The submanifold Dirac operator has been studied for this decade, which is closely related to Frenet-Serret and generalized Weierstrass relations. In this article, we will give a submanifold Dirac operator defined over a surface immersed in E4 with U(1)-gauge field as torsion in the sense of the Frenet-Serret relation, which also has data of immersion of the surface in E4. Mathematics Subject Cl...

The study of magnetic curves, seen as solutions Lorentz equation, has been done mainly in 3-dimensional case, motivated by theoretical physics. Then it was extended higher dimensions, for instance K?hlerian or Sasakian frame. This paper deals the first time literature with Frenet curves dimensional paracontact context. Several classifications are provided here different types on para-Sasakian m...

In this paper smooth trajectories are computed in the Lie group SO(2, 1) as a motion planning problem by assigning a Frenet frame to the rigid body system to optimize the cost function of the elastic energy which is spent to track a timelike curve in Minkowski space. A method is proposed to solve a motion planning problem that minimizes the integral of the Lorentz inner product of Darboux vecto...

This paper proposes a new method for designing a developable surface by constructing a surface pencil passing through a given curve, which is quite in accord with the practice in industry design and manufacture. By utilizing the Frenet trihedron frame, we derive the necessary and sufficient conditions to construct a developable surface through a given curve. Considering the requirements in shoe...

Traffic simulation heavily relies on lane model. This paper presents a novel method to model lanes based on the road axis under the Frenet frame. The road axis is generated from the geographic information system data after curve approximation, discretization, and compression. This lane model couples mileage information with three-dimensional geometric information, so it offers an easy and fast ...

In this paper, position vectors of a time-like curve with respect to standard frame of Minkowski space E31 are studied in terms of Frenet equations. First, we prove that position vector of every time-like space curve in Minkowski space E31 satisfies a vector differential equation of fourth order. The general solution of mentioned vector differential equation has not yet been found. By special c...

We consider the motion of a particle described by an action that is a functional of the Frenet-Serret [FS] curvatures associated with the embedding of its worldline in Minkowski space. We develop a theory of deformations tailored to the FS frame. Both the Euler-Lagrange equations and the physical invariants of the motion associated with the Poincaré symmetry of Minkowski space, the mass and the...

Looking at the Frenet-Serret equations from the viewpoint of dynamical systems, one can prove that when the curvature of a plane curve is given as a function of the radius, the problem of reconstructing this curve is reducible to quadratures. Additionally, two different integration procedures are presented. These methods are illustrated first via the famous lemniscate of Bernoulli, which is imm...