96,727 research outputs found
A coordinate system on a surface: definition, properties and applications
Coordinate systems associated to a finite set of sample points have been extensively studied, especially in the context of interpolation of multivariate scattered data. Notably, Sibson proposed the so-called natural neighbor coordinates that are defined from the Voronoi diagram of the sample points. A drawback of those coordinate systems is that their definition domain is restricted to the convex hull of the sample points. This makes them difficult to use when the sample points belong to a surface. To overcome this difficulty, we propose a new system of coordinates. Given a closed surface , i.e. a -manifold of , the coordinate system is defined everywhere on the surface, is continuous, and is local even if the sampling density is finite. Moreover, it is inherently -dimensional while the previous systems are -dimensional. No assumption is made about the ordering, the connectivity or topology of the sample points nor of the surface. We illustrate our results with an application to interpolation over a surface
A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces
In this paper we present a high-order kernel method for numerically solving
diffusion and reaction-diffusion partial differential equations (PDEs) on
smooth, closed surfaces embedded in . For two-dimensional
surfaces embedded in , these types of problems have received
growing interest in biology, chemistry, and computer graphics to model such
things as diffusion of chemicals on biological cells or membranes, pattern
formations in biology, nonlinear chemical oscillators in excitable media, and
texture mappings. Our kernel method is based on radial basis functions (RBFs)
and uses a semi-discrete approach (or the method-of-lines) in which the surface
derivative operators that appear in the PDEs are approximated using
collocation. The method only requires nodes at "scattered" locations on the
surface and the corresponding normal vectors to the surface. Additionally, it
does not rely on any surface-based metrics and avoids any intrinsic coordinate
systems, and thus does not suffer from any coordinate distortions or
singularities. We provide error estimates for the kernel-based approximate
surface derivative operators and numerically study the accuracy and stability
of the method. Applications to different non-linear systems of PDEs that arise
in biology and chemistry are also presented
Interpolation and scattered data fitting on manifolds using projected PowellāSabin splines
We present methods for either interpolating data or for fitting scattered data on a two-dimensional smooth manifold. The methods are based on a local bivariate Powell-Sabin interpolation scheme, and make use of a family of charts {(UĪ¾ , Ī¾)}Ī¾ā satisfying certain conditions of smooth dependence on Ī¾. If is a C2-manifold embedded into R3, then projections into tangent planes can be employed. The data fitting method is a two-stage method. We prove that the resulting function on the manifold is continuously differentiable, and establish error bounds for both methods for the case when the data are generated by a smooth function
A Novel Asymptotic Solution to the Sommerfeld Radiation Problem: Analytic field expressions and the emergence of the Surface Waves
The well-known "Sommerfeld radiation problem" of a small -Hertzian- vertical
dipole above flat lossy ground is reconsidered. The problem is examined in the
spectral domain, through which it is proved to yield relatively simple integral
expressions for the received Electromagnetic (EM) field. Then, using the Saddle
Point method, novel analytical expressions for the scattered EM field are
obtained, including sliding observation angles. As a result, a closed form
solution for the subject matter is provided. Also, the necessary conditions for
the emergence of the so-called Surface Wave are discussed as well. A complete
mathematical formulation is presented, with detailed derivations where
necessary.Comment: 14 pages, 3 figures, Submitted for publication to "Progress in
Electromagnetics Research" (PIER) at 21/09/201
Helicity Maximization of Structured Light to Empower Nanoscale Chiral Matter Interaction
Structured light enables the characterization of chirality of optically small
nanoparticles by taking advantage of the helicity maximization concept recently
introduced in[1]. By referring to fields with nonzero helicity density as
chiral fields, we first investigate the properties of two chiral optical beams
in obtaining helicity density localization and maximization requirements. The
investigated beams include circularly polarized Gaussian beams and also an
optical beam properly composed by a combination of a radially and an
azi-muthally polarized beam. To acquire further enhancement and localization of
helicity density beyond the diffraction limit, we also study chiral fields at
the vicinity of a spherical dielectric nanoantenna and demon-strate that the
helicity density around such a nanoantenna is a superposition of helicity
density of the illu-minating field, scattered field, and an interference
helicity term. Moreover, we illustrate when the nanoan-tenna is illuminated by
a proper combination of azimuthal and radially polarized beams, the scattered
nearfields satisfy the helicity maximization conditions beyond the diffraction
limit. The application of the concept of helicity maximization to nanoantennas
and generating optimally chiral nearfield result in helici-ty enhancement which
is of great advantage in areas like detection of nanoscale chiral samples,
microsco-py, and optical manipulation of chiral nanoparticles
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