3,178 research outputs found
A discrete Laplace-Beltrami operator for simplicial surfaces
We define a discrete Laplace-Beltrami operator for simplicial surfaces. It
depends only on the intrinsic geometry of the surface and its edge weights are
positive. Our Laplace operator is similar to the well known finite-elements
Laplacian (the so called ``cotan formula'') except that it is based on the
intrinsic Delaunay triangulation of the simplicial surface. This leads to new
definitions of discrete harmonic functions, discrete mean curvature, and
discrete minimal surfaces. The definition of the discrete Laplace-Beltrami
operator depends on the existence and uniqueness of Delaunay tessellations in
piecewise flat surfaces. While the existence is known, we prove the uniqueness.
Using Rippa's Theorem we show that, as claimed, Musin's harmonic index provides
an optimality criterion for Delaunay triangulations, and this can be used to
prove that the edge flipping algorithm terminates also in the setting of
piecewise flat surfaces.Comment: 18 pages, 6 vector graphics figures. v2: Section 2 on Delaunay
triangulations of piecewise flat surfaces revised and expanded. References
added. Some minor changes, typos corrected. v3: fixed inaccuracies in
discussion of flip algorithm, corrected attributions, added references, some
minor revision to improve expositio
Discrete conformal maps and ideal hyperbolic polyhedra
We establish a connection between two previously unrelated topics: a
particular discrete version of conformal geometry for triangulated surfaces,
and the geometry of ideal polyhedra in hyperbolic three-space. Two triangulated
surfaces are considered discretely conformally equivalent if the edge lengths
are related by scale factors associated with the vertices. This simple
definition leads to a surprisingly rich theory featuring M\"obius invariance,
the definition of discrete conformal maps as circumcircle preserving piecewise
projective maps, and two variational principles. We show how literally the same
theory can be reinterpreted to addresses the problem of constructing an ideal
hyperbolic polyhedron with prescribed intrinsic metric. This synthesis enables
us to derive a companion theory of discrete conformal maps for hyperbolic
triangulations. It also shows how the definitions of discrete conformality
considered here are closely related to the established definition of discrete
conformality in terms of circle packings.Comment: 62 pages, 22 figures. v2: typos corrected, references added and
updated, minor changes in exposition. v3, final version: typos corrected,
improved exposition, some material moved to appendice
Analytic Approximations for Transit Light Curve Observables, Uncertainties, and Covariances
The light curve of an exoplanetary transit can be used to estimate the
planetary radius and other parameters of interest. Because accurate parameter
estimation is a non-analytic and computationally intensive problem, it is often
useful to have analytic approximations for the parameters as well as their
uncertainties and covariances. Here we give such formulas, for the case of an
exoplanet transiting a star with a uniform brightness distribution. We also
assess the advantages of some relatively uncorrelated parameter sets for
fitting actual data. When limb darkening is significant, our parameter sets are
still useful, although our analytic formulas underpredict the covariances and
uncertainties.Comment: 33 pages, 14 figure
Motion among random obstacles on a hyperbolic space
We consider the motion of a particle along the geodesic lines of the
Poincar\`e half-plane. The particle is specularly reflected when it hits
randomly-distributed obstacles that are assumed to be motionless. This is the
hyperbolic version of the well-known Lorentz Process studied by Gallavotti in
the Euclidean context. We analyse the limit in which the density of the
obstacles increases to infinity and the size of each obstacle vanishes: under a
suitable scaling, we prove that our process converges to a Markovian process,
namely a random flight on the hyperbolic manifold.Comment: 19 pages, 4 figure
Wavelet methods for a weighted sparsity penalty for region of interest tomography
We consider region of interest (ROI) tomography of piecewise constant functions. Additionally, an algorithm is developed for ROI tomography of piecewise constant functions using a Haar wavelet basis. A weighted ℓp–penalty is used with weights that depend on the relative location of wavelets to the region of interest. We prove that the proposed method is a regularization method, i.e., that the regularized solutions converge to the exact piecewise constant solution if the noise tends to zero. Tests on phantoms demonstrate the effectiveness of the method.FWF, T 529-N18, Mumford-Shah models for tomography IINSF, 1311558, Tomography and Microlocal AnalysisFWF, W 1214, Doktoratskolleg "Computational Mathematics
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