arXiv research

Characterizes concircular helices and surfaces in 3D space.

Study finds all helical surfaces with a constant ratio of principal curvatures.

Researchers create exact minimal surfaces with helical motifs in biological structures.

The study characterizes helices in Euclidean and hyperbolic spaces.

A topologically minimal surface may be isotoped into a normal form with respect to a fixed triangulation. If the intersection with each tetrahedron is simply connected, then the pieces of this normal form are triangles, quadrilaterals, and helicoids. Helical pieces can have any number of positive or negative twists. We…

Characterizes concircular helices in space forms and ruled hypersurfaces.

We introduce semi-helix hyper surfaces of Euclidean spaces. We also provide a local characterization of how these semi-helices are constructed.

The Enneper surface and helix surfaces are unique in their geometric properties.

Helical ribbons arise in many biological and engineered systems, often driven by anisotropic surface stress, residual strain, and geometric or elastic mismatch between layers of a laminated composite. A full mathematical analysis is developed to analytically predict the equilibrium deformed helical shape of an initiall…

The present paper attempts to show an alternative approach with regards to rational Pythagorean-hodograph (PH) curves and especially more natural approach for rational PH helices (i.e. rational helices). It exploits geometric features of rational helices to obtain a simpler construction of these curves and apply this t…

A 3-parameter family of helical tubular surfaces obtained by screw revolving a circle provides a useful pedagogical example of how to study geodesics on a surface that admits a 1-parameter symmetry group, but is not as simple as a surface of revolution like the torus which it contains as a special case. It serves as a …

Study harmonic surfaces in 3D space, proving superposition principle.

In this paper, we consider a regular curve on an oriented surface in Euclidean 3-space with the Darboux frame $\{\mathsf{T},\mathsf{V},\mathsf{U}\}$ along the curve, where $\mathsf{T}$ is the unit tangent vector field of the curve, $\mathsf{U}$ is the surface normal restricted to the curve and $\mathsf{V}=\mathsf{U}\ti…

New type of ruled surfaces studied with properties and examples.

Study on triharmonic curves in 3D spaces, proving their existence and classification.

Plasma toroidal metric singularities in helical devices and tokamaks, giving rise to magnetic surfaces inside the plasma devices are investigated in two cases. In the first we consider the case of a rotational plasma on an helical device with circular cross-section and dissipation. In this case singularities are shown …

Study of curves and surfaces in Riemannian spaces making a constant angle with a parallel transported direction.

Innocent musing on geodesics on the surface of helical pasta shapes leads to a single continuous 4-parameter family of surfaces invariant under at least a 1-parameter symmetry group and which contains as various limits spheres, tori, helical tubes, and cylinders, all useful for illustrating various aspects of geometry …

In this paper, in Euclidean n -space, we investigate the relation between slant helices and spherical helices. Moreover, in E n, we show that a slant helix and the tangent indicatrix of the slant helix have the same axis (or direction). Also, we give the important relations between slant helices, spherical helices in E…

In this study, we give definitions and characterizations of eikonal slant helices, eikonal Darboux helices and non-normed eikonal Darboux helices in 3-dimensional pseudo- Riemannian manifold M . We show that every eikonal slant helix is also an eikonal Darboux helix for timelike and spacelike curves. Furthermore, we ob…

The paper defines Vn-slant helices in a lightlike cone and their curvature functions.

Study classifies polyharmonic helices in various space forms.

In this paper, we find all constant slope surfaces in the Euclidean 3-space, namely those surfaces for which the position vector of a point of the surface makes constant angle with the normal at the surface in that point. These surfaces could be thought as the bi-dimensional analogue of the generalized helices. Some pi…

In this paper, we define slant helices in three dimensional Lie Groups with a bi-invariant metric and obtain a characterization of slant helices. Moreover, we give some relations between slant helices and their involutes, spherical images.

Isophote comprises a locus of the surface points whose normal vectors make a constant angle with a fixed vector. In this paper, isophote curves are studied on timelike surfaces in Minkowski 3-space E31. The axises of spacelike and timelike isophote curves are found via their Darboux frames. Subsequently, the relationsh…

Building on previous results on the quadratic helicity in magnetohydrodynamics (MHD) we investigate particular minimum helicity states. Those are eigenfunctions of the curl operator and are shown to constitute solutions of the quasi-stationary incompressible ideal MHD equations. We then show that these states have inde…

Study on null helices in semi-Riemannian manifolds with special submanifolds.

The paper investigates polyharmonic helices in 3D solvable Lie group Sol_3 and Euclidean spheres.

New invariants defined for volume-preserving flows on 3-manifolds.

Optimized concentric helices minimize the ropelength of non-alternating torus knots.

The helicity of a vector field is a measure of the average linking of pairs of integral curves of the field. Computed by a six-dimensional integral, it is widely useful in the physics of fluids. For a divergence-free field tangent to the boundary of a domain in 3-space, helicity is known to be invariant under volume-pr…

In this work, we studied the properties of the spherical indicatrices of involute curve of a space curve and presented some characteristic properties in the cases that involute curve and evolute curve are slant helices and helices, spherical indicatrices are slant helices and helices and we introduced new representatio…

Solvents can induce helical knots in simulated biopolymer tubes.

Weaved helices form mechanically stable 3D structures.

In this paper, we give some characterizations for spacelike helices in Minkowski space-time. We find the differential equations characterizing the spacelike helices and also give the integral characterizations for these curves in Minkowski space-time.

In this work, we give some new characterizations for inclined curves and slant helices in n-dimensional Euclidean space E^{n}. Morever, we consider the pre-characterizations about inclined curves and slant helices and reconfigure them.

In this paper, we define some new associated curves as integral curves of a vector field generated by Frenet vectors of tangent indicatrix of a curve in Euclidean 3-space. We give some relationships between curvatures of these curves. By using these associated curves, we give some methods to construct helices and slant…

A new algebraic method for computing helicity is developed, by discovering a relationship between helicity of fluid mechanics and algebraic polynomial invariants of knot theory. We have constructed a topological invariant $t^{H\left(\mathcal{L}\right)}$ for a link $\mathcal{L}$ of knots, where $H$ is the helicity of a …

The class of traveling wave solutions of the sine-Gordon equation is known to be in 1-1 correspondence with the class of (necessarily singular) pseudospherical surfaces in Euclidean space with screw-motion symmetry: the pseudospherical helicoids. We explicitly describe all pseudospherical helicoids in terms of elliptic…

New constraints rule out some optimal domains for helicity maximisation.

In this study, we give definitions and characterizations of eikonal slant helix curves, eikonal Darboux helices and non-normed eikonal Darboux helices in three dimensional Riemannian manifold 3 M . We show that every eikonal slant helix is also an eikonal Darboux helix. Furthermore, we obtain that if the curve a is a n…

Study surfaces with constant ratio of principal curvatures in Euclidean and isotropic geometries.

Given a flow on a 3-dimensional integral homology sphere, we give a formula for the Euler characteristic of its transverse surfaces, in terms of boundary data only. We illustrate the formula with several examples, in particular with surfaces of low genus. As an application, we show that for a right-handed flow with an …

Study isotropic Riemannian maps and helices along them.

We prove that any regular Casimir in 3D magnetohydrodynamics is a function of the magnetic helicity and cross-helicity. In other words, these two helicities are the only independent regular integral invariants of the coadjoint action of the MHD group $\text{SDiff}(M)\ltimes\mathfrak X^*(M)$, which is the semidirect pro…

We obtain explicit, isometry-invariant integral formulas for twisting, writhing and helicity, and prove the theorem LINK = TWIST + WRITHE on the 3-sphere and in hyperbolic 3-space. We then use these results to derive upper bounds for the helicity of vector fields and lower bounds for the first eigenvalue of the curl op…

We prove that any regular integral invariant of volume-preserving transformations is equivalent to the helicity. Specifically, given a functional $\mathcal I$ defined on exact divergence-free vector fields of class $C^1$ on a compact 3-manifold that is associated with a well-behaved integral kernel, we prove that $\mat…

Proves helicity is the only regular Casimir for 3D hydrodynamics.