The potential use of satellites in hyperbolic position finding.
Configurations of hyperbolic position-fixing systems using synchronous satellites, showing Loran-like networks on earth resulting from these configurations
HYPERBOLIC POSITION LINE↗
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LORAN-C, an overview
LORAN-C a highly accurate radio navigation positioning system which operates at an assigned frequency of 10 kHz, and provides phase-coded pulses to develop hyperbolic time-difference lines-of-position (LOP's) was evaluated. LORAN-C provides precise time and time interval to within plus or minus 5 microseconds of UTC. The steps taken to plan, install, operate, and maintain the LORAN-C system up to the year 2000 are discussed. Topics included in the discussion were: theory of operation, timing, chain lanning, group repetition interval, coding delay versus emission delay, chain calibration, chart verification, system accuracy, signal reliability, and future developments.
Thrall, W. J.↗
Classical path A and B functions for line broadening of positive ions
A, a and B functions for electron broadening of positive ions with hyperbolic path
Cooper, J.↗
Curvature-controlled geometrical lensing behavior in self-propelled colloidal particle systems
In many biological systems, the curvature of the surfaces cells live on influences their collective properties. Curvature should likewise influence the behavior of active colloidal particles. We show using molecular simulation of self-propelled active particles on surfaces of Gaussian curvature (both positive and negative) how curvature sign and magnitude can alter the system's collective behavior. Curvature acts as a geometrical lens and shifts the critical density of motility-induced phase separation (MIPS) to lower values for positive curvature and higher values for negative curvature, which we explain theoretically by the nature of parallel lines in spherical and hyperbolic space. Curvature also fluidizes dense MIPS clusters due to the emergence of defect patterns disrupting the crystalline order inside the clusters. Using our findings, we engineer three confining surfaces that strategically combine regions of different curvature to produce a host of novel dynamical behaviors, including cyclic MIPS on spherocylinders, directionally biased cyclic MIPS on spherocones, and position dependent cluster fluctuations on metaballs.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Barriers in the transition to global chaos in collisionless magnetic reconnection. II. Field line spectroscopy
The transitional phase from local to global chaos in the magnetic field of a reconnecting current layer is investigated. The identification of the ridges in the field of the finite time Lyapunov exponent as barriers to the field line motion is carried out adopting the technique of field line spectroscopy to analyze the radial position of a field line while it winds its way through partial stochastic layers and to compare the frequencies of the field line motion with the corresponding frequencies of the distinguished hyperbolic field lines that are the nonlinear generalizations of linear X-lines.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Medial axis and local thickness computation using the Fast Sweeping Method
This report describes an efficient and robust voxel-based methodology for computing the medial axis, local thickness, and distance-to-skeleton of arbitrary three-dimensional geometries. It is assumed that the object can be represented by an exact or approximate signed distance function on a discrete grid. The gradient of such function is used to formulate a hyperbolic partial differential equation (PDE) that models the collapse of the position vector in space. By exploiting the causality property of the PDE, the Fast Sweeping Method is able to obtain the solution in a finite number of sweeps independent of the mesh resolution. The intersection of characteristic lines leads to the formation of shocks and a discrete bisector function is used to identify the medial axis. The same PDE approach is used to compute the local thickness inside the object and obtain the distance-to-skeleton field. Multiple examples are given in two and three dimensions along with a resolution study. The methodology has optimal complexity and yields subsecond computational times for geometries with over a million zones on a single core. The methodology is also capable of parallelization across shared and distributed memory architectures.
97 MATHEMATICS AND COMPUTING↗
Compact wide-aperture hyperbolic analyzers
Compact electrostatic analyzers that employ a new geometry are described. They accept collimated beams of ions through relatively wide (non-slit-like) inlet apertures and focus them at various locations, depending on the ion energy per charge, along rectangular position-sensing detectors. Each analyzer has a dynamic range of 10 or more for a fixed deflection voltage. The focal line width at a given energy is a small fraction of the inlet aperture width, and the focal lines are approximately coplanar.
Curtis, C. C.↗
Computation of the inviscid supersonic flow over an external axial corner
A second-order finite-difference procedure is used to evaluate the inviscid supersonic flowfield surrounding an external axial corner composed of swept planar compression surfaces and representing the inlets on existing high-speed aircraft. The governing partial differential equations in conservation-law form are hyperbolic with respect to the axial coordinate and are solved iteratively by means of MacCormack's algorithm. The procedure treats both the peripheral shock wave and vortical singularities as discontinuities. Numerical results are presented for two parametric studies regarding the effects on the flowfield of varying the free-stream Mach number and the leading edge sweep of the horizontal wedge. Results of parametric Mach number study agree with the Mach number independence principle in that as the Mach number increases, such characteristics as shock shape, cross-flow sonic line location, and vortical singularity position approach an asymptote.
Kutler, P.↗
Preliminary estimates of Gulf Stream characteristics from TOPEX data and a precise gravimetric geoid
TOPEX sea surface height data has been used, with a gravimetric geoid, to calculate sea surface topography across the Gulf Stream. This topography was initially computed for nine tracks on cycles 21 to 29. Due to inaccurate geoid undulations on one track, results for eight tracks are reported. The sea surface topography estimates were used to calculate parameters that describe Gulf Stream characteristics from two models of the Gulf Stream. One model was based on a Gaussian representation of the velocity while the other was a hyperbolic representation of velocity or the sea surface topography. The parameters of the Gaussian velocity model fit were a width parameter, a maximum velocity value, and the location of the maximum velocity. The parameters of the hyperbolic sea surface topography model were the width, the height jump, position, and sea surface topography at the center of the stream. Both models were used for the eight tracks and nine cycles studied. Comparisons were made between the width parameters, the maximum velocities, and the height jumps. Some of the parameter estimates were found to be highly (0.9) correlated when the hyperbolic sea surface topography fit was carried out, but such correlations were reduced for either the Gaussian velocity fits or the hyperbolic velocity model fit. A comparison of the parameters derived from 1-year TOPEX data showed good agreement with values derived by Kelly (1991) using 2.5 years of Geosat data near 38 deg N, 66 deg W longitude. Accuracy of the geoid undulations used in the calculations was of order of +/- 16 cm with the accuracy of a geoid undulation difference equal to +/- 15 cm over a 100-km line in areas with good terrestrial data coverage. This paper demonstrates that our knowledge or geoid undulations and undulation differences, in a portion of the Gulf Stream region, is sufficiently accurate to determine characteristics of the jet when used with TOPEX altimeter data. The method used here has not been shown to be more accurate than methods that average altimeter data to form a reference surface used in analysis to obtain the Gulf Stream characteristics. However, the results show the geoid approach may be used in areas where lack of current meandering reduces the accuracy of the average surface procedure.
Rapp, Richard H.↗
A new method for recognizing quadric surfaces from range data and its application to telerobotics and automation
The problem of recognizing and positioning of objects in three-dimensional space is important for robotics and navigation applications. In recent years, digital range data, also referred to as range images or depth maps, have been available for the analysis of three-dimensional objects owing to the development of several active range finding techniques. The distinct advantage of range images is the explicitness of the surface information available. Many industrial and navigational robotics tasks will be more easily accomplished if such explicit information can be efficiently interpreted. In this research, a new technique based on analytic geometry for the recognition and description of three-dimensional quadric surfaces from range images is presented. Beginning with the explicit representation of quadrics, a set of ten coefficients are determined for various three-dimensional surfaces. For each quadric surface, a unique set of two-dimensional curves which serve as a feature set is obtained from the various angles at which the object is intersected with a plane. Based on a discriminant method, each of the curves is classified as a parabola, circle, ellipse, hyperbola, or a line. Each quadric surface is shown to be uniquely characterized by a set of these two-dimensional curves, thus allowing discrimination from the others. Before the recognition process can be implemented, the range data have to undergo a set of pre-processing operations, thereby making it more presentable to classification algorithms. One such pre-processing step is to study the effect of median filtering on raw range images. Utilizing a variety of surface curvature techniques, reliable sets of image data that approximate the shape of a quadric surface are determined. Since the initial orientation of the surfaces is unknown, a new technique is developed wherein all the rotation parameters are determined and subsequently eliminated. This approach enables us to position the quadric surfaces in a desired coordinate system. Experiments were conducted on raw range images of spheres, cylinders, and cones. Experiments were also performed on simulated data for surfaces such as hyperboloids of one and two sheets, elliptical and hyperbolic paraboloids, elliptical and hyperbolic cylinders, ellipsoids and the quadric cones. Both the real and simulated data yielded excellent results. Our approach is found to be more accurate and computationally inexpensive as compared to traditional approaches, such as the three-dimensional discriminant approach which involves evaluation of the rank of a matrix. Finally, we have proposed one other new approach, which involves the formulation of a mapping between the explicit and implicit forms of representing quadric surfaces. This approach, when fully realized, will yield a three-dimensional discriminant, which will recognize quadric surfaces based upon their component surfaces patches. This approach is faster than prior approaches and at the same time is invariant to pose and orientation of the surfaces in three-dimensional space.
Alvertos, Nicolas↗