1,033 research outputs found
Metric-measure boundary and geodesic flow on Alexandrov spaces
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
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 Älanku 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
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
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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