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In this paper, we present a novel method for computing multiple geodesic connections between two arbitrary points on a smooth surface. Our method is based on a homotopy approach that is able to capture the ambiguity of geodesic connections in the presence of positive Gaussian curvature that generates focal curves. Contrary to previous approaches, we exploit focal curves to gain theoretical insights on the number of connecting geodesics and a practical algorithm for collecting these. We consider our method as a contribution to the contemporary debate regarding the calculation of distances in general situations, applying continuous concepts of classical differential geometry which are not immediately transferable in purely discrete settings.
Computing the spectral decomposition of the Laplace‚ÄďBeltrami operator on a manifold M has proven useful for applications such as shape retrieval and geometry processing. The standard operator acts on scalar functions which can be identified with sections of the trivial line bundle M √óR. In this work we propose to extend the discussion to Laplacians on nontrivial real line bundles. These line bundles are in one-to-one correspondence with elements of the first cohomology group of the manifold with Z2 coefficients. While we focus on the case of two-dimensional closed surfaces, we show that our method also applies to surfaces with boundaries.
Denoting by β the rank of the first cohomology group, there are 2β different line bundles to consider and each of these has a naturally associated Laplacian that possesses a spectral decomposition. Using our new method it is possible for the first time to compute the spectra of these Laplacians by a simple modification of the finite element basis functions used in the standard trivial bundle case. Our method is robust and efficient. We illustrate some properties of the modified spectra and eigenfunctions and indicate possible applications for shape processing. As an example, using our method, we are able to create spectral shape descriptors with increased sensitivity in the eigenvalues with respect to geometric deformations and to compute cycles aligned to object symmetries in a chosen homology class.
In the present work, we extend the theoretical and numerical discussion of the well-known Laplace‚ÄďBeltrami operator by equipping the underlying manifolds with additional structure provided by vector bundles. Focusing on the particular class of flat complex line bundles, we examine a whole family of Laplacians including the Laplace‚ÄďBeltrami operator as a special case. To demonstrate that our proposed approach is numerically feasible, we describe a robust and efficient finite-element discretization, supplementing the theoretical discussion with first numerical spectral decompositions of those Laplacians.
Our method is based on the concept of introducing complex phase discontinuities into the finite element basis functions across a set of homology generators of the given manifold. More precisely, given an m-dimensional manifold M and a set of n generators that span the relative homology group Hm-1(M, ∂ M), we have the freedom to choose n phase shifts, one for each generator, resulting in a n-dimensional family of Laplacians with associated spectra and eigenfunctions. The spectra and absolute magnitudes of the eigenfunctions are not influenced by the exact location of the paths, depending only on their corresponding homology classes.
Employing our discretization technique, we provide and discuss several interesting computational examples highlighting special properties of the resulting spectral decompositions. We examine the spectrum, the eigenfunctions and their zero sets which depend continuously on the choice of phase shifts.