Hyperboloidal Surfaces Research Articles

Generalized dark-bright vector soliton solution to the mixed coupled nonlinear Schrödinger equations.

We have constructed a dark-bright N-soliton solution with 4N+3 real parameters for the physically interesting system of mixed coupled nonlinear Schrödinger equations. Using this as well as an asymptotic analysis we have investigated the interaction between dark-bright vector solitons. Each colliding dark-bright one-soliton at the asymptotic limits includes more coupling parameters not only in the polarization vector but also in the amplitude part. Our present solution generalizes the dark-bright soliton in the literature with parametric constraints. By exploiting the role of such coupling parameters we are able to control certain interaction effects, namely beating, breathing, bouncing, attraction, jumping, etc., without affecting other soliton parameters. Particularly, the results of the interactions between the bound state dark-bright vector solitons reveal oscillations in their amplitudes under certain parametric choices. A similar kind of effect was also observed experimentally in the BECs. We have also characterized the solutions with complicated structure and nonobvious wrinkle to define polarization vector, envelope speed, envelope width, envelope amplitude, grayness, and complex modulation. It is interesting to identify that the polarization vector of the dark-bright one-soliton evolves on a spherical surface instead of a hyperboloid surface as in the bright-bright case of the mixed coupled nonlinear Schrödinger equations.

Dual universe and hyperboloidal relative velocity surface arising from extended special relativity

In two recent and completely independent articles (Hill and Cox in Proc R Soc A 468:4174, 2012; Vieira in Rev Bras Ensino F’s 34(3):1–15, 2012), the authors propose an extension of the Lorentz transformations of special relativity which are applicable to velocities in excess of the speed of light and do not involve the need to introduce imaginary quantities which are difficult to reconcile with everyday experience. These independent derivations, obtained from entirely distinct perspectives, mean that there is now some commonality of agreement in the basic equations underlying superluminal motion. One consequence of the new theories is that the standard special relativity formula for the addition of relative velocities also applies for velocities in excess of the speed of light. The new theories are based on the assumption that for any two inertial frames separated by an infinite relative velocity, the product of the two measured velocities for the same particle must necessarily be the square of the speed of light. Here, we amplify the major physical consequences embodied in the theory, including the surprising and novel idea of the co-existence of two “worlds”, such that in a subluminal world, everything is travelling with speeds less than the speed of light, while in the superluminal world, everything is travelling with speeds greater than the speed of light. We also establish the remarkable result that the reciprocal surface for the relative velocity formula can be re-orientated as a fully axially symmetric hyperboloidal surface, the full physical implications of which are not altogether transparent.

Discretization of asymptotic line parametrizations using hyperboloid surface patches

Two-dimensional affine A-nets in 3-space are quadrilateral meshes that discretize surfaces parametrized along asymptotic lines. The characterizing property of A-nets is planarity of vertex stars, so for generic A-nets the elementary quadrilaterals are skew. We classify the simply connected affine A-nets that can be extended to continuously differentiable surfaces by gluing hyperboloid surface patches into the skew quadrilaterals. The resulting surfaces are called "hyperbolic nets" and are a novel piecewise smooth discretization of surfaces parametrized along asymptotic lines. It turns out that a simply connected affine A-net has to satisfy one combinatorial and one geometric condition to be extendable - all vertices have to be of even degree and all quadrilateral strips have to be "equi-twisted". Furthermore, if an A-net can be extended to a hyperbolic net, then there exists a 1-parameter family of such C^1-surfaces. It is briefly explained how the generation of hyperbolic nets can be implemented on a computer. The article uses the projective model of Pluecker geometry to describe A-nets and hyperboloids.

Refraction of C-line vortices

The refraction and reflection of a plane electromagnetic wave at an interface is governed by the well-known Fresnel coefficients. The incident wave, as a single plane wave, has a Fourier transform, which is a δ-function in wavevector space k. And the transmitted and reflected waves are similar with their δ-functions on different k’s. We study instead the refraction and reflection of the natural elaboration of a plane wave; a ‘differentiated plane wave’; the first derivative of a plane wave with respect to its direction, whose Fourier transform is, in addition, the derivative of a δ-function. The motivation is that unlike a plane wave this wave has features, such as C-line vortices, and, being so primitive, exact analytical expressions and universal forms are obtained. The refracted and reflected waves are also differentiated plane waves. This construction is loosely connected with the analysis of beam shifts. The wave field geometry is naturally expressed in terms of its C-lines, or vortices of circular polarization, suitably defined. For a single such wave, there are two straight C-lines parallel to the propagation direction, one right-handed and one left-handed. When superposed with another such wave (the reflected wave), differently polarized and in a different direction, each straight C-line becomes a three-dimensional coiled shape on a hyperboloid surface. Finally, separately, for scalar waves, a local ‘law of vortex refraction’ is derived, independent of any specific model of the incident wave.

Intraocular Lens Implant with Mirrors and Intraocular Three Lenses Implant with a Changed Curvature Radius

Two different intraocular lens implants are suggested for age macula degeneration (AMD) eye disease that causes the loss of central vision. An intraocular Lens implant that includes two mirrors with a continuity change in their curvature radius and an intraocular hyperboloid surface lens implant closed by two lenses also with a continuity change in their curvature radiuses and includes an additional lens between the two. Thus, both eyes can see an increased image of a near central object projected on the retina, removed from the degenerated macula and also see the peripheral objects images decreasing continuity according to the continuity change in the curvature radiuses of the mirrors and lenses.

Determination of first-order derivatives of skew ray at aspherical surface

A set of algorithms is proposed for determining the first-order derivatives of a skew ray with respect to all the independent variables of an aspherical boundary surface in a general 3D optical system. Ellipsoidal, paraboloidal, hyperboloidal, and cylindrical boundary surfaces are given as examples. The proposed method has important applications in the analysis of aspherical surfaces and enables a detailed understanding of a wide variety of optical effects, including reflection and refraction, the modulation transfer function, and aberrations.

Skew ray tracing and sensitivity analysis of hyperboloid optical boundary surfaces

One of the most popular mathematical tools in fields of robotics, mechanisms and computer graphics is the 4×4 homogeneous transformation matrix. Our group's previous application of the homogeneous transformation matrix to flat and spherical optical boundaries has been extended to hyperboloid surfaces for: (1) skew ray tracing to determine the paths of reflected/refracted skew rays; (2) sensitivity analysis for direct mathematical expression of differential changes of incident points and reflected/refracted vectors with respect to changes in incident light sources and boundary geometric parameters; (3) a sensitivity analysis-based merit function derived directly from mathematical expression of catadioptric imaging system components. The presented methodology is highly suited to digital implementation and offers direct and rapid analytical statement of ray path, chief ray, marginal rays and merit functions of optical systems.

Using two global positioning system satellites to improve wireless fidelity positioning accuracy in urban canyons

It is well known that a global positioning system (GPS) receiver needs to ‘see’ at least four satellites to provide a three-dimensional fix solution. But in difficult environments such as an urban canyon, the number of ‘visible’ satellites is often not enough. Wireless fidelity (WiFi) signals have been utilised for positioning mainly based on the fingerprinting technology. However, the accuracy of WiFi positioning outdoors is from several tens of metres to more than one hundred metres. This study proposes a new methodology to integrate WiFi positioning technology and GPS to improve positioning accuracy in urban canyons. When only two GPS satellites are visible, the pseudorange observations can be used to generate a time difference of arrival (TDOA) measurement. The TDOA generates a hyperboloid surface which can be intersected with the surface of the Earth and shows the possible location of the user on that line of position. Integrating this method with the WiFi fingerprinting technology can increase the positioning performance significantly. The test results show that the positioning accuracy can be improved by more than 50% if the new method can be applied.

Force on a Conducting Tip near a Metallic Surface Coated with a Polarizable Dielectric Layer: Theory and Experiment

In the present study, we have derived a formula for the electrostatic force between the conducting tip and a metallic sample coated with a thin layer of polarizable dielectric of known dielectric constant by exploiting the fact that the tip-sample configuration in scanning probe microscopes can be represented as confocal hyperboloid surfaces of revolution in a prolate spheroidal coordinate system. The general behavior of the dependence of the force on tip-sample separation and the thickness of the dielectric film has been experimentally verified using a conducting atomic force microscope. The results are in better agreement with prolate spheroidal coordinate system over a wide range of tip-sample distances than the results obtained by the widely used sphere-on-plane approximation. [DOI: 10.1380/ejssnt.2011.206]

Asymptotics of Schwarzschild black hole perturbations

We study linear gravitational perturbations of Schwarzschild spacetime by solving numerically Regge–Wheeler–Zerilli equations in a time domain using hyperboloidal surfaces and a compactifying radial coordinate. We stress the importance of including the asymptotic region in the computational domain in studies of gravitational radiation. The hyperboloidal approach should be helpful in a wide range of applications employing black hole perturbation theory.

Gravitational perturbations of Schwarzschild spacetime at null infinity and the hyperboloidal initial value problem

We study gravitational perturbations of Schwarzschild spacetime by solving a hyperboloidal initial value problem for the Bardeen–Press equation. Compactification along hyperboloidal surfaces in a scri-fixing gauge allows us to have access to the gravitational waveform at null infinity in a general setup. We argue that this hyperboloidal approach leads to a more accurate and efficient calculation of the radiation signal than the common approach where a timelike outer boundary is introduced. The method can be generalized to study perturbations of Kerr spacetime using the Teukolsky equation.

Exact and geometrical optics energy trajectories in twisted beams

Energy trajectories, that is, integral curves of the Poynting (current) vector, are calculated for scalar Bessel and Laguerre–Gauss beams carrying orbital angular momentum. The trajectories for the exact waves are helices, winding on cylinders for Bessel beams and hyperboloidal surfaces for Laguerre–Gauss beams. In the geometrical optics approximations, the trajectories for both types of beam are overlapping families of straight skew rays lying on hyperboloidal surfaces; the envelopes of the hyperboloids are the caustics: a cylinder for Bessel beams and two hyperboloids for Laguerre–Gauss beams.

Deployable membrane structure based on the Bennett linkage

Deplayable structures based on the alternative forms of the Bennett linkage have been proposed and investigated by some researchers, which can be fully folded to a bundle and deployed to a flat surface. For aerospace applications, a deployable membrane structure based on the alternative form of the Bennett linkage is proposed and developed in the current paper. When fully deployed, four bars of the structure are not coplanar and the membrane tensioned by them can form a hyperboloidal surface. The compatibility between the structure and the membrane during the process of deployment is investigated, and preferred ranges of main design parameters are proposed to ensure the compatibility. The method to construct the assembly of the structure is also considered. In addition, three physical models were manufactured to verify the analytical investigations of the current paper.

Dirac's inspired point form and hadron form factors

Noticing that the “point-form” approach referred to in many recent works implies physics described on hyperplanes, an approach inspired from the Dirac's one, which involves a hyperboloid surface, is presented. A few features pertinent to this new approach are emphasized. Consequences as for the calculation of form factors are discussed.

Relativistic quantum mechanics: a Dirac's point-form inspired approach

This paper describes a tentative relativistic quantum mechanics approach inspired by Dirac's point-form, which is based on the physics description on a hyperboloid surface. It is mainly characterized by a non-standard relation of the constituent momenta of some system to its total momentum. Contrary to instant- and front-form approaches, where it takes the form of a 3-dimensional δ ( ⋯ ) function, the relation is given here by a Lorentz-scalar constraint. Thus, in the c.m. frame, the sum of the constituent momenta, which differs from zero off-energy shell, has no fixed direction, in accordance with the absence of preferred direction on a hyperboloid surface. To some extent, this gives rise to an extra degree of freedom entering the description of the system of interest. The development of a consistent formalism within this picture is described. Comparison with other approaches is made.

Development of a Geant4 solid for stereo mini-jet cells in a cylindrical drift chamber

Stereo mini-jet cells will be indispensable components of a future e + e − linear collider central tracker such as JLC-CDC. There is, however, no official Geant4 solid available at present to describe such geometrical objects, which had been a major obstacle for us to develop a full Geant4-based simulator with stereo cells built in. We have thus extended Geant4 to include a new solid ( TwistedTubs), which consists of three kinds of surfaces: two end planes, inner and outer hyperboloidal surfaces, and two so-called twisted surfaces that make slant and twisted φ-boundaries. Design philosophy and its realization in the Geant4 framework are described together with algorithmic details. We have implemented stereo cells with the new solid, and tested them using geantinos and Pythia events ( e + e −→ ZH at s =350 GeV). The performance was found reasonable: the stereo cells consumed only 25% more CPU time than ordinary axial cells.

Marching cube algorithm: review and trilinear interpolation adaptation for image-based dosimetric models

Current internal organ dose assessment methodologies utilize three-dimensional (3D) medical images of the body to model organ shapes and tissue interfaces. These models are coupled to computer programs that measure radionuclide energy deposition or chord-length distributions directly within these images. Previous studies have shown that the rectangular shape of image voxels generates voxel effects that alter the outcome of these calculations. To minimize voxel effects, the present study proposes to use the Marching Cube (MC) algorithm to generate isosurfaces delineating tissue interfaces from the gray-level images. First, a review of the different techniques surrounding the MC algorithm is presented. Next, an adaptation of the algorithm is proposed in which a trilinear interpolation of the gray levels is used to generate a hyperboloid surface within the MCs. This new technique is shown to solve the classic ambiguity problem of the MC algorithm and also to reduce the data size inherent to the triangulated surface. It also provides a simple algorithm to accurately measure distances within the image. The technique is then tested with a mathematical model of trabecular bone. The trilinear interpolation method is shown to remove voxel effects and to produce reliable chord-length distributions across image regions. The technique is thus recommended for use with digital medical images needed for internal radiation transport simulations. The current study is performed for a single isosurface that separates two media within the same image, but it is proposed that the technique can be extended to multiple isosurfaces that delineate several organs or organ regions within 3D tomographic voxels of human anatomy.

Baseline resolution of 1,2H and 3,4He isotope mixtures with a Paul ion trap mass spectrometer

A Paul ion trap mass spectrometer with fractional resolution of m/Δm=1565 was constructed and tested. The major improvement in this new trap over that previously reported with m/Δm=324 [Rev. Sci. Instrum. 73, 2157 (2002)] was due to tighter machining tolerances of the hyperboloidal surfaces, with minor improvement due to better stability of the amplitude of the radio frequency trapping voltage. The present resolution is shown to be sufficient to separate clearly (to baseline) the following masses: 4He+ and D2+ (0.0256 u separation), 3He+ and H3+ (0.0074 u), 3He+ and HD+ (0.0059 u), and 4He2+ and H2+ (0.0143 u separation). Resolution of 4He+ and HT+ (0.0043 u separation) should also be possible, but was not carried out. Resolution of H3+ and HD+ (0.0015 u separation) is possible with peaks of comparable intensity, or with deconvolution.

Spin-dependent Bohm trajectories associated with an electronic transition in hydrogen

The Bohm causal theory of quantum mechanics with spin-dependence is used to determine electron trajectories when a hydrogen atom is subjected to (semi-classical) radiation. The transition between the 1s ground state and the 2p0 state is examined. It is found that transitions can be identified along Bohm trajectories. The trajectories lie on invariant hyperboloid surfaces of revolution in R^3. The energy along the trajectories is also discussed in relation to the hydrogen energy eigenvalues.

On the Three Laws of Gearing

Three laws of gearing are presented in terms of a three link 1-dof spatial direct contact mechanism. The first law of gearing defines the instantaneous relationship between an infinitesimal displacement of an output body to an infinitesimal angular displacement of an input body for a specified tooth contact normal. A system of cylindroidal coordinates are introduced to facilitate a universal methodology to parameterize the kinematic geometry of generalized motion transmission between skew axes. The second law of gearing establishes a relation between the instantaneous gear ratio and the apparent radii of the hyperboloidal pitch surface in contact as parameterized using a system of cylindroidal coordinates. The third law of gearing establishes an instantaneous relationship for the relative curvature of two conjugate surfaces in direct contact and shows that this relation is independent of the tooth profile geometry. These three laws of gearing along with the system of cylindroidal coordinates establish, in part, a generalized geometric theory comparable to the existing theory for planar kinematics.