We embark in a program of studying the problem of better approximating surfaces by triangulations(triangular meshes) by considering the approximating triangulations as finite metric spaces and the target smooth surface as their Haussdorff-Gromov limit. This allows us to define in a more natural way the relevant elements, constants and invariants s.a. principal directions and principal values, G...

Using fine electromagnetic signals to measure observables of other fields like curvature and torsion of a space, and the corresponding value of their integrals of the action of perception of curvature through electronic signals that detect curvature on a curved surface, it is designed and constructed a sensor of curvature of accelerometer type that detects and curvature measures in 2 and 3-dime...

Our principal goal is to study the prescribed curvature problem in a manifold with density. In particular, we consider the Euclidean 3-space R3 with a positive density function eφ, where φ = −x2 − y2, (x, y, z) ∈ R3 and construct all the helicoidal surfaces in the space by solving the second-order non-linear ordinary differential equation with the weighted Gaussian curvature and the mean curvat...

We prove that any strongly regular Weingarten surface in Euclidean space carries locally geometric principal parameters. The basic theorem states that any strongly regular Weingarten surface is determined up to a motion by its structural functions and the normal curvature function satisfying a geometric differential equation. We apply these results to the special Weingarten surfaces: minimal su...

Significant local structure of terrain surfaces can be described by structural lines which are connected sets of points where the surface is approximately cylindrical, i.e., the ratio of the principal curvatures is large. At each point curvature is maximal in the curvature direction associated with the curvature with larger absolute value. These lines form the skeleton of the surface for constr...

Estimating principal curvatures and principal directions of a surface from a polyhedral approximation with a large number of small faces, such as those produced by iso-surface construction algorithms, has become a basic step in many computer vision algorithms. Particularly in those targeted at medical applications. In this paper we describe a method to estimate the tensor of curvature of a surf...

This work concerns the approximation of the shape operator of smooth surfaces in R from polyhedral surfaces. We introduce two generalized shape operators that are vector-valued linear functionals on a Sobolev space of vector fields and can be rigorously defined on smooth and on polyhedral surfaces. We consider polyhedral surfaces that approximate smooth surfaces and prove two types of approxima...

In many geometry processing applications, the estimation of differential geometric quantities such as curvature or normal vector field is an essential step. In this paper, we investigate a new class of estimators on digital shape boundaries based on Integral Invariants. More precisely, we provide both proofs of multigrid convergence of principal curvature estimators and a complete experimental ...