Geodesic Curvature Research Articles

Relation Between the Low Frequency GAM and the Neoclassical Flow

Since the magnetic field strength is not constant on the magnetic flux surface, the flow also varies so that the density compression occurs along the poloidal direction. Since the inhomogeneous flow causes the density compression in the poloidal direction, the parallel flow is also perturbed. In this study, we investigate the effects of the parallel flow perturbation on the geodesic acoustic mode (GAM) when it is described by the kinetic approach. Using the continuity equation, it is shown that the flow perturbation in the geodesic curvature direction is balanced by the lowest-order term of the density perturbation in , and the flow perturbation in the parallel direction is balanced by the higher-order terms of the density perturbation. Since the density perturbation includes both the perpendicular and parallel flow perturbation contributions, the GAM frequency obtained by the kinetic approach has the parallel flow perturbation contribution, which is 1/ term in the GAM frequency equation. The low frequency branch of the dispersion relation is also discussed to demonstrate the connection between the GAM theory and neoclassical theory for the first time. It is shown that the flow perturbation in the geodesic curvature direction is balanced mostly by the parallel flow perturbation. It means that the flow in the flux surface is divergence free approximately as in the usual transport ordering. Thus, the poloidal flow goes to the neoclassical flow when the low frequency branch is taken.

Efficient and robust discrete conformal equivalence with boundary

We describe an efficient algorithm to compute a discrete metric with prescribed Gaussian curvature at all interior vertices and prescribed geodesic curvature along the boundary of a mesh. The metric is (discretely) conformally equivalent to the input metric. Its construction is based on theory developed in [Gu et al. 2018b] and [Springborn 2020], relying on results on hyperbolic ideal Delaunay triangulations. Generality is achieved by considering the surface's intrinsic triangulation as a degree of freedom, and particular attention is paid to the proper treatment of surface boundaries. While via a double cover approach the case with boundary can be reduced to the case without boundary quite naturally, the implied symmetry of the setting causes additional challenges related to stable Delaunay-critical configurations that we address explicitly. We furthermore explore the numerical limits of the approach and derive continuous maps from the discrete metrics.

Motion of Charged Spinning Particles in a Unified Field

Using a geometry wider than Riemannian one, the parameterized absolute parallelism (PAP) geometry, we derived a new curve containing two parameters. In the context of the geometrization philosophy, this new curve can be used as a trajectory of charged spinning test particle in any unified field theory constructed in the PAP space. We show that imposing certain conditions on the two parameters, the new curve can be reduced to a geodesic curve giving the motion of a scalar test particle or/and a modified geodesic giving the motion of neutral spinning test particle in a gravitational field. The new method used for derivation, the Bazanski method, shows a new feature in the new curve equation. This feature is that the equation contains the electromagnetic potential term together with the Lorentz term. We show the importance of this feature in physical applications.

A Length-Constrained Ideal Curve Flow

Abstract A recent article [1] considered the so-called ‘ideal curve flow’, a sixth-order curvature flow that seeks to deform closed planar curves to curves with least variation of total geodesic curvature in the L2 sense. It was critical in the analysis in that article that there was a length bound on the evolving curves. It is natural to suspect therefore that the length-constrained ideal curve flow should permit a more straightforward analysis, at least in the case of small initial ‘energy’. In this article we show this is indeed the case, with suitable initial data providing a flow that exists for all time and converges smoothly and exponentially to a multiply-covered round circle of the same length and winding number as the initial curve.

A spin-less particle on a rotating curved surface in Minkowski space

In Minkowski space , we derive the effective Schrödinger equation describing a spin-less particle confined to a rotating curved surface . Using the thin-layer quantization formalism to constrain the particle on , we obtain the relativity-corrected geometric potential , and a novel effective potential related to both the Gaussian curvature and the geodesic curvature of the rotating surface. The Coriolis effect and the centrifugal potential also appear in the equation. Subsequently, we apply the surface Schrödinger equation to a rotating cylinder, sphere and torus surfaces, in which we find that the interplays between the rotation and surface geometry can contribute to the energy spectrum based on the potentials they offer.

Isoperimetric inequalities and geometry of level curves of harmonic functions on smooth and singular surfaces

We investigate the logarithmic convexity of the length of the level curves for harmonic functions on surfaces and related isoperimetric type inequalities. The results deal with smooth surfaces, as well as with singular Alexandrov surfaces (also called surfaces with bounded integral curvature), a class which includes for instance surfaces with conical singularities and surfaces of CAT(0) type. Moreover, we study the geodesic curvature of the level curves and of the steepest descent for harmonic functions on surfaces with non-necessarily constant Gaussian curvature K. Such geodesic curvature functions turn out to satisfy certain Laplace-type equations and inequalities, from which we infer various maximum and minimum principles. The results are complemented by a number of growth estimates for the derivatives L' and L'' of the length of the level curve function L, as well as by examples illustrating the presentation. Our work generalizes some results due to Alessandrini, Longinetti, Talenti, Ma–Zhang and Wang–Wang.

Spectral invariants of Dirichlet-to-Neumann operators on surfaces

We obtain a complete asymptotic expansion for the eigenvalues of the Dirichlet-to-Neumann maps associated with Schrödinger operators on Riemannian surfaces with boundary. For the zero potential, we recover the well-known spectral asymptotics for the Steklov problem. For non-zero potentials, we obtain new geometric invariants determined by the spectrum. In particular, for constant potentials, which give rise to the parameter-dependent Steklov problem, the total geodesic curvature on each connected component of the boundary is a spectral invariant. Under the constant curvature assumption, this allows us to obtain some interior information from the spectrum of these boundary operators.

Comparison of private flux region instability in conventional and super-X divertor configurations

Understanding turbulence in the divertor leg of tokamaks is essential to predict the heat deposition profile on the divertor plate. This in turn is important for evaluating advanced divertor configurations, such as the super-X divertor. Within the divertor region, the private flux region is of interest because it is relatively unaffected by turbulence extending from the outboard midplane, so instabilities in this region could have a particularly pronounced effect on transport. These instabilities are modeled using the Arbitrary Topology Equation Reader (ArbiTER) eigenvalue code. Eigenmodes are examined further by comparing physics models to determine the fundamental mechanisms behind their formation and quantifying the effect of individual terms. This analysis is conducted on both conventional and super-X divertors to compare these effects. The resulting analysis reveals the presence of a geodesic curvature driven instability that is significantly more pronounced in the super-X configuration.

On the Space of $$C^1$$ Regular Curves on Sphere with Constrained Curvature

In this paper, we prove that the \(C^0\) and \(C^1\) topologies are the same on the set of \(C^1\) regular curves in the 2-sphere whose tangent vectors are Lipschitz continuous, and the a.e. existing geodesic curvatures are essentially bounded in an open interval. Besides, we study the subset consisting of curves that start and end at given points with given directions, and prove that this subset is a Banach manifold.

Geodesic Hermite Spline Curve on Triangular Meshes

Curves on a polygonal mesh are quite useful for geometric modeling and processing such as mesh-cutting and segmentation. In this paper, an effective method for constructing C1 piecewise cubic curves on a triangular mesh M while interpolating the given mesh points is presented. The conventional Hermite interpolation method is extended such that the generated curve lies on M. For this, a geodesic vector is defined as a straightest geodesic with symmetric property on edge intersections and mesh vertices, and the related geodesic operations between points and vectors on M are defined. By combining cubic Hermite interpolation and newly devised geodesic operations, a geodesic Hermite spline curve is constructed on a triangular mesh. The method follows the basic steps of the conventional Hermite interpolation process, except that the operations between the points and vectors are replaced with the geodesic. The effectiveness of the method is demonstrated by designing several sophisticated curves on triangular meshes and applying them to various applications, such as mesh-cutting, segmentation, and simulation.

Isometric Correspondences of 3D Models with Geodesic Curves Constrained Functional Maps

<p indent="0mm">For the isometric correspondences between 3D models problem, a method of isometric correspondences based on the functional maps framework is proposed, which can simultaneously detect two isometric correspondences in both direct and symmetric directions by adding geodesic curves constraints. Firstly, two symmetric point pairs on each model are selected, and four corresponded point pairs in both direct and the symmetric directions are constructed among the symmetric point pairs. Secondly, it theoretically proves that the geodesic curves between the four pairs of corresponded points on the two models are isometric correspondences, and then using the above four pairs of corresponded points to obtain six pairs of corresponded geodesic curves. Finally, the above pairs of corresponded geodesic curves are added as geometric descriptor constraints in the functional maps framework, which can compute two isometric correspondences in both direct and symmetric directions. At the same time, it theoretically proves that the functional maps representation matrix of a reflection intrinsic symmetry is symmetric and orthogonal, and by adding it as a regularized item of the representation matrix to the functional maps framework based on geodesic curves constraint to detect the intrinsic symmetry of the 3D model. Experiments on the open non-rigid 3D model databases demonstrate that the geodesic distance error of this method is smaller than previous works in the detection results of isometric correspondences and intrinsic symmetry of the 3D models.

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.

Stress-dominated growth of two-dimensional materials on nonplanar substrates

Curved features are ubiquitous on solid surfaces, but the effect of surface curvatures on growth of two-dimensional (2D) materials has not yet been established. Using a newly developed method based on the Metropolis algorithm and taking graphene as a prototype, we find that a curved feature on substrates can result in a variety of topological defects in 2D materials. As the feature's size increases by just nanometers, the defects can vary from adatoms, dislocation pairs, and grain boundary scars to long-range grain boundaries, in contrast to previously reported defect-free modes of rigid colloidal crystals growing on spheres. We identify an important role of curvature-induced lattice stress in lowering the growth rate over the curved features and driving a plastic instability in the materials. When the feature's size increases to several nanometers, the stress effect is compromised by an enhanced effect of geodesic curvature, yielding long-range grain boundaries as a result of increased local growth rate on the feature with respect to that on flat regions. We further provide a ‘phase diagram’ of defects that helps to guide a rational choice of geometrical parameters of features towards the growth of high-quality 2D materials as well as controllable creation of topological defects.

Existence of curves with constant geodesic curvature in a Riemannian 2-sphere

We prove the existence of immersed closed curves of constant geodesic curvature in an arbitrary Riemannian 2-sphere for almost every prescribed curvature. To achieve this, we develop a min-max scheme for a weighted length functional.

Some Special Ruled Surfaces Generated by a Direction Curve according to the Darboux Frame and their Characterizations

In this work, we consider the Darboux frame T , V , U of a curve lying on an arbitrary regular surface and we construct ruled surfaces having a base curve which is a V -direction curve. Subsequently, a detailed study of these surfaces is made in the case where the directing vector of their generatrices is a vector of the Darboux frame, a Darboux vector field. Finally, we give some examples for special curves such as the asymptotic line, geodesic curve, and principal line, with illustrations of the different cases studied.

Concentration solutions to the singularly prescribed Gaussian and geodesic curvatures problem

We consider the following Liouville-type equation with exponential Neumann boundary condition:{−Δu˜=ε2K(x)e2u˜,x∈D,∂u˜∂n+1=εκ(x)eu˜,x∈∂D, where D⊂R2 is the unit disk, ε2K(x)>0 and εκ(x)>0 stand for the prescribed Gaussian curvature and geodesic curvature of the boundary, respectively. We prove the existence of concentration solutions if κ(x)+K(x)+κ(x)2 (x∈∂D) has a local extremum point, which is a new result for exponential Neumann boundary problems.

On a surface pencil with a common new type of special surface curve in Galilean space G 3

In this study, we investigate a new type of surface curve called a D-type special curve. We also show that this special curve is more general than a geodesic curve or an asymptotic curve. Then, we give the necessary and sufficient conditions for a curve to be the new D-type special curve using Frenet frame in Galilean space. Moreover, we investigate some results by taking account of a D-type special curve as a helix, a Salkowski curve and an anti-Salkowski curve. After then, for the sake of visualizing this study, we plot some examples for this surface pencil (i.e. surface family).

Demystifying autoparallels in alternative gravity

Autoparallel curves along with geodesic curves can arise as trajectories of physical test bodies. We explicitly derive autoparallels as effective post-Riemannian geometric constructs, and at the same time we argue \emph{against} postulating autoparallels as fundamental equations of motion for test bodies in alternative gravity theories.

Cyclic parallel structure Jacobi operator for real hypersurfaces in complex two-plane Grassmannians

In this paper, from the property of Killing for structure Jacobi tensor $\mathbb {R}_{\xi }$, we introduce a new notion of cyclic parallelism of structure Jacobi operator$R_{\xi }$ on real hypersurfaces in the complex two-plane Grassmannians. By virtue of geodesic curves, we can give the equivalent relation between cyclic parallelism of $R_{\xi }$ and Killing property of $\mathbb {R}_{\xi }$. Then, we classify all Hopf real hypersurfaces with cyclic parallel structure Jacobi operator in complex two-plane Grassmannians.

The Sub-Riemannian Limit of Curvatures for Curves and Surfaces and a Gauss-Bonnet Theorem in the Group of Rigid Motions of Minkowski Plane with General Left-Invariant Metric

The group of rigid motions of the Minkowski plane with a general left-invariant metric is denoted by E 1 , 1 , g λ 1 , λ 2 , where λ 1 ≥ λ 2 &gt; 0 . It provides a natural 2 -parametric deformation family of the Riemannian homogeneous manifold Sol 3 = E 1 , 1 , g 1 , 1 which is the model space to solve geometry in the eight model geometries of Thurston. In this paper, we compute the sub-Riemannian limits of the Gaussian curvature for a Euclidean C 2 -smooth surface in E 1 , 1 , g L λ 1 , λ 2 away from characteristic points and signed geodesic curvature for the Euclidean C 2 -smooth curves on surfaces. Based on these results, we get a Gauss-Bonnet theorem in the group of rigid motions of the Minkowski plane with a general left-invariant metric.