Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDiscrete Koenigs nets and discrete isothermic surfaces
281
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We discuss discretization of Koenigs nets (conjugate nets with equal Laplace invariants) and of isothermic surfaces. Our discretization is based on the notion of dual quadrilaterals: two planar quadrilaterals are called dual, if their corresponding sides are parallel, and their non-corresponding diagonals are parallel. Discrete Koenigs nets are defined as nets with planar quadrilaterals admitting dual nets. Several novel geometric properties of discrete Koenigs nets are found; in particular, two-dimensional discrete Koenigs nets can be characterized by co-planarity of the intersection points of diagonals of elementary quadrilaterals adjacent to any vertex; this characterization is invariant with respect to projective transformations. Discrete isothermic nets are defined as circular Koenigs nets. This is a new geometric characterization of discrete isothermic surfaces introduced previously as circular nets with factorized cross-ratios.
rate research
Read More
Asymptotic net is an important concept in discrete differential geometry. In this paper, we show that we can associate affine discrete geometric concepts to an arbitrary non-degenerate asymptotic net. These concepts include discrete affine area, mean curvature, normal and co-normal vector fields and cubic form, and they are related by structural and compatibility equations. We consider also the particular cases of affine minimal surfaces and affine spheres.
We give an elaborated treatment of discrete isothermic surfaces and their analogs in different geometries (projective, Mobius, Laguerre, Lie). We find the core of the theory to be a novel projective characterization of discrete isothermic nets as Moutard nets. The latter belong to projective geometry and are nets with planar faces defined through a five-point property: a vertex and its four diagonal neighbors span a three dimensional space. Analytically this property is equivalent to the existence of representatives in the space of homogeneous coordinates satisfying the discrete Moutard equation. Restricting the projective theory to quadrics, we obtain Moutard nets in sphere geometries. In particular, Moutard nets in Mobius geometry are shown to coincide with discrete isothermic nets. The five-point property in this particular case says that a vertex and its four diagonal neighbors lie on a common sphere, which is a novel characterization of discrete isothermic surfaces. Discrete Laguerre isothermic surfaces are defined through the corresponding five-plane property which requires that a plane and its four diagonal neighbors share a common touching sphere. Equivalently, Laguerre isothermic surfaces are characterized by having an isothermic Gauss map. We conclude with Moutard nets in Lie geometry.
The main result of this paper is a discrete Lawson correspondence between discrete CMC surfaces in R^3 and discrete minimal surfaces in S^3. This is a correspondence between two discrete isothermic surfaces. We show that this correspondence is an isometry in the following sense: it preserves the metric coefficients introduced previously by Bobenko and Suris for isothermic nets. Exactly as in the smooth case, this is a correspondence between nets with the same Lax matrices, and the immersion formulas also coincide with the smooth case.
In this paper we are interested in defining affine structures on discrete quadrangular surfaces of the affine three-space. We introduce, in a constructive way, two classes of such surfaces, called respectively indefinite and definite surfaces. The underlying meshes for indefinite surfaces are asymptotic nets satisfying a non-degeneracy condition, while the underlying meshes for definite surfaces are non-degenerate conjugate nets satisfying a certain natural condition. In both cases we associate to any of these nets several discrete affine invariant quantities: a metric, a normal and a co-normal vector fields, and a mean curvature. Moreover, we derive structural and compatibility equations which are shown to be necessary and sufficient conditions for the existence of a discrete quadrangular surface with a given affine structure.
Discrete conformal structure on polyhedral surfaces is a discrete analogue of the conformal structure on smooth surfaces, which includes tangential circle packing, Thurstons circle packing, inversive distance circle packing and vertex scaling as special cases and generalizes them to a very general context. Glickenstein conjectured the rigidity of discrete conformal structures on polyhedral surfaces, which includes Luos conjecture on the rigidity of vertex scaling and Bowers-Stephensons conjecture on the rigidity of inversive distance circle packings on polyhedral surfaces as special cases. We prove Glickensteins conjecture using a variational principle. We further study the deformation of discrete conformal structures on polyhedral surfaces by combinatorial curvature flows. It is proved that the combinatorial Ricci flow for discrete conformal structures, which is a generalization of Chow-Luos combinatorial Ricci flow for circle packings and Luos combinatorial Yamabe flow for vertex scaling, could be extended to exist for all time and the extended combinatorial Ricci flow converges exponentially fast for any initial data if the discrete conformal structure with prescribed combinatorial curvature exists. This confirms another conjecture of Glickenstein on the convergence of the combinatorial Ricci flow and provides an effective algorithm for finding discrete conformal structures with prescribed combinatorial curvatures. The relationship of discrete conformal structures on polyhedral surfaces and 3-dimensional hyperbolic geometry is also discussed. As a result, we obtain some new convexities of the co-volume functions for some generalized 3-dimensional hyperbolic tetrahedra.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
