Examinando por Materia "Geometry, differential"
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Ítem Gabriel-constrained Parametric Surface Triangulation(2008-10) Ruíz, Óscar E.; Cadavid, Carlos; Lalinde, Juan G.; Serrano, Ricardo; Peris-Fajarnés, Guillermo; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEThe Boundary Representation of a 3D manifold contains FACES (connected subsets of a parametric surface S : R2−R3) -- In many science and engineering applications it is cumbersome and algebraically difficult to deal with the polynomial set and constraints (LOOPs) representing the FACE -- Because of this reason, a Piecewise Linear (PL) approximation of the FACE is needed, which is usually represented in terms of triangles (i.e. 2-simplices) -- Solving the problem of FACE triangulation requires producing quality triangles which are: (i) independent of the arguments of S, (ii) sensitive to the local curvatures, and (iii) compliant with the boundaries of the FACE and (iv) topologically compatible with the triangles of the neighboring FACEs -- In the existing literature there are no guarantees for the point (iii) -- This article contributes to the topic of triangulations conforming to the boundaries of the FACE by applying the concept of parameter independent Gabriel complex, which improves the correctness of the triangulation regarding aspects (iii) and (iv) -- In addition, the article applies the geometric concept of tangent ball to a surface at a point to address points (i) and (ii) -- Additional research is needed in algorithms that (i) take advantage of the concepts presented in the heuristic algorithm proposed and (ii) can be proved correctÍtem Hessian Eigenfunctions for Triangular Mesh Parameterization(SCITEPRESS, 2016) Mejía, Daniel; Ruíz, Oscar; Cadavid, Carlos A.; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEHessian Locally Linear Embedding (HLLE) is an algorithm that computes the nullspace of a Hessian functional H for Dimensionality Reduction (DR) of a sampled manifold M -- This article presents a variation of classic HLLE for parameterization of 3D triangular meses -- Contrary to classic HLLE which estimates local Hessian nullspaces, the proposed approach follows intuitive ideas from Differential Geometry where the local Hessian is estimated by quadratic interpolation and a partition of unity is used to join all neighborhoods -- In addition, local average triangle normals are used to estimate the tangent plane TxM at x ∈ M instead of PCA, resulting in local parameterizations which reflect better the geometry of the surface and perform better when the mesh presents sharp features -- A high frequency dataset (Brain) is used to test our algorithm resulting in a higher rate of success (96.63%) compared to classic HLLE (76.4%)Ítem Manifold Learning with Orthogonal Geodesic Grids(2014) Ruíz, Óscar E.; Cadavid, Carlos; Ebratt, Roberto; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEIn Reverse Engineering, it is capital to find a parametric trimmed surface which approximates a triangular mesh (2-manifold with border) M in R3 -- This article proposes and implements a quasi isometry f: M -> R2 which allows a parameterization of M -- We consider quasi - developable 2- manifolds M in R3 -- f(p) = (u,w) with (u,w) being the coordinates of p in M under a grid of geodesic curves Ci(u) and Cj(w) on M -- We seek that the geodesic curves Ci(u) and Cj(w) be orthogonal to each other on M -- This means, that the Ci(u) should not cross each other, and each Ci(u) should intersect each Cj(w) in perpendicular mannerÍtem On a minimal factorization conjecture(Elsevier, 2007-08) Cadavid, Carlos A.; Vélez, Juan D.; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAELet be a proper holomorphic map from a connected complex surface S onto the open unit disk D⊂C, with 0∈D as its unique singular value, and having fiber genus g>0 -- Assume that in case g⩾2, admits a deformation whose singular fibers are all of simple Lefschetz type -- It has been conjectured that the factorization of the monodromy f∈M around ϕ (0) in terms of righ-thanded Dehn twists induced by the monodromy of has the least number of factors among all possible factorizations of f as a product of righthanded Dehn twists in the mapping class group (see [M. Ishizaka, One parameter families of Riemann surfaces and presentations of elements of mapping class group by Dehn twists, J. Math. Soc. Japan 58 (2) (2006) 585–594]) -- In this article, the validity of this conjecture is established for g=1Ítem PL-Geodesics on PL-continuous partial meshes(2001) Ruíz, Óscar E.; Cadavid M., Carlos A.; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEGeometric characteristics of 2-manifolds embedded in R3 space have been analyzed from the point of view of differential geometry and topology -- In the past, results relevant to these areas have been found for C∞ curves and surfaces -- However, current scientific, industrial, entertainment and medical applications, and availability of more powerful point sampling systems, press for characterization of discrete counterparts for the continuous properties and characteristics evaluated previously in C∞ curves and surfaces -- Recent works have presented estimation methods for properties such as the principal and rotated quadrics of point sampled surfaces -- The present article uses the findings of previous investigations to propose and implement a method for evaluation of planarity of surfaces -- It is based on:(i) Estimation of a C0 partial mesh fitting sets of planar or grid sample points -- (ii) Evaluation of the piecewise - linear (PL) version of families of geodesic curves on the mesh -- (iii) Diagnostic of the property of planarity based on the behavior of the families of geodesic curves -- The present work can be applied in the area of design and manufacturing of products based on sheet materials, such as apparel, metal stamping, thin structures, etcÍtem A q-exponential statistical Banach manifold(ELSEVIER, 2013-02) Quiceno Echavarría, Héctor Román; Loaiza Ossa, Gabriel Ignacio; department:Universidad EAFIT. Escuela de Ciencias. Grupo de Investigación Análisis Funcional y Aplicaciones; Héctor R. Quiceno (hquiceno@eafit.edu.co); Gabriel Loaiza (gloaiza@eafit.edu.co); Análisis Funcional y AplicacionesLetµbe a given probability measure andMµ the set ofµ-equivalent strictly positive probability densities -- In this paper we construct a Banach manifold on Mµ, modeled on the space L∞(p · µ) where p is a reference density, for the non-parametric q-exponential statistical models (Tsallis’s deformed exponential), where 0 < q < 1 is any real number -- This family is characterized by the fact that when q → 1, then the non-parametric exponential models are obtained and the manifold constructed by Pistone and Sempi is recovered, up to continuous embeddings on the modeling space -- The coordinate mappings of the manifold are given in terms of Csiszár’s Φ-divergences; the tangent vectors are identified with the one-dimensional q-exponential models and q-deformations of the score functionÍtem A Riemannian geometry in the q-Exponential Banach manifold induced by q-Divergences(Springer Berlin Heidelberg, 2013) Loaiza Ossa, Gabriel Ignacio; Quiceno Echavarría, Héctor Román; Universidad EAFIT. Escuela de Ciencias. Grupo de Investigación Análisis Funcional y Aplicaciones; Gabriel Loaiza (gloaiza@eafit.edu.co); Héctor R. Quiceno (hquiceno@eafit.edu.co); Análisis Funcional y AplicacionesFor the family of non-parametric q-exponential statistical models, in a former paper, written by the same authors, a differentiable Banach manifold modelled on Lebesgue spaces of real random variables has been built -- In this paper, the geometry induced on this manifold is characterized by q-divergence functionals -- This geometry turns out to be a generalization of the geometry given by Fisher information metric and Levi-Civita connections -- Moreover, the classical Amari´s α-connections appears as special case of the q−connections (q) -- The main result is the expected one, namely the zero curvature of the manifold