Position vectors of a spacelike general helices in Minkowski Space $\e_1^3$

TL;DR

This paper investigates the position vectors of spacelike general helices in Minkowski space using Frenet equations, deriving differential equations and parametric forms, with examples for different normal vectors.

Contribution

It introduces a third-order differential equation approach to find position vectors of spacelike general helices in Minkowski space, providing explicit parametric representations.

Findings

Derived a third-order differential equation for position vectors.

Obtained parametric forms of general helices from intrinsic equations.

Provided examples for helices with spacelike and timelike principal normals.

Abstract

In this paper, position vector of a spacelike general helix with respect to standard frame in Minkowski space E$^3_1$ are studied in terms of Frenet equations. First, a vector differential equation of third order is constructed to determine position vector of an arbitrary spacelike general helix. In terms of solution, we determine the parametric representation of the general helices from the intrinsic equations. Moreover, we give some examples to illustrate how to find the position vectors of a spacelike general helices with a spacelike and timelike principal normal vector from the intrinsic equations.