Spherical Curves Research Articles

Direction Curves Associated with Darboux Vectors Fields and Their Characterizations

In this paper, we consider the Darboux frame of a curve α lying on an arbitrary regular surface and we use its unit osculator Darboux vector D ¯ o , unit rectifying Darboux vector D ¯ r , and unit normal Darboux vector D ¯ n to define some direction curves such as D ¯ o -direction curve, D ¯ r -direction curve, and D ¯ n -direction curve, respectively. We prove some relationships between α and these associated curves. Especially, the necessary and sufficient conditions for each direction curve to be a general helix, a spherical curve, and a curve with constant torsion are found. In addition to this, we have seen the cases where the Darboux invariants δ o , δ r , and δ n are, respectively, zero. Finally, we enrich our study by giving some examples.

Nullcone dual worldsheets and caustics of spacelike fronts in null de Sitter sphere

In physics, the application of the regular cases of spacelike fronts has been studied extensively; however, under the gravitational influence, the spacelike fronts can be singular. In this paper, we will consider the traveling trajectories of a set of parallel light rays as spacelike fronts with singularities in null de Sitter sphere. It is of great significance to investigate and explore the spacelike fronts which may have singularities in null de Sitter sphere. We also characterize the singularities of the nullcone dual worldsheets generated by spacelike fronts via the spherical curvature. Then we define the nullcone caustics and analyze the geometric properties of the nullcone caustics at singular points. Moreover, we establish the dual relationship between spacelike fronts and the nullcone dual worldsheets.

On the geometric dynamics of the charged point-particle propagated through the spherical optical fiber

In this manuscript, we basically focus on the geometric dynamics of the charged particle propagation through the spherical frame. To be able to investigate this process, we suppose that the trajectory of the linearly polarizable light coincides with the path of the point-particle having charges, in which this action is exerted to take place along with the optical fiber designated by the spherical frame on the unit two-sphere. We prove in an elegant fashion that two famous parallel transportations i.e. Fermi–Walker parallel transportation and Rytov parallel transportation are applicable for our case so that we can obtain the geometric phase of the charged point-particle. Later, we present a description of spherical electromagnetic curves on the unit sphere by using the fundamental definition of the Lorentz force and electromagnetic force of the charged point-particle. We further apply our findings to compute the optical magnetic-torque, optical linear and angular momentum, energy-exchanges rate, Poynting vector, electromagnetic force, etc. to comprehend the physical dynamics of the moving charged particle.

On geodesically reversible Finsler manifolds

A Finsler manifold is said to be geodesically reversible if the reversed curve of any geodesic remains a geometrical geodesic. Well-known examples of geodesically reversible Finsler metrics are Randers metrics with closed [Formula: see text]-forms. Another family of well-known examples are projectively flat Finsler metrics on the [Formula: see text]-sphere that have constant positive curvature. In this paper, we prove some geometrical and dynamical characterizations of geodesically reversible Finsler metrics, and we prove several rigidity results about a family of the so-called Randers-type Finsler metrics. One of our results is as follows: let [Formula: see text] be a Riemannian–Finsler metric on a closed surface [Formula: see text], and [Formula: see text] be a small antisymmetric potential on [Formula: see text] that is a natural generalization of [Formula: see text]-form (see Sec. 1). If the Randers-type Finsler metric [Formula: see text] is geodesically reversible, and the geodesic flow of [Formula: see text] is topologically transitive, then we prove that [Formula: see text] must be a closed [Formula: see text]-form. We also prove that this rigidity result is not true for the family of projectively flat Finsler metrics on the [Formula: see text]-sphere of constant positive curvature.

Poncelet plectra: harmonious curves in cosine space

It has been shown that the family of Poncelet N-gons in the confocal pair (elliptic billiard) conserves the sum of cosines of its internal angles. Curiously, this quantity is equal to the sum of cosines conserved by its affine image where the caustic is a circle. We show that furthermore, (i) when N = 3, the cosine triples of both families sweep the same planar curve: an equilateral cubic resembling a plectrum (guitar pick). We also show that (ii) the family of triangles excentral to the confocal family conserves the same product of cosines as the one conserved by its affine image inscribed in a circle; and that (iii) cosine triples of both families sweep the same spherical curve. When the triple of log-cosines is considered, this curve becomes a planar, plectrum-shaped curve, rounder than the one swept by its parent confocal family.

Smarandache curves of Anti-Salkowski curve according to the spherical indicatrix curve of the unit darboux vector

In this paper, we have defined special Smarandache curves according to Sabban frame formed by the unit Darboux vector of Anti-Salkowski curve. Next, the Sabban frame belonging to these curves have been constituted. Last, the geodesic curvatures of these Smarandache curves have been calculated and an example for each curve has been illustrated.

Monte Carlo simulations and mean-field modeling of electric double layers at weakly and moderately charged spherical macroions.

Monte Carlo simulations are employed to determine the differential capacitance of an electric double layer formed by small size-symmetric anions and cations in the vicinity of weakly to moderately charged macroions. The influence of interfacial curvature is deduced by investigating spherical macroions, ranging from flat to moderately curved. We also calculate the differential capacitance using a previously developed mean-field model where, in addition to electrostatic interactions, the excluded volumes of the ions are taken into account using either the lattice-gas or the Carnahan-Starling equation of state. For both equations of state, we compare the mean-field model for arbitrary curvature with a recently developed second-order curvature expansion. Our Monte Carlo simulations predict an increase in the differential capacitance with growing macroion curvature if the surface charge density is small, whereas for moderately charged macroions the differential capacitance passes through a local minimum. Both mean-field models tend to somewhat overestimate the differential capacitance as compared with Monte Carlo simulations. At the same time, they do reproduce the curvature dependence of the differential capacitance, especially for small surface charge density. Our study suggests that the quality of mean-field modeling does not worsen when weakly or moderately charged macroions exhibit spherical curvature.

A Delaunay-type classification result for prescribed mean curvature surfaces in 𝕄2(κ) × ℝ

The purpose of this paper is to study immersed surfaces in the product spaces $\mathbb{M}^2(\kappa)\times\mathbb{R}$, whose mean curvature is given as a $C^1$ function depending on their angle function. This class of surfaces extends widely, among others, the well-known theory of surfaces with constant mean curvature. In this paper we give necessary and sufficient conditions for the existence of prescribed mean curvature spheres, and we describe complete surfaces of revolution proving that they behave as the Delaunay surfaces of CMC type.

Enveloids and involutoids of spherical Legendre curves

We define ϕ-enveloids for one-parameter families of spherical Legendre curves. The ϕ-enveloids are not only the generalizations of isogonal trajectories of regular spherical curves but also the generalizations of envelopes of spherical curves. We also consider the relationships between enveloids of spherical Legendre curves and plane Legendre curves. Furthermore, we generalize the notions of involutes and evolutes to the notions of involutoids and evolutoids of spherical curves from the view point of ϕ-enveloids. Then we give the duality theorem for involutoids and evolutoids. Finally, we give the relationships among involutes, evolutes, 0-enveloids (envelopes) and π/2-enveloids (normal envelopes).

Full-Waveform Inversion of High-Frequency Teleseismic Body Waves Based on Multiple Plane-Wave Incidence: Methods and Practical Applications

ABSTRACTTeleseismic full-waveform inversion has recently been applied to image subwavelength-scale lithospheric structures (typically a few tens of kilometers) by utilizing hybrid methods in which an efficient solver for the 1D background model is coupled with a full numerical solver for a small 3D target region. Among these hybrid methods, the coupling of the frequency–wavenumber technique with the spectral element method is one of the most computationally efficient ones. However, it is normally based on a single plane-wave incidence, and thus cannot synthesize secondary global phases generated at interfaces outside the target area. To remedy the situation, we propose to use a multiple plane-wave injection method to include secondary global phases in the hybrid modeling. We investigate the performance of the teleseismic full-waveform inversion based on single and multiple plane-wave incidence through an application in the western Pyrenees and compare it with previously published images and the inversion based on a global hybrid method. In addition, we also test the influence of Earth’s spherical curvature on the tomographic results. Our results demonstrate that the teleseismic full-waveform inversion based on a single plane-wave incidence can reveal complex lithospheric structures similar to those imaged using a global hybrid method and is reliable for practical tomography for small regions with an aperture of a few hundred kilometers. However, neglecting the Earth’s spherical curvature and secondary phases leads to errors on the recovered amplitudes of velocity anomalies (e.g., about 2.8% difference for density and VS, and 4.2% for VP on average). These errors can be reduced by adopting a spherical mesh and injecting multiple plane waves in the frequency–wavenumber-based hybrid method. The proposed plane-wave teleseismic full-waveform inversion is promising for mapping subwavelength-scale seismic structures using high-frequency teleseismic body waves (>1 Hz) including coda waves recorded at large N seismic arrays.

Quadratically pinched hypersurfaces of the sphere via mean curvature flow with surgery

We study mean curvature flow in \({\mathbb {S}}_K^{n+1}\), the round sphere of sectional curvature \(K>0\), under the quadratic curvature pinching condition \(|A|^{2} < \frac{1}{n-2} H^{2} + 4 K\) when \(n\ge 4\) and \(|A|^{2} < \frac{3}{5}H^{2}+\frac{8}{3}K\) when \(n=3\). This condition is related to a famous theorem of Simons (Ann Math 2(88):62–105, 1968), which states that the only minimal hypersurfaces satisfying \(\vert A\vert ^2<nK\) are the totally geodesic hyperspheres. It is related to but distinct from “two-convexity”. Notably, in contrast to two-convexity, it allows the mean curvature to change sign. We show that the pinching condition is preserved by mean curvature flow, and obtain a “cylindrical” estimate and corresponding pointwise derivative estimates for the curvature. As a result, we find that the flow becomes either uniformly convex or quantitatively cylindrical in regions of high curvature. This allows us to apply the surgery apparatus developed by Huisken and Sinestrari (Invent Math 175(1):137–221, 2009) (cf. Haslhofer and Kleiner, Duke Math J 166(9):1591–1626, 2017). We conclude that any smoothly, properly immersed hypersurface \({\mathcal {M}}\) of \({\mathbb {S}}_K^{n+1}\) satisfying the pinching condition is diffeomorphic to \({\mathbb {S}}^n\) or to the connected sum of a finite number of copies of \({\mathbb {S}}^1\times {\mathbb {S}}^{n-1}\). If \({\mathcal {M}}\) is embedded, then it bounds a 1-handlebody. The results are sharp when \(n\ge 4\).

New approach to optical spherical [formula omitted]t- flux by Heisenberg optical Landau Lifshitz condition

In this article, we study spherical optical Lorentz flux of spherical St − optical flows with a spherical frame on the spherical optical Heisenberg space SH2. We get optical new conditions for spherical St − optical Lorentz Heisenberg flux with spherical fields. Thus, we define spherical Lorentz Heisenberg flux for optical magnetic fields. We provide new construction for optical spherical curvatures of spherical St − flows with spherical Landau Lifshitz condition. Additionally, magnetic optical flux surface is constructed in certain uniform optical surfaces with numerical and analytical results by spherical Landau Lifshitz condition.

A quasi complex-lamellar solution for a hemispherically bounded cyclonic flowfield

This work describes an inviscid solution for a cyclonic flowfield evolving in a hemispherical chamber configuration. In this context, a vigorously swirling motion is triggered by a gaseous stream that is introduced tangentially to the inner circumference of the chamber's equatorial plane. The resulting updraft spirals around while sweeping the chamber wall, reverses direction while approaching the headend, and then tunnels itself out through the inner core portion of the chamber. Our analysis proceeds from the Bragg–Hawthorne formulation, which proves effective in the treatment of steady, incompressible, and axisymmetric motions. Then using appropriate boundary conditions, we are able to obtain a closed-form expression for the Stokes streamfunction in both spherical and cylindrical coordinates. Other properties of interest are subsequently deduced, and these include the principal velocities and pressure distributions, vorticity, swirl intensity, helicity density, crossflow velocity, and the location of the axial mantle; the latter separates the outer annular updraft from the inner, centralized downdraft. Due in large part to the overarching spherical curvature, the cyclonic motion also exhibits a polar mantle across which the flow becomes entirely radial inward. Along this lower and shorter spherical interface, the polar velocity vanishes while switching angular direction. Interestingly, both polar and axial mantles coincide in the exit plane where the ideal outlet size, prescribed by the mantle position, is found to be approximately 70.7% of the chamber radius. We thus recover the same mantle fraction of the cyclonic flow analog in a right-cylindrical chamber where an essentially complex-lamellar motion is established.

Optical spherical electromotive density with some fractional applications with Laplace transform in spherical Heisenberg space [formula omitted

In this paper, we construct a new theory of spherical electromotive force of spherical Ss −magnetic flows with the spherical frame in spherical Heisenberg space SH2. Also, we obtain some optical conditions of spherical Ss −magnetic Lorentz flux by using optical spherical fields in spherical Heisenberg space SH2. Moreover, we give some applications of spherical Heisenberg electromotive forces fEϕαH,fEϕtH,fEϕsH for spherical vector fields. Additionally, we compute representation of spherical curvatures of spherical Ss −magnetic flows by considering Heisenberg spherical ferromagnetic spin in spherical Heisenberg space SH2. Finally, spherical Heisenberg electromotive forces fEϕαH,fEϕtH,fEϕsH are given in a static and uniform magnetic surface with the q -Homotopy analysis transform method in spherical Heisenberg space SH2.

Regularity of optimal transport maps on locally nearly spherical manifolds

Given a compact connected n-dimensional Riemannian manifold, we investigate the smoothness of the optimal transport map between the smooth densities with respect to the squared Riemannian distance cost. The optimal map is characterized by exp(gradu), where the potential function u satisfies a Monge–Ampère type equation. Delanoë [7] showed the smoothness of u on the Riemannian surfaces when the scalar curvature is close to 1 in C 2 norm. In this work, we study the regularity issue on Riemannian manifolds with curvature sufficiently close to curvature of round sphere in C 2 norm in all dimensions and prove that the 𝒞-curvature on such Riemannian manifolds satisfies an improved Ma-Trudinger-Wang condition and the Jacobian of the exponential map is positive. As a consequence, we imply the smoothness of the optimal transport map by the continuity method.

Space Curves Related by a Transformation of Combescure

In this paper, the curves associated with the Combescure transform are discussed. With the help of the fact that these curves have a common Frenet frame, an equivalence relation is defined. The equivalence classes obtained by this equivalence relation have been examined for some special curves and it has been obtained that all curves in the equivalence class of a helix curve are also helix curves. This is also true for k-slant helix curves. The important part of this paper consists of the useful construction method to obtain a curve from the given curve α with the help of Combescure transformation. With this method, some special curves such as Bertrand, Mannheim, Salkowski, anti-Salkowski or spherical curve can be obtained from any curve related by a Combescure transform. For example, we obtain an example of Mannheim curves explicitly obtained from an anti-Salkowski curve. It is not easy to find an example of Mannheim curves except circular helix in the literature. In general, the condition for being Bertrand anti-Salkowski or spherical curve of the curve β obtained from the given curve α with the help of Combescure transformation were obtained.

Natural and conjugate mates of Frenet curves in three-dimensional Lie group

In this study, we introduce the natural mate and conjugate mate of a Frenet curve in a three dimensional Lie group $ \mathbb{G} $ with bi-invariant metric. Also, we give some relationships between a Frenet curve and its natural mate or its conjugate mate in $ \mathbb{G} $. Especially, we obtain some results for the natural mate and the conjugate mate of a Frenet curve in $ \mathbb{G} $ when the Frenet curve is a general helix, a slant helix, a spherical curve, a rectifying curve, a Salkowski (constant curvature and non-constant torsion), anti-Salkowski (non-constant curvature and constant torsion), Bertrand curve. Finally, we give nice graphics with numeric solution in Euclidean 3-space as a commutative Lie group.

Arrow diagrams on spherical curves and computations

We give a definition of an integer-valued function [Formula: see text] derived from arrow diagrams for the ambient isotopy classes of oriented spherical curves. Then, we introduce certain elements of the free [Formula: see text]-module generated by the arrow diagrams with at most [Formula: see text] arrows, called relators of Type ([Formula: see text]) (([Formula: see text]), ([Formula: see text]), ([Formula: see text]) or ([Formula: see text]), respectively), and introduce another function [Formula: see text] to obtain [Formula: see text]. One of the main results shows that if [Formula: see text] vanishes on finitely many relators of Type ([Formula: see text]) (([Formula: see text]), ([Formula: see text]), ([Formula: see text]) or ([Formula: see text]), respectively), then [Formula: see text] is invariant under the deformation of type RI (strong RI[Formula: see text]I, weak RI[Formula: see text]I, strong RI[Formula: see text]I[Formula: see text]I or weak RI[Formula: see text]I[Formula: see text]I, respectively). The other main result is that we obtain new functions of arrow diagrams with up to six arrows explicitly. This computation is done with the aid of computers.

Spherical Principal Curves.

This paper presents a new approach for dimension reduction of data observed on spherical surfaces. Several dimension reduction techniques have been developed in recent years for non-euclidean data analysis. As a pioneer work, (Hauberg 2016) attempted to implement principal curves on Riemannian manifolds. However, this approach uses approximations to process data on Riemannian manifolds, resulting in distorted results. This study proposes a new approach to project data onto a continuous curve to construct principal curves on spherical surfaces. Our approach lies in the same line of (Hastie and Stuetzle et al. 1989) that proposed principal curves for data on euclidean space. We further investigate the stationarity of the proposed principal curves that satisfy the self-consistency on spherical surfaces. The results on the real data analysis and simulation examples show promising empirical characteristics of the proposed approach.

Spherical magnetic flux flows with fractional Heisenberg spherical ferromagnetic spin of optical spherical flux density with fractional applications

In this paper, we construct a new approach of spherical magnetic Lorentz flux of spherical [Formula: see text]-magnetic flows of particles by the spherical frame in [Formula: see text] spherical space. Eventually, we obtain some optical conditions of spherical [Formula: see text]-magnetic Lorentz flux by using directional spherical fields. Moreover, we determine spherical magnetic Lorentz flux for spherical vector fields. Also, we give new construction for spherical curvatures of spherical [Formula: see text]-magnetic flows by considering Heisenberg spherical ferromagnetic spin. The approximate solution is expressed by a table and some graphics. Finally, the magnetic flux surface is demonstrated in a static and uniform magnetic surface by using the analytical and numerical results.