The main goal of this study is to bring together the spinors, which have a major place in several disciplines from mathematics physics, and Positional Adapted Frame (PAF) new type frame that attracts attention many researchers. In accordance with purpose, we introduce spinor representations for trajectories endowed PAF Euclidean 3-space $\mathbb{E}^3$, construct equations vectors. Then, find re...

A rotation matrix can be generated by a unit timelike quaternion in the three dimensional Minkowski space. In this paper, we correspond columns of this rotation matrix to Frenet vectors of a nonlightlike curve and we find a reformulation of Frenet’s formulas of nonlightlike curves in terms of the timelike quaternions in the Minkowski 3-space. Moreover, we do same things for the parallel transpo...

We consider a unit speed curve α in the Euclidean ndimensional space E and denote the Frenet frame of α by {V1, . . . ,Vn}. We say that α is a cylindrical helix if its tangent vector V1 makes a constant angle with a fixed direction U . In this work we give different characterizations of such curves in terms of their curvatures. AMS Mathematics Subject Classification (2000): 53C40, 53C50

In this article, a new type of ruled surfaces in a Lorentz 3-space R31 is obtained by a strictly connected time-like oriented line moving with Frenet s frame along a space-like curve. These surfaces are classified into time-like and space-like surfaces. The wellknown theorems due to Bonnet and Chasles in the 3-dimensional Euclidean space are proved for a time-like ruled surface. 2005 Published ...

The role played by torsion of magnetic vortex line curves or filaments, in the equilibrium state of magnetars is investigated. When the magnetars equilibrium equations are written in terms Frenet-Serret frame it is shown that in regions of the magnetic star where the Frenet torsion is constant it induces an oscillation in the vortex filaments. By solving the magnetar equilibrium equation we sho...

In this work, we describe a Frenet frame in 4-dimensional Euclidean space and call as parallel transport (PTF). PTF is an alternative approach to defining moving frame. This obtained by rotating the tangent vector first binormal of unit speed curve euler angle then give curvature functions according curve. Also, introduce $(k,m)$-type slant helices 4-Space. Additionally, obtain characterization...

General offset curves are treated in the context of Minkowski geometry, the geometry of the two-dimensional plane, stemming from the consideration of a strictly convex, centrally symmetric given curve as its unit circle. Minkowski geometry permits us to move beyond classical confines and provides us with a framework in which to generalize the notion of Pythagorean-hodograph curves in the case o...

In recent years, nonlinear model predictive control has been extensively used for solving automotive motion and planning tasks. order to formulate the problem, different coordinate systems can be with advantages. We propose compare formulations MPC related optimization involving a Cartesian Frenet frame in single program. specify costs collision avoidance constraints more advantageous frame, de...

In this study, the effect of fractional derivatives on curves, whose application area is increasing day by day, investigated. While investigating effect, conformable derivative, which best suits algebraic structure differential geometry, selected. As a result, many special curves and Frenet frame previously obtained using classical have been redefined with help derivatives.

In this paper, we present a physics-based deformable model framework for the quantification of shape and motion parameters of the Left Anterior Descending (LAD) coronary artery in the heart’s local frame of reference. We define the long-axis of the heart as the local frame of reference. The shape of the LAD is modeled as a parametric curved axis with Frenet-Serret frame. The motion of the LAD (...