1,033 research outputs found

    Metric-measure boundary and geodesic flow on Alexandrov spaces

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    We relate the existence of many infinite geodesics on Alexandrov spaces to a statement about the average growth of volumes of balls. We deduce that the geodesic flow exists and preserves the Liouville measure in several important cases. The developed analytic tool has close ties to integral geometry

    O zakrivljenostima na trokutnim mreĆŸama

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    A face-based curvature estimation on triangle meshes is presented in this paper. A flexible disk is laid on the mesh around a given triangle. Such a bent disk is used as a geodesic neighborhood of the face for approximating normal and principal curvatures. The radius of the disk is free input data in the algorithm. Its influence on the curvature values and the stability of estimated principal directions are investigated in the examples.U članku je prikazana procjena zakrivljenosti na trokutnim mreĆŸama, bazirana na stranicama. Gipki disk poloĆŸen je na mreĆŸu oko danog trokuta. Takav prilagodljiv disk koristi se kao geodetska okolina stranice za aproksimaciju normalnih i glavnih zakrivljenosti. Polumjer diska je nezavisni ulazni podatak u algoritmu. U primjerima se istraĆŸuje njegov utjecaj na vrijednosti zakrivljenosti i na stabilnost procijenjenih glavnih smjerova

    Discrete spherical means of directional derivatives and Veronese maps

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    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 general construction for obtaining discrete spherical means of directional derivatives. The construction is based on using the Minkowski's existence theorem and Veronese maps. Approximating the directional derivatives by appropriate finite differences allows one to obtain finite difference operators with good rotation invariance properties. In particular, we use discrete circular and spherical means to derive discrete approximations of various linear and nonlinear first- and second-order differential operators, including discrete Laplacians. A practical potential of our approach is demonstrated by considering applications to nonlinear filtering of digital images and surface curvature estimation

    On the estimation of the curvatures and bending rigidity of membrane networks via a local maximum-entropy approach

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    We present a meshfree method for the curvature estimation of membrane networks based on the Local Maximum Entropy approach recently presented in (Arroyo and Ortiz, 2006). A continuum regularization of the network is carried out by balancing the maximization of the information entropy corresponding to the nodal data, with the minimization of the total width of the shape functions. The accuracy and convergence properties of the given curvature prediction procedure are assessed through numerical applications to benchmark problems, which include coarse grained molecular dynamics simulations of the fluctuations of red blood cell membranes (Marcelli et al., 2005; Hale et al., 2009). We also provide an energetic discrete-to-continuum approach to the prediction of the zero-temperature bending rigidity of membrane networks, which is based on the integration of the local curvature estimates. The Local Maximum Entropy approach is easily applicable to the continuum regularization of fluctuating membranes, and the prediction of membrane and bending elasticities of molecular dynamics models
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