Spherical Curves Research Articles

X-ray phase-contrast imaging of strong shocks on OMEGA EP.

The ongoing improvement in laser technology and target fabrication is opening new possibilities for diagnostic development. An example is x-ray phase-contrast imaging (XPCI), which serves as an advanced x-ray imaging diagnostic in laser-driven experiments. In this work, we present the results of the XPCI platform that was developed at the OMEGA EP Laser-Facility to study multi-Mbar single and double shocks produced using a kilojoule laser driver. Two-dimensional radiation-hydrodynamic simulations agree well with the shock progression and the spherical curvature of the shock fronts. It is demonstrated that XPCI is an excellent method to determine with high accuracy the front position of a trailing shock wave propagating through an expanding CH plasma that was heated by a precursor Mbar shock wave. The interaction between the rarefaction wave and the shock wave results in a clear signature in the radiograph that is well reproduced by radiation-hydrodynamic simulations.

Foliations of asymptotically flat manifolds by stable constant mean curvature spheres

Foliations of asymptotically flat manifolds by stable constant mean curvature spheres

Some Rigidity Theorems of Closed Geodesic Polygons and Spherical Curves in Metric Spaces with Curvature Bounded Below

This paper examines characterizations of closed curves in a geodesic metric space with curvature bounded below, including closed geodesic polygons and closed spherical curves bounding surfaces isometric to convex polygons and circles in the model space.

Salkowski curves and spherical epicycloids

The relationship between Salkowski curves, a family of slant helices with constant curvature and non-constant torsion, and the family of spherical epicycloid curves is studied. It is shown that, for some values of the parameter defining the Salkowski curve, the curve is the image by a shear transformation along the z-axis of a spherical epicycle. Therefore, the projection of both curves on the xy-plane is the same. This result can be extended to the whole family of Salkowski curves if some parameter defining the spherical epicycle is allowed to be a complex imaginary number.

Optical path length difference of ray paths inside a reference sphere: a new approach for determining wavefront aberrations in axially symmetrical systems with an object at infinity

A new, to the best of our knowledge, method is proposed to analyze the wavefront aberrations in an axially symmetrical system with an object at infinity. The study commences by determining the optical path length (OPL) difference between two ray paths from the object to the reference sphere. The proposed method is then used to explore the claim in J. Opt. Soc. Am. A 19, 1187 (2002)JOAOD60740-323210.1364/JOSAA.19.001187 that wavefront aberrations do not depend on the radius of the reference sphere. It is found that this claim does not hold when the image plane is non-Gaussian. The proposed approach is applied to determine the primary wavefront aberrations of an axially symmetrical system with an object at infinity. The numerical values of the primary aberrations (spherical, coma, astigmatism, and field curvature) are in excellent agreement with those determined by Zemax simulations. The superiority of the proposed method over Zemax in determining the distortion aberration is confirmed through raytracing. Overall, the proposed method provides a useful contribution to the development of optical systems for applications requiring high-resolution imaging.

The flow-geodesic curvature and the flow-evolute of spherical curves

The flow-geodesic curvature and the flow-evolute of spherical curves

Smooth morphing of immersed spherical curves and application to ruled surfaces

In this paper, we realize a rapid, smooth iterative morphing algorithm between two immersed and closed spherical curves of arbitrary turning numbers. By compensating in turning numbers, we linearly interpolate between geodesic curvatures and the side lengths of their accurate spherical polygons approximation. Our algorithm is essentially based on a geometrical closure condition for a flow of spherical polygons involving the variation of both sides and exterior angles. A curvature smoothing flow using discrete geodesic curvature is adopted for smooth curve generation. To demonstrate the good properties of the method, a morphing of spherical curves and ruled surfaces is illustrated.

An electromechanically driven dielectric elastomer based tunable reflector

Deformable optics offer numerous advantages over conventional optical assemblies, including compactness, cost-effectiveness, efficiency, and flexibility. This study focuses on a reflector based on dielectric elastomer actuators (DEAs) with an internal fluid (air) coupling. DEAs are a class of electroactive materials adept at accommodating substantial actuation strains and rapid responses. Fluid distributed between the active and passive parts remains entirely enclosed by the device and transmits actuation pneumatically. Dynamic maneuvers conducted through a series of controlled electrical signals demonstrate proper control over optical characteristics. However, DEs exhibit inherent flaws in dynamic actuation, referred to as instabilities, which are mitigated by applying an initial pre-stretch. The study identifies optimal parameters that confer stability to the reflector: minimum to no creep, zero residual vibrations, and low viscous losses. An analytical framework is developed to assess device performance, focusing on the spherical curvature assumption that closely resembles the behavior of tunable spherical reflectors. Additionally, an optical bench setup is employed to demonstrate the relationship between focal length and applied pressure. Notably, this paper underscores the potential of a DE-based variable focal length reflector to function effectively within a dynamic environment.

Host-Guest Interaction Mediated Interfacial Co-Assembly of Cyclodextrin and Bottlebrush Surfactants for Precisely Tunable Photonic Supraballs.

Investigations of host-guest interactions at water-oil (w/o) interfaces are limited in single emulsion systems producing simple self-assembled objects with limited uses. Here, within hierarchically ordered water-in-oil-in-water (w/o/w) multiple emulsion droplets, interfacial self-assembly of (polynorbornene-graft-polystyrene)-block-(polynorbornene-graft-polyethylene glycol) (PNPS-b-PNPEG) bottlebrush block copolymers can be precisely controlled through host-guest interactions. α-Cyclodextrin (α-CD) in the aqueous phase can thread onto PEG side chains of the bottlebrush surfactants adsorbed at the w/o interface, leading to dehydration and collapsed chain conformation of the PEG block. Consequently, spherical curvature of the w/o internal droplets increases with the increased asymmetry of the bottlebrush molecules, producing photonic supraballs with precisely tailored structural parameters as well as photonic bandgaps. This work provides a simple but highly effective strategy for precise manipulation of complex emulsion systems applicable in a variety of applications, such as photonic pigments, cosmetic products, pesticides, artificial cells, etc.

Helicoidal minimal surfaces in the 3-sphere: an approach via spherical curves

We prove an existence and uniqueness theorem about spherical helicoidal (in particular, rotational) surfaces with prescribed mean or Gaussian curvature in terms of a continuous function depending on the distance to its axis. As an application in the case of vanishing mean curvature, it is shown that the well-known conjugation between the belicoid and the catenoid in Euclidean three-space extends naturally to the 3-sphere to their spherical versions and determine in a quite explicit way their associated surfaces in the sense of Lawson. As a key tool, we use the notion of spherical angular momentum of the spherical curves that play the role of profile curves of the minimal helicoidal surfaces in the 3-sphere.

Total Torsion and Spherical Curves Bending

Total Torsion and Spherical Curves Bending

Characterization of dual curves using the theory of infinitesimal bending

In this paper, we generalize the main ideas and statements of the theory of infinitesimal bending in Euclidean 3‐space to dual curves in the dual 3‐space . The basic condition we introduce is the invariance of the dual arc length with appropriate precision. Necessary and sufficient conditions for the dual field to be an infinitesimal bending field are given. Some useful formulas and facts about dual arc length and dual bending field are obtained. An explicit characterization of the dual spherical curve bending is presented, and the corresponding infinitesimal bending field is decomposed into dual vectors of the dual Frenet frame. Finally, some examples are given to illustrate our results.

Influence of reference sphere on test accuracy of Ritchey–Common method

AbstractThe authors analyse the influence of reference sphere on the test accuracy of Ritchey–Common method during testing of flat mirror surface. A simulation model is established to simulate the influence of different radii of curvature of reference sphere on system wavefront. Then, the difference between the flat mirror surface result with reference sphere surface error to the actual flat mirror surface is examined by this model. The test accuracy of flat mirror result can be effectively improved by eliminating the influence of the surface error of reference sphere. When the surface accuracy of the reference sphere is better than 0.01 λ, the influence on RMS test accuracy is within 0.01 λ. R–C test path is built to test a 100 mm‐diameter flat mirror surface, and its results are compared with those of the flat mirror surface without surface error of reference sphere and with the test result of the interferometer. When the surface accuracy of the actual used area of reference sphere is better than 0.01 λ, the effect on flat mirror RMS accuracy is within 0.01 λ. The RMS test accuracy can reach 0.01 λ, when the surface error of reference sphere is retained. The experiment verifies the correctness of the simulation results and guarantees the improvement in the test accuracy of R–C method.

On harmonic and asymptotically harmonic Finsler manifolds

In the present paper we introduce and investigate some types of harmonic Finsler manifolds and find out the interrelation between them. We give some characterizations of such spaces in terms of the Finsler mean curvature of geodesic spheres and the Finsler Laplacian of the distance function induced by the Finsler metric. We investigate some properties of the Finsler mean curvature of geodesic spheres of different radii. We prove that certain harmonic Finsler manifolds are of isotropic S-curvature and isotropic Ricci curvature. Moreover, we give some examples of non-Riemannian Finsler harmonic manifolds of constant flag curvature and constant S-curvature. Finally, we provide a technique to construct harmonic Finsler manifolds of Randers type.

Spacelike Ac-Slant Curves with Non-Null Principal Normal in Minkowski 3-Space

In this paper, we define a spacelike ac-slant curve whose scalar product of its acceleration vector and a unit non-null fixed direction is a constant in Minkowski 3-space. Furthermore, we give a characterization depending on the curvatures of the spacelike ac-slant curve. After that, we get the relationship between a spacelike ac-slant curve and several distinct types of curves, such as spacelike Lorentzian spherical curves, spacelike helices, spacelike slant helices, and spacelike Salkowski curves, enhancing our understanding of its geometric properties in Minkowski 3-space. Finally, we used Mathematica, a symbolic computation software, to support the notions of an ac-slant curve with attractive images.

Cholesteric Spherical Reflectors with Tunable Color from Single-Domain Cellulose Nanocrystal Microshells.

The wavelength- and polarization-selective Bragg reflection of visible light exhibited by films produced by drying cholesteric liquid crystal (CLC) suspensions of cellulose nanocrystals (CNCs) render these biosourced nanoparticles highly potent for many optical applications. While the conventionally produced films are flat, the CLC-derived helical CNC arrangement would acquire new powerful features if given spherical curvature. Drying CNC suspension droplets does not work, because the onset of kinetic arrest in droplets of anisotropic colloids leads to severe buckling and loss of spherical shape. Here, these problems are avoided by confining the CNC suspension in a spherical microshell surrounding an incompressible oil droplet. This prevents buckling, ensures strong helix pitch compression, and produces single-domain cholesteric spherical reflector particles with distinct visible color. Interestingly, the constrained shrinkage leads to spontaneous puncturing, leaving every particle with a single hole through which the inner oil phase can be extracted for recycling. By mixing two different CNC types at varying fractions, the retroreflection color is tuned throughout the visible spectrum. The new approach adds a versatile tool in the quest to utilize bioderived CLCs, enabling spherically curved particles with the same excellent optical quality and smooth surface as previously obtained only in flat films.

The Geometry of Low- and High-Level Perceptual Spaces.

Low-level features are typically continuous (e.g., the gamut between two colors), but semantic information is often categorical (there is no corresponding gradient between dog and turtle) and hierarchical (animals live in land, water, or air). To determine the impact of these differences on cognitive representations, we characterized the geometry of perceptual spaces of five domains: a domain dominated by semantic information (animal names presented as words), a domain dominated by low-level features (colored textures), and three intermediate domains (animal images, lightly texturized animal images that were easy to recognize, and heavily texturized animal images that were difficult to recognize). Each domain had 37 stimuli derived from the same animal names. From 13 participants (9F), we gathered similarity judgments in each domain via an efficient psychophysical ranking paradigm. We then built geometric models of each domain for each participant, in which distances between stimuli accounted for participants' similarity judgments and intrinsic uncertainty. Remarkably, the five domains had similar global properties: each required 5-7 dimensions, and a modest amount of spherical curvature provided the best fit. However, the arrangement of the stimuli within these embeddings depended on the level of semantic information: dendrograms derived from semantic domains (word, image, and lightly texturized images) were more "tree-like" than those from feature-dominated domains (heavily texturized images and textures). Thus, the perceptual spaces of domains along this feature-dominated to semantic-dominated gradient shift to a tree-like organization when semantic information dominates, while retaining a similar global geometry.

Free Boundary Minimal Annuli Immersed in the Unit Ball

We construct a family of compact free boundary minimal annuli immersed in the unit ball B3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {B}^3$$\\end{document} of R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document}, the first such examples other than the critical catenoid. This solves a problem formulated by Nitsche in 1985. These annuli are symmetric with respect to two orthogonal planes and a finite group of rotations around an axis, and are foliated by spherical curvature lines. We show that the only free boundary minimal annulus embedded in B3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {B}^3$$\\end{document} foliated by spherical curvature lines is the critical catenoid; in particular, the minimal annuli that we construct are not embedded. On the other hand, we also construct families of non-rotational compact embedded capillary minimal annuli in B3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {B}^3$$\\end{document}. Their existence solves in the negative a problem proposed by Wente in 1995.

Timelike spherical curves according to equiform Bishop frame in 3-dimensional Minkowski space

In this paper, we study the equiform Bishop formulae for the equiform timelike curves in 3-dimensional Minkowski space where the equiform timelike spherical curves are defined according to the equiform Bishop frame. We establish a necessary and sufficient condition for an equiform timelike curve to be an equiform timelike spherical curve. Furthermore, we give certain characterizations of equiform spherical curves in 3-dimensional Minkowski space, which are timelike with an equiform spacelike principal normal vector.

A Schur’s theorem via a monotonicity and the expansion module

Abstract In this paper we present a monotonicity which extends a classical theorem of A. Schur comparing the chord length of a convex plane curve with that of a space curve of smaller curvature. The main result establishes a Schur’s Theorem for spherical curves, which extends the well-known Cauchy’s Arm Lemma. We also extend the result by allowing the curve with the smaller curvature being any curve in 𝕊 n {\mathbb{S}^{n}} .