The Geometrical Characterizations of the Bertrand Curves of the Null Curves in Semi-Euclidean 4-Space

According to the Frenet equations of the null curves in semi-Euclidean 4-space, the existence conditions and the geometrical characterizations of the Bertrand curves of the null curves are given in this paper. The examples and the graphs of the Bertrand pairs with two different conditions are also given in order to supplement the conclusion of this paper more intuitively.

Motions of Curves in 4-Dimensional Galilean Space G4

In this paper, we investigate the flow of curve and its equiform geometry in 4-dimensional Galilean space. We obtain that the Frenet equations and curvatures of inextensible flows of curves and its equiformly invariant vector fields and intrinsic quantities are independent of time. We find that the motions of curves and its equiform geometry can be defined by the inviscid and stochastic Burgers’ equations in 4-dimensional Galilean space.

Frenet curves in 3-dimensional $ \delta $-Lorentzian trans Sasakian manifolds

<abstract><p>In this paper, we give some characterizations of Frenet curves in 3-dimensional $ \delta $-Lorentzian trans-Sasakian manifolds. We compute the Frenet equations and Frenet elements of these curves. We also obtain the curvatures of non-geodesic Frenet curves on 3-dimensional $ \delta $-Lorentzian trans-Sasakian manifolds. Finally, we give some results for these curves.</p></abstract>

Geometrical particles and Legendrian dualities related to null curves

In this paper, we investigate the special properties of geometrical particles with null paths in de Sitter 3-space–time, new Frenet equations and an important invariant associated with null paths are presented. By means of unfolding theory, the local topological structure of the lightlike dual surfaces is revealed. It is found that the lightlike dual surface has some singularities whose types can be determined by the invariant. Based on the theory of Legendrian dualities on pseudospheres and the theory of contact manifolds, it is shown that there exists the [Formula: see text]-dual relationship between the lightlike transversal trajectory of the particle and the lightlike dual surface. In addition, an interesting and important fact mentioned is that the contact of lightlike transversal trajectory with lightcone quadric and the contact of lightlike transversal trajectory with null hyperplane have the same order when they are related to the same type of singularities of the lightlike dual surface.

On Null Curves in Minkowski 3-Space and Its Fractal Folding

In this paper, a form for Frenet equations of all null curves in Minkowski 3-space has been presented. New types of foldings of curves are obtained. The connection between folding, deformation and Frenet equations of curves are also deduced.

Serret-Frenet formulas for octonionic curves

In this paper, we dene spatial octonionic curves (SOC) in R7 and octonionic curves (OC) in R8 by using octonions. Firstly, we determine Serret-Frenet equations, and curvatures of the SROC in R7. Then, Serret-Frenet equations for the OC in R8 are calculated with the help of Serret-Frenet equations of SOC in R7.

Spiralling liquid jets: verifiable mathematical framework, trajectories and peristaltic waves

The dynamics of a jet of an inviscid incompressible liquid spiralling out under the action of centrifugal forces is considered with both gravity and the surface tension taken into account. This problem is of direct relevance to a number of industrial applications, ranging from the spinning disc atomization process to nanofibre formation. The mathematical description of the flow by necessity requires the use of a local curvilinear non-orthogonal coordinate system centred around the jet’s baseline, and we present the general formulation of the problem without assuming that the jet is slender. To circumvent the inconvenience inherent in the non-orthogonality of the local coordinate system, the orthonormal Frenet basis is used in parallel with the local non-orthogonal basis, and the equation of motion, with the velocity considered with respect to the local coordinate system, is projected onto the Frenet basis. The variation of the latter along the baseline is then described by the Frenet equations which naturally brings the baseline’s curvature and torsion into the equations of motion. This technique allows one to handle different line-based non-orthogonal curvilinear coordinate systems in a straightforward and mathematically transparent way. An analysis of the slender-jet approximation that follows the general formulation shows how a set of ordinary differential equations describing the jet’s trajectory can be derived in two cases: $\mathit{We}=O(1)$ and $\unicode[STIX]{x1D716}\mathit{We}=O(1)$ as $\unicode[STIX]{x1D716}\rightarrow 0$, where $\unicode[STIX]{x1D716}$ is the ratio of characteristic length scales across and along the jet and $\mathit{We}$ is the Weber number. A one-dimensional model for the propagation of nonlinear peristaltic disturbances along the jet is derived in each of these cases. A critical review of the work published on this topic is presented showing where errors typically occur and how to identify and avoid them.

Motions of Curves in the Pseudo-Galilean SpaceG31

We study the flows of curves in the pseudo-Galilean 3-space and its equiform geometry without any constraints. We find that the Frenet equations and intrinsic quantities of the inelastic flows of curves are independent of time. We show that the motions of curves in the pseudo-Galilean 3-space and its equiform geometry are described by the inviscid and viscous Burgers’ equations.

Characterizations of Spacelike Slant Helices In Minkowski 3-Space

In this paper, we investigate tangent indicatrix, principal normal indicatrix and binormal indicatrix of a spacelike curve with spacelike, timelike and null principal normal vector in Minkowski 3-space E3 1 and we construct their Frenet equations and curvature functions. Moreover, we obtain some differential equations which characterize for a spacelike curve to be a slant helix by using the Frenet apparatus of spherical indicatrix of the curve. Also related examples and their illustrations are given. Mathematics Subject Classification 2010: 53A04, 53C50.