Related papers: Discrete channel surfaces
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…
We detail the theory of Discrete Riemann Surfaces. It takes place on a cellular decomposition of a surface, together with its Poincar\'e dual, equipped with a discrete conformal structure. A lot of theorems of the continuous theory follow…
We study Christoffel and Darboux transforms of discrete isothermic nets in 4-dimensional Euclidean space: definitions and basic properties are derived. Analogies with the smooth case are discussed and a definition for discrete Ribaucour…
In earlier papers we introduced a representation of isotopy classes of compact surfaces embedded in the three-sphere by so called rectangular diagrams. The formalism proved useful for comparing Legendrian knots. The aim of this paper 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…
A Darboux transformation for polarized space curves is introduced and its properties are studied, in particular, Bianchi permutability. Semi-discrete isothermic surfaces are described as sequences of Darboux transforms of polarized curves…
Given a trivalent graph in the 3-dimensional Euclidean space, we call it a discrete surface because it has a tangent space at each vertex determined by its neighbor vertices. To abstract a continuum object hidden in the discrete surface, we…
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…
We provide a convincing discretisation of Demoulin's $\Omega$-surfaces along with their specialisations to Guichard and isothermic surfaces with no loss of integrable structure.
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…
In this paper, we study some relationships existing between some particular mathematical structures: discrete surfaces coming from discrete topology and mathematical morphology, poset-based connected manifolds coming from discrete topology,…
We survey structure-preserving discretizations of minimal surfaces in Euclidean space. Our focus is on a discretization defined via parallel face offsets of polyhedral surfaces, which naturally leads to a notion of vanishing mean curvature…
Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. In this survey we discuss the following two fundamental Discretization Principles: the…
A canal surface is an envelope of a one parameter family of spheres. In this paper we present an efficient algorithm for computing the implicit equation of a canal surface generated by a rational family of spheres. By using Laguerre and Lie…
Inspired by the Weierstrass representation of smooth affine minimal surfaces with indefinite metric, we propose a constructive process producing a large class of discrete surfaces that we call discrete affine minimal surfaces. We show that…
We present a geometric framework for discrete classical field theories, where fields are modeled as "morphisms" defined on a discrete grid in the base space, and take values in a Lie groupoid. We describe the basic geometric setup and…
We describe and study geometric properties of discrete circular and spherical means of directional derivatives of functions, as well as discrete approximations of higher order differential operators. For an arbitrary dimension we present a…
This paper develops a discrete theory of real Riemann surfaces based on quadrilateral cellular decompositions (quad-graphs) and a linear discretization of the Cauchy-Riemann equations. We construct a discrete analogue of an antiholomorphic…
We define and discuss the notion of pseudospherical surfaces in asymptotic coordinates on time scales. Thus we extend well known notions of discrete pseudospherical surfaces and smooth pseudosperical surfaces on more exotic domains (e.g,…
While a generic smooth Ribaucour sphere congruence admits exactly two envelopes, a discrete R-congruence gives rise to a 2-parameter family of discrete enveloping surfaces. The main purpose of this paper is to gain geometric insights into…