Confocal Paraboloids Research Articles

Геометрические места точек, равноотстоящих от двух заданных геометрических фигур. Часть 5: геометрические места точек, равноудаленных от сферы и плоскости

In this paper have been investigated the loci equidistant from sphere and plane, and properties of obtained surfaces have been studied. Four options for possible mutual arrangement of plane and sphere have been considered: the plane passes through the center of the sphere; the plane intersects the sphere; the plane is tangent to the sphere; the plane passes outside the sphere.
 In all options of the mutual arrangement of the sphere and the plane, the loci are two surfaces - two coaxial confocal paraboloids of revolution. The general properties of the obtained paraboloids of revolution have been studied: foci and vertices positions, axes of rotation, the distance from the sphere center to the vertices of the paraboloids, the distance between the vertices of the paraboloids, and the position of the directorial planes have been defined.
 Have been derived equations for the surfaces of the loci equidistant from the sphere and the plane: various paraboloids of revolution.
 The loci in each of the four options for the possible mutual arrangement of the plane and the sphere are as follows. 1. The original plane passes through the sphere center – two coaxial confocal multidirectional paraboloids of revolution symmetric relative to the original plane. 2. The initial plane intersects the sphere – two coaxial confocal multidirectional but not symmetrical paraboloids of revolution, since the circle of intersection of the plane and the sphere does not coincide with the diameter of the sphere great circle. 3. The plane is tangent to the sphere – a paraboloid of revolution and a straight line (more precisely, a second order zero-quadric – a cylindrical surface with zero radius) passing through the tangency point of the plane and the sphere and the sphere center. 4. The plane passes outside the sphere – the equidistant loci will be two coaxial confocal unidirectional paraboloids of revolution.

Mersenne Beam-Compressor with Field-Flattener Optics as Main Telescope Design for Extremely Large Telescopes and Telescope Arrays

The Mersenne beam-compressor is the fundamental arrangement for two mirror reflecting telescopes. The Mersenne beam-compressor consists of two confocal paraboloids. The design produces an aberration-free, collimated output beam, which makes it easy to relocate the beam to any distant location. This property of Mersenne beam compressors makes them near-ideal for telescopes with requirements for ultra-stable instruments and interferometric applications. The only problem against such application is the Petzval field curvature that is induced in the output beam. This becomes very pronounced at very high compression ratios, which are required for interferometry with large aperture telescopes. The high compression ratio is a problem because, such large aperture telescopes are also required to deliver good performance at Nasmyth and Coude locations, to be feasible. This is not possible at high compression ratios using conventional design of Mersenne beam-compressor, due to prohibitively strong field curvature. Here, we propose powered and un-powered field-flattener optics to address the problem of field curvature in Mersenne beam-compressor. After passing through this set of optics, the output beam is completely aberration free. This arrangement can produce very high compression ratios, which are useful for various special applications. We also illustrate the performance of such a system using example designs. The performance analysis shows that such a design can give considerably high performance at distant locations of foci as compared to existing relay systems.

High-efficiency confocal paraboloids coupler design for natural light illumination systems

Recently, natural light illumination has become a hot topic in renewable energy research. Natural light illumination systems (NLISs) are used to guide sunlight for indoor illumination. The collector of NLISs is Lightbrick which consists of multi-prism. This paper proposes a novel design of confocal paraboloids couplers, which are important optical devices for NLISs. The coupler is designed with a prism optical structure and confocal paraboloids to collect a significant amount of inlet light from Lightbrick and compress it to one outlet light source. The prism structure causes the inlet light direction to coincide with a previous inlet light source, and the confocal paraboloid structure compresses two inlet light sources into one outlet. This newly designed confocal paraboloids coupler has high coupling efficiency, and its volume is smaller when the sunlight collection area is larger.

Analytic Modeling of Magnetic Field in the Magnetosheath and Outer Magnetosphere

AbstractMagnetosheath is a region surrounded by the bow shock and the magnetopause. According to statistical analysis, the boundaries can be treated as nonconfocal paraboloids of rotations. It is not easy to meet boundary conditions simultaneously on both sides. This problem is treated here and its solution is presented. A linear combination of harmonics that satisfy the condition on the magnetopause are selected in order to meet the condition on the bow shock. The method allows to calculate surface currents on the magnetopause. It is applied for typical conditions in solar wind. A modified magnetic dipole is used as a model of the magnetospheric field. Its modification is found by the method of potentials using cylindrical harmonics. Magnetopause currents are compared with results from a model with confocal paraboloids, which is commonly used in literature.

Design of high-efficiency confocal paraboloid optical coupler for daylighting

A set of high-efficiency optical coupler design methods that can be used to collect multiple light sources for compression and integrate them into one light source outlet is proposed. It adopts parabolic confocal principle and high reflectivity coating and completes the structural design with the formulation concept. In addition to direct sunlight collection, the optical coupler is also coupled to the exit source of the collection subsystem of a natural light illumination system (NLIS). The concept of NLIS is to collect sunlight and direct it into indoor lighting. Due to the numerous light sources and the nonparallel light exiting from the light collection subsystem, it is inconvenient for space utilization and back-end transmission. Therefore, the actual NLIS structure is considered as a coupler to improve the efficiency of this design. The optical software used in this study was Fred. Finally, coupling 900 light sources still maintains 50.28% coupler efficiency. The results show that this optical coupler can provide perfect use in the application, and also solve the export problem, and finally achieve high coupler efficiency.

Far-field reflector problem and intersection of paraboloids

In this article, we propose a numerical approach to the far field reflector problem which is an inverse problem arising in geometric optics. Caffarelli et al. (Contemp Math 226:13---32, 1999) proposed an algorithm that involves the computation of the intersection of the convex hull of confocal paraboloids. We show that computing this intersection amounts to computing the intersection of a power diagram (a generalization of the Voronoi diagram) with the unit sphere. This allows us to provide an algorithm that computes efficiently the intersection of confocal paraboloids using the exact geometric computation paradigm. Furthermore, using an optimal transport formulation, we cast the far field reflector problem into a concave maximization problem. This allows us to numerically solve the far field reflector problem with up to 15k paraboloids. We also investigate other geometric optic problems that involve union of confocal paraboloids and also intersection and union of confocal ellipsoids. In all these cases, we show that the computation of these surfaces is equivalent to the computation of the intersection of a power diagram with the unit sphere.

Calculation of permittivity tensors for invisibility devices by effective medium approach in general relativity

Permittivity tensors of arbitrary shaped invisibility devices were obtained using an effective medium approach in general relativity. As special cases, analytical expressions for the permittivity tensors of invisibility cloaks for the elliptic cylinder, prolate spheroid, and the confocal paraboloid geometry are derived. In the case of a elliptic cylinder, it was found that the point of infinite light speed in the electromagnetic space becomes two points in physical space in the zz component of the permittivity tensor. This result is different from the case of a perfect cylinder in which there is a line of cloak at which the speed of light becomes infinite. In the cases of prolate spheroid and confocal paraboloid, the point of infinite light speed in the electromagnetic space becomes a line in physical space for the first two tensor components and the third component of the permittivity tensor becomes singular at the line of cloak.

ELID grinding characteristics of large stamping die for Wolter mirror

An ultra-precision ELID grinding of large stamping dies of Wolter mirror for X-ray telescope was presented in this paper. The large stamping dies (S55C) with confocal paraboloid and hyperboloid was ground by ELID arc-enveloped grinding. In this ELID grinding system, cast iron fiber bonded (CIFB) diamond wheels were controlled by 3-dimentional ways to scan the work-piece and generate required surfaces. Its grinding characteristics such as attainable form accuracy, surface roughness were investigated. Furthermore, some measures to improve form accuracy were discussed and verified such as truing, compensating, and on-machine measuring.

A Minkowski-style theorem for focal functions of compact convex reflectors

This paper continues the study of a class of compact convex hypersurfaces in R n , n ≥ 1 , \mathbf {R}^n, ~n \geq 1, which are boundaries of compact convex bodies obtained by taking the intersection of (solid) confocal paraboloids of revolution. Such hypersurfaces are called reflectors. In R 3 \mathbb {R}^3 reflectors arise naturally in geometrical optics and are used in design of light reflectors and reflector antennas. They are also important in rendering problems in computer graphics. The notion of a focal function for reflectors plays a central role similar to that of the Minkowski support function for convex bodies. In this paper the basic question of when a given function is a focal function of a convex reflector is answered by establishing necessary and sufficient conditions. In addition, some smoothness properties of reflectors and of the associated directrix hypersurfaces are also etablished.

Stagnant film model of the effect of natural convection on the dendrite operating state

We develop a simple model of the influence of natural convection on the selection of the operating state (dendrite tip velocity, V, and tip radius, ϱ) for dendritic growth of a pure material. We hypothesize that the important aspects of natural convection can be accounted for by considering the global convection that would occur in the vicinity of a sphere of radius R that characterizes the size of a dendritic array that is growing from a point source. We estimate the thickness, δ, of a stagnant boundary layer surrounding this sphere by matching the value of the Nusselt number obtained from the heat transfer literature. We solve the steady-state problem of a paraboloidal dendrite at temperature T M growing toward a confocal paraboloid at temperature T ∞ located at a distance δ from the tip. This results in a new transcendental equation, that depends on the gravitational acceleration, g, and transport properties of the melt, for the dimensionless supercooling, S, in terms of the Peclet number, Pe. By assuming that the selection parameter σ is not directly affected by natural convection, we are able to compute values of V and ϱ as functions of the supercooling, Δ T, for various values of g. Results are quite insensitive to the precise choice of R and in remarkably good agreement with the data of Glicksman et al. for growth of succinonitrile on Earth and in microgravity. The value of g at which our model indicates that convective effects become significant has a much stronger dependence (∼ Δ T 9.5) on supercooling than that (∼ Δ T 5.5) calculated by using the analogy of Gill et al. between natural convection and forced convection.

Confocal Paraboloids: Some Comments

Confocal Paraboloids: Some Comments

Cameras and filters for nebular photography

The parameters of filter-cameras for nebular photography are outlined and a method of using confocal paraboloids to give the basic camera a higher resolution is investigated. Some astronomical problems requiring filter-cameras with large fields of view are mentioned. Two systems are described which increase the small field of the basic camera to 300 and then 900. Comments are then made on the design and construction of interference filters for this, type of photography.

An Experimental Study of Near-Field Cassegrainian Antennas*

The near-field Cassegrainian antenna is a double-reflector system that employs, in its simplest form, confocal paraboloids. Unlike the standard Cassegrain which employs a hyperboloidal subreflector illuminated by a spherical wave, the near-field device is fed by a uniform phase front. Experimental data on noise performance, gain, and radiation patterns have been obtained at a frequency of 6 gc using two 16-foot paraboloids (focal length-to-diameter ratios of 0.375 and 0.25) in both standard and near-field configurations. Using the shallow antenna, zenith noise temperatures of 10°K and 6°K were obtained for the standard and near-field systems, respectively; at an elevation angle of 10° the antenna temperatures were 50°K and 20°K. Using the deep secondary reflector, zenith noise temperatures of 4°K were obtained for both configurations; at 10° above the horizon, however, the standard Cassegrain has an antenna temperature of 30°K and the near-field device 13°K. In all cases, the antenna efficiencies are not far above 50 per cent. Discussion of noise produced by various methods of mounting subreflectors is included. Since noise produced by transmission lines and antenna environment is closely related to these experiments, it is discussed in detail in appendices.

Loss Measurements of the Beam Waveguide

The diffraction loss of a new low-loss waveguide for millimeter and shorter wavelength, called the beam waveguide, was measured. The loss measurements were made using a resonator technique. The beam waveguide resonator derived from the beam waveguide consists of confocal paraboloids and is itself a very useful millimeter and sub-millimeter wave circuit component having already found application in some optical masers. Measurements made to determine the reflection loss of the resonator end plates also resulted in information on the loss of 90/spl deg/ bends in the beam waveguide. The results of the loss measurements made on the beam waveguide, in the frequency range near 9 Gc, are in good agreement with theoretical values given by Goubau.

The Choice of Suitable Gap Forms for the Study of Corona Breakdown and the Field Along the Axis of a Hemispherically Capped Cylindrical Point-to-Plane Gap

The various corona gap forms suitable for the convenient investigation of corona phenomena of both signs in the laboratory that are capable of yielding quantitative data involving the Townsend integral for the thresholds of the manifold phenomena are discussed. For laboratory study the most convenient form of gap is the hemispherically capped cylindrical point-to-plane system. Analysis of the potential fall along the axis in such gaps by means of electrolytic model studies indicates that the essential parameter is L/r, the ratio of gap length, L, from point surface to plane relative to the point radius, r. The model study potentials are, however, incapable of giving accurate values of the field strength. These fields must be obtained by computation. Laboratory studies of corona indicate the desirability of relatively large values of L/r. Practical considerations based on studies at atmospheric pressure indicate point radii r of 0.025 and 0.05 cm with a ratio L/r=160 to be those giving the greatest flexibility and most widely separated thresholds at convenient potentials. The use of such point systems to standardize data is recommended. On this basis the field along the axis for the hemispherically capped cylindrical point-to-plane gap as computed by E. E. Dodd is given and here applied to the calculation of the point-to-plane Townsend integral for pre-onset burst pulse thresholds in air as observed by H. W. Bandel. The value is compared with the values obtained by W. N. English for confocal paraboloids where the fields are accurately known. The results are satisfactorily consistent. Discrepancies in the fixing of burst pulse thresholds by various observers are discussed.

The Apparent Breakdown of Meek's Streamer Criterion in Divergent Gaps due to the Failure of Townsend's Ionization Function

Meek's criterion postulates that a streamer will propagate if the positive ion space charge field close to the anode is equal to $K$ times the impressed field, where $K$ should be a constant independent of gap configuration and pressure. Measurements in confocal paraboloid gaps, designed to substantiate this, revealed that $K$ was not constant, but decreased with decreasing pressures. Morton showed that on a negatively charged smaller electrode Townsend's ionization functions do not represent the ionization produced in divergent fields. Consequently it was believed that the calculations of the space charge field of the positive ions produced in such gaps must be inaccurate in view of this erroneous assumption. A series of measurements on the dark current $i$ in the same gap showed that it was not represented by $i={i}_{0} \mathrm{exp} [\ensuremath{\int}\ensuremath{\alpha}\mathrm{dx}]$ at lower pressures. The trend and the apparent quantitative variation of $K$ with pressure is explained if the apparent ${i}_{0}$ versus voltage curves are extrapolated to streamer onset for each pressure. Thus for practical purposes under the conditions studied $K$ is sensibly constant in conformity with Meek's theory. Per contra, the applications of Meek's theory to gaps with very divergent fields where the currents can be expected to deviate markedly from those computed by the relation $i={i}_{0} \mathrm{exp} [\ensuremath{\int}\ensuremath{\alpha}\mathrm{dx}]$ are not justified, thus placing a practical limitation on the use of Meek's theory to streamer studies. Roughly this phenomenon appears when the change of the field exceeds two percent change over an average electron free path.

Threshold Field Studies of Various Positive Corona Phenomena

Use of the oscillograph to study the discharge between confocal paraboloid electrodes in air at atmospheric pressure has given the threshold field strength at which the various positive corona phenomena begin. Some effects of humidity and of accumulated oxides in the gap are described. Current-potential curves are given which show the more prominent effects of humidity and accumulated oxides on the discharge after corona onset. From the field data it is possible to verify Meek's theory attending the formation of pre-onset streamers.

Positive-Point-to-Plane Discharge in Air at Atmospheric Pressure

Investigations have been carried out on current-voltage characteristics of positive-point-to-plane discharge. All currents start abruptly and increase at first linearly and then more steeply with voltage, until finally breakdown into a spark occurs. The corona process in air is shown to be a discontinuous one, involving the alternate propagation and space charge quenching of positive streamers out from the point. Photoelectric ionization in the gas is shown to account for the rapid self-propagation of streamers. Attachment of electrons to form negative ions and subsequent detachment of these electrons in the high field near the point provides means of starting new streamers for self-maintaining corona process. Fields at onset are calculated by the use of confocal paraboloids as electrodes, and it is shown that the fields necessary for such corona process are available. Introduction of a radioactive source into the gap indicates that the sharp cut-off of the corona current as the voltage is decreased is due to lack of electrons for initiating new streamers. Offset voltage marks the point at which the self-perpetuating ionization process begins. The initial current rise is found to increase as the gap distance is decreased and this is shown to be caused by the change in field distribution and consequent further extension of streamers into the gap. Below a certain minimum gap distance, depending on the point geometry, the initial streamers propagate completely across the gap and this introduction of the highly efficient ionization at the cathode causes spark breakdown. The initial current-voltage slopes are expressed as a resistance which is found to increase linearly with the gap and to account for approximately 1 percent of the total voltage drop in the gap. Extrapolation of resistance to zero resistance gives the minimum gap at which stable corona will flow. Above the initial current rise, the current increases as the square of the difference between the applied and offset voltage. Evidence indicates that this is caused by a superposition upon the otherwise linear rise in current with voltage, of an increase proportional to the current itself. Visual observation near breakdown indicates that this increase proportional to the current is associated with the utilization by later streamers of the residual paths of their predecessors. This becomes more pronounced as the current and voltage increase; the effect of earlier streamers in this way increases the current in later streamers. Spark breakdown occurs when the amalgamated streamers succeed in crossing to the plate.

Confocal Paraboloids

Proceedings of the London Mathematical SocietyVolume s1-19, Issue 1 p. 129-143 Articles Confocal Paraboloids A. G. Greenhill, A. G. GreenhillSearch for more papers by this author A. G. Greenhill, A. G. GreenhillSearch for more papers by this author First published: November 1887 https://doi.org/10.1112/plms/s1-19.1.129AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Volumes1-19, Issue1November 1887Pages 129-143 RelatedInformation