arXiv research

Integrable nets described with curvature relations to pseudospherical surfaces.

Defines CAMC discrete nets and their properties.

We study discrete curvatures computed from nets of curvature lines on a given smooth surface, and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties of the smooth limit surface and the shape regularity of the discrete net.

We study local and global approximations of smooth nets of curvature lines and smooth conjugate nets by respective discrete nets (circular nets and planar quadrilateral nets) with infinitesimal quads. It is shown that choosing the points of discrete nets on the smooth surface one can obtain second-order approximation g…

Study properties of Weingarten surfaces using curvature nets.

Analytic plane curves determine unique conformal coordinates.

Circular nets with spherical parameter lines have geometric properties related to Darboux cyclides and terminating Laplace sequences.

2x2 Lax representation for circular nets with constant negative Gauss curvature.

Cyclidic nets are introduced as discrete analogs of curvature line parametrized surfaces and orthogonal coordinate systems. A 2-dimensional cyclidic net is a piecewise smooth $C^1$-surface built from surface patches of Dupin cyclides, each patch being bounded by curvature lines of the supporting cyclide. An explicit de…

An embedded cubic graph consisting of segments of geodesics such that the angles at any vertex are equal to $2π/3$ is a closed local minimal net. This net is regular if all segments of geodesics are equal. The problem of classification of closed local minimal nets on surfaces of constant negative curvature has been for…

Stable nets on convex hypersurfaces maintain their shape under small perturbations.

Isothermic nets created from special maps for smooth surfaces.

We consider a general theory of curvatures of discrete surfaces equipped with edgewise parallel Gauss images, and where mean and Gaussian curvatures of faces are derived from the faces' areas and mixed areas. Remarkably these notions are capable of unifying notable previously defined classes of surfaces, such as discre…

We show that the discrete principal nets in quadrics of constant curvature that have constant mixed area mean curvature can be characterized by the existence of a Königs dual in a concentric quadric.

Derives Ribaucour coordinates for curves and submanifolds, smoothing curvature line nets.

Geodesic nets with three vertices have at most one balanced vertex.

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 congruences is given. Surfaces of constant mean curvature are special among all iso…

New method efficiently learns positive-definite curvature for neural nets.

New Bäcklund transformations for discrete pseudospherical surfaces of revolution are found.

Closed geodesic nets on surfaces have limited branch points

We consider conformally flat hypersurfaces in four dimensional space forms with their associated Guichard nets and Lamé's system of equations. We show that the symmetry group of the Lamé's system, satisfying Guichard condition, is given by translations and dilations in the independent variables and dilations in the dep…

New discretizations of principal curvature lines discovered.

Principal binets generalize curvature line surfaces to square lattices and are a discrete integrable system.

Study on knotted spheres in 4D space with curvature and surface properties.

Knot theory is the study of isotopy classes of embeddings of the circle $S^1$ into a 3-manifold, specifically $R^3$. The Fáry-Milnor Theorem says that any curve in $R^3$ of total curvature less than $4π$ is unknotted. More generally, a (finite) graph consists of a finite number of edges and vertices. Given a topologica…

We investigate geometric aspects of the the Bäcklund transform of principal contact element nets. A Bäcklund transform exists if and only if it the principal contact element net is of constant negative Gaussian curvature (a pseudosphere). We describe an elementary construction of the Bäcklund transform and prove its co…

Study curvature loci of 3-manifolds in R^6 and R^5.

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 the…

Paper tackles optimization challenges in deep neural nets, presenting Newton-based methods.

The problem of determining the {\it Bonnet hypersurfaces in} $R^{n+1}$, for $n>1$, is studied here. These hypersurfaces are by definition those that can be isometrically mapped to another hypersurface or to itself (as locus) by at least one nontrivial isometry preserving the mean curvature. The other hypersurface and/o…

Discrete linear Weingarten surfaces in space forms are characterized as special discrete $Ω$-nets, a discrete analogue of Demoulin's $Ω$-surfaces. It is shown that the Lie-geometric deformation of $Ω$-nets descends to a Lawson transformation for discrete linear Weingarten surfaces, which coincides with the well-known L…

In this paper are studied the nets of principal curvature lines on surfaces embedded in Euclidean $3-$space near their end points, at which the surfaces tend to infinity. This is a natural complement and extension to smooth surfaces of the work of Garcia and Sotomayor (1996), devoted to the study of principal curvature…

We define a new notion of total curvature, called net total curvature, for finite graphs embedded in Rn, and investigate its properties. Two guiding principles are given by Milnor's way of measuring the local crookedness of a Jordan curve via a Crofton-type formula, and by considering the double cover of a given graph …

Sharp bounds on scalar curvature spectrum and rigidity theorems.

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 transformation group principle (smooth geometric objects and their discretizations are invar…

Integrable discretization preserves key properties of confocal quadrics.

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, the Cantor set). In particular, we present a new expression for the discrete Gauss…

We propose a unified definition for discrete analogues of constant mean curvature surfaces in spaces of constant curvature as a special case of discrete special isothermic nets. Bäcklund transformations and Lawson's correspondence are discussed. It is shown that the definition generalizes previous definitions and a con…

In the first part, we give a self contained introduction to the theory of cyclic systems in n-dimensional space which can be considered as immersions into certain Grassmannians. We show how the (metric) geometries on spaces of constant curvature arise as subgeometries of Moebius geometry which provides a slightly new v…

New discrete cmc surfaces defined from sphere packings and combinatorics.

Skew parallelogram nets factorize, encompassing discrete differential geometry.

The study constructs non-planar nets and webs on quadrics and Minkowski space.

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…

Preconditioned SGLD improves training of deep neural networks by adapting to curvature.

New bounds explain deterministic non-smooth deep nets without large Lipschitz constants.

Our work connects parameter magnitudes and Hessian eigenspaces in deep neural nets.

Classically, isothermic surfaces are characterized as those surfaces which are "divisible into infinitesimal squares by their curvature lines". This characterization is the direct analogue to the definition of discrete isothermic nets. In order to understand the relations between the discrete and the smooth theory bett…

Investigates the connection between quadrics and Christoffel duals, and zero mean curvature surfaces.