad-graphs, i.e. on surface cell decompositions with quadrilateral faces, is given. The notion of integrability laid in the basis of the classification is the three-dimensional…

We present the first steps of a procedure which discretises surface theory in classical projective differential geometry in such a manner that underlying integrable structure is preserved. We propose a canonical frame in terms of which the…

We present in this article a survey of recent results in value distribution theory for the Gauss maps of several classes of immersed surfaces in space forms, for example, minimal surfaces in Euclidean $n$-space ($n$=3 or 4), improper affine…