Spherical Curves Research Articles

Optical modeling for geometric phase for the Hasimoto transformations on unit sphere

In this paper, the key issue we address here is the Hasimoto transformation in three-dimensional Euclidean space, which allows us to specify the magnetic trajectory for the spherical curve in R3. We, then show the motion of the spherical curve is equal to the nonlinear heat system. Lastly, we solve this nonlinear heat system numerically and then give a physical application of it.

On an Explicit Characterization of Spherical Curves in Dual Lorentzian 3-SpaceD13

In this study, we developed a mathematical framework for studying spherical and null curves in dual Lorentzian 3-spaceD13. Considering the causal character of the dual curve, sufficient and necessary conditions for spacelike curves to be dual Lorentzian spherical were given. Also, new sufficient and necessary conditions for non-null curves to be hyperbolic dual spherical were presented. Then, an insight on curves with null-principal normal is reported.

Spherepc: An R Package for Dimension Reduction on a Sphere

Dimension reduction is a technique that can compress given data and reduce noise. Recently, a dimension reduction technique on spheres, called spherical principal curves (SPC), has been proposed. SPC fits a curve that passes through the middle of data with a stationary property on spheres. In addition, a study of local principal geodesics (LPG) is considered to identify the complex structure of data. Through the description and implementation of various examples, this paper introduces an R package [spherepc](https://CRAN.R-project.org/package=spherepc) for dimension reduction of data lying on a sphere, including existing methods, SPC and LPG.

Diffusion dynamics in branched spherical structure

Diffusion on a spherical surface with trapping is a common phenomenon in cell biology and porous systems. In this paper, we study the diffusion dynamics in a branched spherical structure and explore the influence of the geometry of the structure on the diffusion process. The process is a spherical movement that occurs only for a fixed radius and is interspersed with a radial motion inward and outward the sphere. Two scenarios govern the transport process in the spherical cavity: free diffusion and diffusion under external velocity. The diffusion dynamics is described by using the concepts of probability density function (PDF) and mean square displacement (MSD) by Fokker–Planck equation in a spherical coordinate system. The effects of dead ends, sphere curvature, and velocity on PDF and MSD are analyzed numerically in detail. We find a transient non-Gaussian distribution and sub-diffusion regime governing the angular dynamics. The results show that the diffusion dynamics strengthens as the curvature of the spherical surface increases and an external force is exerted in the same direction of the motion.

A quasiconformal Hopf soap bubble theorem

We show that any compact surface of genus zero in {mathbb {R}}^3 that satisfies a quasiconformal inequality between its principal curvatures is a round sphere. This solves an old open problem by H. Hopf, and gives a spherical version of Simon’s quasiconformal Bernstein theorem. The result generalizes, among others, Hopf’s theorem for constant mean curvature spheres, the classification of round spheres as the only compact elliptic Weingarten surfaces of genus zero, and the uniqueness theorem for ovaloids by Han, Nadirashvili and Yuan. The proof relies on the Bers-Nirenberg representation of solutions to linear elliptic equations with discontinuous coefficients.

Measure of overlap between two arbitrary ellipses on a sphere.

Various packing problems and simulations of hard and soft interacting particles, such as microscopic models of nematic liquid crystals, reduce to calculations of intersections and pair interactions between ellipsoids. When constrained to a spherical surface, curvature and compactness lead to non-trivial behaviour that finds uses in physics, computer science and geometry. A well-known idealized isotropic example is the Tammes problem of finding optimal non-intersecting packings of equal hard disks. The anisotropic case of elliptic particles remains, on the other hand, comparatively unexplored. We develop an algorithm to detect collisions between ellipses constrained to the two-dimensional surface of a sphere based on a solution of an eigenvalue problem. We investigate and discuss topologically distinct ways two ellipses may touch or intersect on a sphere, and define a contact function that can be used for construction of short- and long-range pair potentials.

The Amphoteric Nature of Phospholipid Bilayer Verified by the Surface Energy

Background The surface energy the phospholipidic spherical membrane vs. pH, see Fig. 4A helped us to understand amphoteric character of cartilage surface (introduced together first time in literature) [4]. Methods The samples of articular cartilage were taken from the knees of an aged one to two-years-old ox. Friction coefficient (cartilage/cartilage) pair and the interfacial energy of a spherical lipid bilayer curves vs pH have “a bell-shaped curve” expressing amphoteric cartilage nature by the isoelectric point (IEP). Results It was found that the lowest energy values cover working conditions at pH 7.3 (±) 1.0 of natural joints, see Fig.4 A, B. The obtained “a bell-shaped curve” for (cartilage/cartilage) pair friction and spherical PLs bilayer is a novelty in this paper. Conclusions The observations have led to the conclusion that friction coefficient (cartilage/cartilage) pair and surface energy of a spherical lipid bilayer curves vs pH have (IEP) and expressing amphoteric surfaces character. The surface energy corresponds well with friction coefficient phospholipid amphoteric bilayer surface.

Finslerian Viewpoint to the Rectifying, Normal, and Osculating Curves

The theory of Finsler metric was introduced by Paul Finsler, in 1918. The author defines this metric using the Minkowski norm instead of the inner product. Therefore, this geometry is a more general metric and includes the Riemannian metric. In the present work, using the Finsler metric, we investigate the position vector of the rectifying, normal and osculating curves in Finslerian 3-space $\mathbb{F}^{3}$. We obtain the general characterizations of these curves in $\mathbb{F}^{3}$. Furthermore, we show that rectifying curves are extremal curves derived from the Finslerian spherical curve. We also plotted various examples by using the Randers metrics.

Full Laplace spectrum of distance spheres in symmetric spaces of rank one

We use Lie-theoretic methods to explicitly compute the full spectrum of the Laplace--Beltrami operator on homogeneous spheres which occur as geodesic distance spheres in (compact or noncompact) symmetric spaces of rank one, and provide a single unified formula for all cases. As an application, we find all resonant radii for distance spheres in the compact case, i.e., radii where there is bifurcation of embedded constant mean curvature spheres, and show that distance spheres are stable and locally rigid in the noncompact case.

Bertrand and Mannheim Curves of Spherical Framed Curves in a Three-Dimensional Sphere

We investigated differential geometries of Bertrand curves and Mannheim curves in a three-dimensional sphere. We clarify the conditions for regular spherical curves to become Bertrand and Mannheim curves. Then, we concentrate on Bertrand and Mannheim curves of singular spherical curves. As singular spherical curves, we considered spherical framed curves. We define Bertrand and Mannheim curves of spherical framed curves. We give conditions for spherical framed curves to become Bertrand and Mannheim curves.

Effect of 0.02% and 0.01% atropine on astigmatism: a two-year clinical trial

BackgroundTo evaluate the effects of 0.02% and 0.01% atropine eye drops on ocular and corneal astigmatism over 2 years.MethodsA prospective clinic-controlled trail. The cohort study assessed 400 myopic children and divided them into three groups: 138 and 142 children were randomized to use either 0.02% or 0.01% atropine eye drops, respectively. They wore single-vision (SV) spectacles, with one drop of atropine applied to both eyes once nightly. Control children (n = 120) only wore SV spectacles. Spherical equivalent refractive errors (SER) and corneal curvature were measured every 4 months. The SER and corneal curvature were assessed by cycloplegic autorefraction and IOLMaster. Ocular and corneal astigmatism were calculated by Thibos vector analysis and then split into its power vector components, J0 (with-the-rule astigmatism) and J45 (oblique).ResultsAfter 2 years, the ocular astigmatism increased by -0.38 ± 0.29 D, -0.47 ± 0.38 D, -0.41 ± 0.35 D in the 0.02%, 0.01% atropine groups and control group, respectively (p = 0.15). The corresponding corneal astigmatism increased by -0.20 ± 0.34 D, -0.28 ± 0.35 D and -0.26 ± 0.26 D (p = 0.18). The ocular astigmatism J0 increased by 0.19 ± 0.28 D, 0.22 ± 0.36 D, 0.18 ± 0.31 D in the 0.02% atropine, 0.01% atropine and control groups, respectively (p = 0.65). The corresponding corneal astigmatism J0 increased by -0.05 ± 0.34 D, -0.11 ± 0.37 D and -0.13 ± 0.30 D (p = 0.23). There was a small but significant increase in ocular astigmatism (including J0) (all P < 0.05), but there were no changes in the ocular astigmatism J45 and corneal astigmatism (including J0 and J45) in the three groups over time (all p > 0.05). However, there were no significant differences in the changes in ocular astigmatism (including J0) among the three groups.ConclusionsTreatment with 0.02% and 0.01% atropine had no clinically significant effect on ocular and corneal astigmatism over 2 years.Trial registrationThe First Affiliated Hospital of Zhengzhou University, ChiCTR-IPD-16008844. Registered 14/07/2016.

Direction Curves of Spherical Indicatrices of A New Framed Curve

In this study, direction curves of spherical indicatrix curves (primarily tangent indicatrix) for a curve in three dimensional Euclidean space were investigated. Mentioned curve was given with a new alternative frame which is {N,C,W} frame. It is a new aspect that researching relations between direction curves and the main curve via that new alternative frame. The harmonic curvatures and geodesic curvatures of spherical image of principal normal indicatrix of direction curves were obtained. The general helix, slant helix and C-slant helix characterizations of direction curves were given. Finally these curves with their figures were exemplified.

Isoperimetry for asymptotically flat 3-manifolds with positive ADM mass

Let $$(M^3, g)$$ be an asymptotically flat 3-manifold with positive ADM mass. In this paper, we show that each leaf of the canonical foliation consisting of stable constant mean curvature spheres is the unique isoperimetric surface for the volume it encloses. Our proof is based on “fill-in” argument and sharp isoperimetric inequality on asymptotically flat 3-manifold with non-negative scalar curvature.

Constant mean curvature spheres in homogeneous three-spheres

We give a complete classification of the immersed constant mean curvature spheres in a three-sphere with an arbitrary homogeneous metric, by proving that for each $H \in \mathbb{R}$, there exists a constant mean curvature $H$ sphere in the space that is unique up to an ambient isometry.

Comparison of the Clinical Outcomes between Combined Femtosecond Laser-Assisted In Situ Keratomileusis and Corneal Cross Linking versus Combined Small-Incision Lenticule Extraction and Corneal Cross Linking.

Purpose The purpose of this study was to compare femtosecond laser-assisted in situ keratomileusis with prophylactic cross linking (FS-LASIK Xtra) and small-incision lenticule extraction with prophylactic cross linking (SMILE Xtra) in terms of their postoperative clinical outcomes. Methods In this retrospective study, 24 patients (48 eyes) with myopia and myopia astigmatism were recruited from 2017 to 2018. All patients underwent comprehensive ophthalmic examinations preoperatively and follow-up visits at one and three months. Four patients (eight eyes) in each group were followed up for 12 months. The following were assessed at each visit: uncorrected distance visual acuity (UDVA), manifest refractive spherical equivalent (MRSE), keratometry values, biomechanical properties, anterior and posterior curvature, and corneal pachymetry. Results The FS-LASIK Xtra and SMILE Xtra groups both included 24 eyes of 12 patients. At 1, 3, and 12 months after surgery, a UDVA of 20/20 or better was achieved for 91.7%, 91.7%, and 87.5% of individuals, respectively, in the FS-LASIK group and 95.8%, 100%, and 100% in the SMILE group, respectively. For 75% of eyes in the FS-LASIK Xtra group and 100% of eyes in the SMILE Xtra group, the achieved MRSE was within ±1.00D of attempted correction. The LASIK Xtra group had a significantly higher mean MRSE than the SMILE Xtra group at 3 and 12 months after surgery (P=0.006, 0.01), and the SMILE Xtra group had significantly higher K1 and K2 values than the FS-LASIK group at one month after surgery (P=0.024, 0.023). Corneal thickness decreased significantly at one month postoperatively and started to increase over the next 12 months in both groups (P=0.001). The biomechanical properties showed no significant intergroup differences at the 12-month follow-up. Conclusions FS-LASIK Xtra and SMILE Xtra are safe and effective in the correction of myopia and myopia astigmatism, and both procedures have the same effect on postoperative corneal morphology and biomechanics.

Active Optics—Advances of Cycloid-like Variable Curvature Mirrors for the VLTI Array

Elasticity theory and active optics led us to the discovery of three geometrical configurations of variable curvature mirrors (VCMs) that are either cycloid-like or tulip-like thickness distributions. Cycloid-like VCMs are generated by a uniform load—air pressure—applied over the mirror rear surface, and reacts without any bending moment along its circular contour. This particular VCM configuration is of practical interest because it smoothly generates accurate optical curvatures, varying from plane at rest to spherical curvatures up to f/2.9 over 16-mm aperture under 6.5-bar air pressure. Starting from the thin plate theory of elasticity and modeling with NASTRAN finite element analysis, one shows that 3-D optimizations—using a non-linear static flexural option—provide an accurate cycloid-like thickness distribution. VCM elasticity modeling in quenched stainless steel–chromium substrates allows the obtaining of diffraction-limited optical surfaces: Rayleigh’s criterion is achieved over a zoom range from flat to f/3.6 over 13-mm clear aperture up to 6-bar loading. These VCMs were originally developed and built at the Marseille Observatory in 1975 and implemented as a cat’s-eye mirror of IR Fourier-transform interferometers for laboratory recording of fast events in gas molecular spectroscopy. Later, for high-angular resolution astronomy with the ESO VLTI array—an interferometer made of 8 m Unit Telescopes (UTs) and 1.8 m Auxiliary Telescopes (ATs)—such VCMs were inevitable components to provide in a 3″ co-phased field-of-view since 1998. They were implemented (1) as cat’s eye mirrors of the height delay-lines beam recombination lab and (2) as ATs mirror-pair for output pupil conjugation of the movable x–y baseline. From the ESO-AMU approved convention of making 10 VCM spares up to 2024, the present modeling should provide a diffraction-limited extended field-of-view. It is pure coincidence that present results from modeling with an outer collarette are identical to results from analytic theory without collarette.

Global uniqueness of large stable CMC spheres in asymptotically flat Riemannian 3-manifolds

Let (M,g) be a complete Riemannian 3-manifold that is asymptotic to Schwarzschild with positive mass and whose scalar curvature vanishes. We unconditionally characterize the large, embedded stable constant mean curvature (CMC) spheres in (M,g).

The forward kinematics of dual rolling contact of one dual unit sphere on another with dual Darboux frame based equation

In this paper, we investigate the forward kinematics of dual rolling contact motion without sliding of one dual unit sphere $\tilde{S}_{m}^{2}$ on the fixed sphere $\tilde{S}_{f}^{2}$ along their dual spherical curves, which correspond to ruled surfaces generated by the straight lines in the real line space $\mathbb{E}^{3}$. We adopt a dual Darboux frame method to develop instantaneous kinematics of dual rolling motion. We obtain some new kinematic equations of rolling motion of the moving sphere $\tilde{S}_{m}^{2}$ with regards to dual unit vectors, dual rolling velocity, and dual geometric invariants. Namely, the dual translational velocity equation of an arbitrary dual point and the equation of the dual angular velocity on the moving sphere $\tilde{S}_{m}^{2}$ are derived. The equation represented by geometric invariants can be handily generalized to suit arbitrary dual spherical curve on $\tilde{S}_{m}^{2}$ and can be differentiated to any order.

Characterization of Rectifying Curves by Their Involutes and Evolutes

A rectifying curve is a twisted curve with the property that all of its rectifying planes pass through a fixed point. If this point is the origin of the Cartesian coordinate system, then the position vector of the rectifying curve always lies in the rectifying plane. A remarkable property of these curves is that the ratio between torsion and curvature is a nonconstant linear function of the arc-length parameter. In this paper, we give a new characterization of rectifying curves, namely, we prove that a curve is a rectifying curve if and only if it has a spherical involute. Consequently, rectifying curves can be constructed as evolutes of spherical twisted curves; we present an illustrative example of a rectifying curve obtained as the evolute of a spherical helix. We also express the curvature and the torsion of a rectifying spherical curve and give necessary and sufficient conditions for a curve and its involute to be both rectifying curves.

A NOTE FOR A GENERALIZATION OF THE DIFFERENTIAL EQUATION OF SPHERICAL CURVES

The differential equation characterizing a spherical curve in $\mathbb{R}^3$ expresses the radius of curvature of the curve in terms of its torsion. In this paper we give a generalization of this equation for a curve lying in an arbitrary surface in $\mathbb{R}^3$. Moreover we establish the analogue of the Frenet equations for a curve lying in a surface of $\mathbb{R}^3$. We have also revisited some formulas for the geodesic torsion of a curve lying in a surface of $\mathbb{R}^3$.