Spherical Curves Research Articles

Diameters of spherical Alexandrov spaces and curvature one orbifolds

Let G be a closed, non-transitive subgroup of O(n+1), where n ≥ 2, and let Q = S/G. We will show that for each n there is a lower bound for the diameter of Q. If G is finite then Q is an orbifold of constant curvature one and an explicit lower bound can be given. For Coxeter groups, the resulting lower bound is independent of dimension. Otherwise, Q is a spherical Alexandrov space and we will show existence of a lower bound. In the process, we will compute some examples of quotient spaces and their diameters.

Stresses in Thin Coatings from Curvature Measurements on Non-Planar Substrates

AbstractMechanical elastic and inelastic properties of thin coatings are often studied by means of the curvature measurement technique in combination with the Stoney formula. It is then implicitly assumed that the elastic substrate is initially flat which implies that the curvatures theoretically remain constant over the substrate. If the substrate has a slight initial curvature, a different situation arises. In this case, the curvatures will vary over the substrate. It has recently been shown that in spite of this, the stresses that appear in the thin coating will remain constant over the surface of the substrate. Therefore, measured curvatures can generally not be used to extract layer stresses without a proper compensation for the initial curvature. A general method for the extraction of mechanical properties from curvature measurements on non-planar substrates is outlined. The method is valid for linear as well as non-linear shell theories. The compensation needed to evaluate a coating on a circular substrate with spherical initial curvature is studied for all relevant parameters. The results are particularly discussed in relation to curvature measurements on silicon wafers.

Some characterizations of Lorentzian spherical space-like curves with the timelike and the null principal normal

In [4] the authors have characterized Lorentzian spherical space-like curves in 3-dimensional Minkowski space with the space-like normal. In this paper we shall characterize the Lorentzian spherical space-like curves in the same space with the timelike and the null principle normal.

Off-axis aberrations of a wide-angle schematic eye model.

A schematic eye model based on anatomical data, which had been previously designed to reproduce image quality on axis, has been transformed into a wide-angle model by simply adding a spherical image surface that plays the role of the retina. This model captures the main features of the wide-angle optical design of the human eye with minimum complexity: four conic optical surfaces plus a spherical image surface. Seidel aberrations (spherical aberration, coma, astigmatism, field curvature, and distortion), longitudinal and transverse chromatic aberrations, and overall monochromatic spot diagrams have been computed for this eye model and for field angles ranging from 0 degree to 60 degrees by both finite and third-order ray tracing. The modulation transfer function for each field angle has been computed as well. In each case our results have been compared with average experimental data found in the literature, showing a reasonably good agreement. The agreement between the model and experimental data is better off axis, mainly at moderate (10 degrees-40 degrees) field angles, than on axis. The model has been applied to simulate a variety of experimental methods in which image aberrations are estimated from measurements taken in the object space. Our results suggest that for some types of aberration, these methods may yield biased estimates.

Unrolling and rolling of curves in non-convex surfaces

The notion of unrolling of a spherical curve is proved to coincide with its development into the tangent plane. The development of a curve in an arbitrary surface in the Euclidean 3-space is then studied from the point of view of unrolling. The inverse operation, called the rolling of a curve onto a surface, is also analysed and the relationship of such notions with the functional defined by the square of curvature is stated. An application to the construction of nonlinear splines on Riemannian surfaces is suggested.

A Triangular Nomogram for Spherical Symmetric Coupler Curves and its Application to Mechanism Design

This paper presents a triangular nomogram for spherical symmetric coupler curves generated by a spherical crank-rocker with special dimensions. The paper also illustrates the application of this nomogram for spherical mechanism design. The expression of symmetric coupler curves traced by the spherical crank-rocker is derived. A computer program is developed to calculate the coordinates of the coupler curve points for the prescribed dimensions of the mechanism. According to the classified properties of coupler curves obtained, a triangular nomogram for spherical symmetric coupler curves can then be constructed by using three design parameters, i.e., length of crank, length of fixed link, and coupler angle. The triangular nomogram provides an alternative selection for dimensional synthesis of spherical path generators. The synthesized mechanisms can also be used to provide initial estimates for optimal synthesis of spherical path generators. A design example is presented.

An ellipsoidal grid gas proportional scintillation counter

Gas Proportional Scintillation Counters using curved grids for solid angle and reflection compensation have been described in the recent literature. They allow large radiation windows with diameters of 25 mm keeping at the same time the good energy resolutions characteristic of those X-ray detectors. However, the grids used have a spherical curvature, which does not correspond to the optimal curvature. In the present work we have calculated by computer simulation an improved shape for the curved grid. This shape can be well fitted to an ellipsoid of revolution, with a large eccentricity. A detector was designed with such an ellipsoidal grid and a radiation window 40 mm in diameter, filled with pure xenon at 927 Torr coupled to an EMI D676QB VUV photomultiplier tube having a 2″ diameter window. For the experiments envisaged, detection of solar X-rays in the 20–80 keV energy range, a 7 cm thick drift region was used, leading to efficiencies from 80% to 20%, respectively. Such a thick drift region reduces the performance mainly for soft X-rays. For 22 keV X-rays the energy resolution obtained, for a broad X-ray beam entering the full 40 mm diameter detector window, is 6.0%. Results are presented showing the variation of the energy resolution with the window diameter and a performance, for ellipsoidal grids superior to that for spherical grids. A discussion of the results obtained is presented.

Representation of flat Lagrangian $H$-umbilical submanifolds in complex Euclidean spaces

The author proved earlier that, a Lagrangian //-umbilical submanifold in complex Euclidean «-space with n>2 is either a complex extensor, a Lagrangian pseudo-sphere, or a flat Lagrangian //-umbilical submanifold. Explicit descriptions of complex extensors and of Lagrangian pseudo-spheres are given earlier. The purpose of this article is to complete the investigation of Lagrangian //-umbilical submanifolds in complex Euclidean spaces by establishing the explicit description of flat Lagrangian //-umbilical submanifolds in complex Euclidean spaces. 1. Statements of theorems. We follow the notation and definitions given in (2). In order to establish the complete classification of Lagrangian //-umbilical submanifolds in Cn we need to introduce the notion of special Legendre curves as follows. Let z: I^S2n~ι cCbeaunit speed Legendre curve in the unit hypersphere S2n~1 (centered at the origin), i.e., — z(s) is a unit speed curve in S2n~1 satisfying the condition: (z'(s iz(s)) = 0 identically. Since = z(s) is a spherical unit speed curve, = 0 identically. Hence, z(s), iz(s% z'(s iz'(s) are orthonormal vector fields defined along the Legendre curve. Thus, there exist normal vector fields P 3, ...,/> along the Legendre curve such that (1.1) Φ), ι'Φ), z'(s iz\s PM iPsis), > Pn(s), iPn(s) form an orthonormal frame field along the Legendre curve. By taking the derivatives of = 0 and of =0, we obtain =0 and = — 1, respectively. Therefore, with respect to an orthonormal frame field chosen above, z can be expressed as

Families of surfaces: focal sets, ridges and umbilics

The contact between a surface in Euclidean space and the family of spheres carries a good deal of useful geometrical information about the surface. The centres of those spheres having degenerate tangency with the surface (in Arnold's notation of type A2) form the focal set, and the set of points on the surface where there is a contact of type A3, the so-called ridge curve, is of some interest in the field of computer vision, as a robust feature of the surface. These ridges come together at umbilics, where the contact between the surface and the unique sphere of curvature is of type D. A dual notion, that of a subparabolic curve is also of some interest. In this paper we describe the way in which the ridge and subparabolic curves can evolve in a generic 1-parameter family of surfaces.

Effects of earth curvature on two-dimensional ray tracing in underwater acoustics

The objective of this paper is to investigate the effects of earth curvature on two-dimensional ray tracing. This is done by comparing numerical solutions of different ray equations. From numerical results, it is concluded that (1) at very long ranges, a 2-D ray path constructed without earth curvature correction may have less turning points than an actual ray; (2) computed travel times of eigenrays may be longer than measurements, and this travel-time error will increase with launch angle; (3) the error of launch angles of eigenrays decreases with launch angle; and (4) both the travel-time error and the launch-angle error increase with propagation range. Conclusions (2) and (3) are consistent with a published result in ocean acoustic tomography. Within a 1000 km range, the effect of earth flattening is not significant, and spherical earth is a good approximation. A set of 2-D ray equations with spherical earth curvature correction is presented in this paper, which can considerably extend the range of validity of 2-D ray tracing while being as simple as the traditional 2-D equations.

On Explicit Equations of the Burmester Curves for the PPP-P Case in Spherical Motion

In this paper, an analytic treatment to investigate the Burmester curves of the PPP-P case in spherical motion is presented. With a rotation angle of a coupler about a pole and an instant centre associated with an inflection point specified, a geometric construction is established to determine the corresponding spherical centre-points and circle-points. According to the geometric construction, the explicit equations of spherical Burmester curves for the PPP-P case are directly derived by using spherical trigonometry, without the use of differential technique. The explicit equations obtained give a clear insight into the geometric properties and special cases of the spherical Burmester curves, which include degenerated curves and symmetric curves. A design application is presented.

UHV electron microscope and simultaneous STM observation of gold stepped surfaces

A miniaturized scanning tunneling microscope (STM) was put in a UHV electron microscope to simultaneously observe specimen surfaces in transmission electron microscope images and STM images. Gold (111) stepped surfaces were observed with a gold STM tip with a spherical apex curvature of 16 nm. The tip positions were measured from the electron microscope images recorded on a VTR during the STM observation. We found that the tip position changed coincidentally with the STM profile and that the shortest distance between the stepped (111) surface of the island and the spherical tip apex was kept constant.

Closed form solution to the position workspace of Stewart parallel manipulators

A novel methodology is presented for the workspace analysis of Stewart parallel manipulators. The workspace is defined that enables a description to be made in a unified framework of both the position and orientation capabilities of the mobile platform. Given the orientation capability of the mobile platform, the piecewise closed solution to the workspace boundary is formulated. It is indicated for the first time that the workspace boundary is in fact the cap of twelve envelope surfaces, each of which is generated by a family of spherical surfaces having the movable center located on a two-branch closed spherical curve. Two examples are given to illustrate the effectiveness of this approach.

Self-focusing and beam attenuation in laser materials processing

During laser materials processing, the melt pool can form a convex or concave surface depending on the processing conditions and surface tension of the liquid metal. The curved surface can act as a lens to refract the laser rays. Temperature distributions below the substrate's surface are calculated for various values of the attenuation coefficient and radii of curvature of the free surface of the melt pool. For convex spherical curvature and a large penetration depth of the laser beam, the subsurface temperatures are found to be extremely high, exceeding the surface temperature severalfold. This is because of self-focusing, that is further focusing of the laser beam, that causes high subsurface laser irradiance.

Stiffening of fluid membranes and entropy loss of membrane closure: Two effects of thermal undulations

The problem of membrane softening by thermal undulations is revisited. In contrast to general belief, fluid membranes are predicted to be stiffened, not softened, by their undulations. Equal values of the effective bending rigidity are calculated from the interplay of local mean curvature modes (hats) on the basically flat membrane and from the coupling of spherical harmonic modes with spherical curvature. In addition, a conjecture is made on the entropy of membrane closure. It relies on a similarity of membrane closure to periodic boundary conditions.

Invariants of curves and fronts via Gauss diagrams

We use a notion of chord diagrams to define their representations in Gauss diagrams of plane curves. This enables us to obtain invariants of generic plane and spherical curves in a systematic way via Gauss diagrams. We define a notion of invariants are of finite degree and prove that any Gauss diagram invariants are of finite degree. In this way we obtain elementary combinatorial formulas for the degree 1 invariants J ± and St of generic plane curves introduced by Arnold [1] and for the similar invariants J ± S and St S of spherical curves. These formulas allow a systematic study and an easy computation of the invariants and enable one to answer several questions stated by Arnold. By a minor modification of this technique we obtain similar expressions for the generalization of the invariants J ± and St to the case of Legendrian fronts. Different generalizations of the invariants and their relations to Vassiliev knot invariants are discussed.

Gauss–Bonnet's Theorem and Closed Frenet Frames

Abstract. Our main result is that integrated geodesic curvature of a (nonsimple) closed curve on the unit two-sphere equals a half integer weighted sum of the areas of the connected components of the complement of the curve. These weights that gives a spherical analogy to the winding number of closed plane curves are found using Gauss–Bonnet’s theorem after cutting the curve into simple closed sub-curves. If the spherical curve is the tangent indicatrix of a space curve we obtain a new short proof of a formula for integrated torsion presented in an unpublished manuscript by C. Chicone and N. J. Kalton. Applying our result to the principal normal indicatrix we generalize a theorem by Jacobi stating that a simple closed principal normal indicatrix of a closed space curve with nonvanishing curvature bisects the unit two-sphere to nonsimple principal normal indicatrices. Some errors in the literature are corrected. We show that a factorization of a knot diagram into simple closed sub-curves defines an immersed disc with the knot as boundary and use this to give an upper bound on the unknotting number of knots.

On Jacobi’s Theorems

In this paper, J. Jacobi’s Theorems [9] have been considered for the spherical curves drawn on the unit dual sphere during the closed space motions. The integral invariants of the ruled surfaee corresponding, in the line space, to the spherical curve dtawn by a fixed point on the moving unit dual sphere during the one-parameter closed motion were calculated with a different approach from the area vector used by H. R. Müller [11], In addition, the ruled surfaces corresponding to the curves drawn by the unit tangent vector, principal normal vector or a unit vector on the osculating plane of the mentioned curve, were seen to be cones with this approach.

Electron crystallographic study of Bi 4(Sr 0.75La 0.25) 8Cu 5O y structure

Structure of Bi 4(Sr 0.75La 0.25) 8Cu 5O y was studied by a image-processing technique based on the combination of high-resolution electron microscopy and electron diffraction. Positions of heavy atoms were obtained by image deconvolution and then positions of oxygen were obtained by phase extension. Both image deconvolution and phase extension are based on the direct method developed in X-ray crystallography. The electron diffraction intensity used in phase extension was corrected by a kind of empirical method to reduce the distortions caused by various effects including the Ewald sphere curvature, dynamic scattering, etc. The efficiency of the image-processing technique and the empirical method of electron diffraction intensity correction is discussed.

THE DEGENERATED SPHERICAL BURMESTER CURVES OF THE PP-P-P CASE

In this paper an analytic method to investigate the degenerated spherical Burmester curves of the PP-P-P case is presented. (Provided that an instant centre and two poles associated with two rotation angles of a coupler are specified, a geometric construction is established to determine the spherical centre- and circle-point.) According to the geometric construction, the explicit equations of spherical Burmester curves for the PP-P-P case are thus derived, without the use of differentiation, but by using spherical cross ratio. By assigning proper parameter values to equations obtained, six types of degenerated spherical Burmester curves of the PP-P-P case are obtained directly, which include three different forms of the degenerated curve: (1) a point, (2) a great circle and a spherical Thales ellipse and (3) three great circles.