Position Vectors of Curves in Isotropic Space $I^3$ and Their Relation to the Frenet Frame

Abstract

This paper investigates position vectors of arbitrary curves in isotropic 3-space (denoted by I^3). We first establish the relationship between a curve’s position vector and the Frenet frame. Then, we derive a natural representation of any curve’s position vector using curvature and torsion. Furthermore, we define various curves within isotropic space, including straight lines, plane curves, helices, general helices, Salkowski curves, and anti-Salkowski curves. Finally, graphical illustrations accompany illustrative examples to elucidate the discussed concepts.