Some characterizations of rectifying curves on a smooth surface in Euclidean 3-space

TL;DR

This paper studies the properties of rectifying curves on smooth surfaces in Euclidean 3-space, focusing on their invariance under isometries and how their position vectors deviate along various directions.

Contribution

It provides new conditions for the invariance of rectifying curves under isometries and analyzes how their position vectors deviate along tangent and normal directions.

Findings

Conditions for invariance of rectifying curves under isometry.

Deviations of position vectors along tangent vectors.

Deviations along surface normal and binormal directions.

Abstract

In this paper, we investigate sufficient condition for the invariance of a rectifying curve on a smooth surface immersed in Euclidean 3-space under isometry by using Darboux frame $\left\lbrace T, P, U\right\rbrace$. Further, we find the deviations of the position vector of a rectifying curve on the smooth surface along any tangent vector $T = a\phi_u + b\phi_v$ with respect to the isometry. We also find the deviations of the position vector of a rectifying curve on the smooth surface along the unit normal $U$ to the surface and along $P (= U \times T)$ with respect to the isometry.