A Study on $f$-Rectifying Curves in Euclidean $n$-Space

Abstract

A rectifying curve in the Euclidean $n$-space $\mathbb{E}^n$ is defined as an arc-length parametrized curve $\gamma$ in $\mathbb{E}^n$ such that its position vector always lies in its rectifying space (i.e., the orthogonal complement of its principal normal vector field) in $\mathbb{E}^n$. In this paper, in analogy to this, we introduce the notion of an $f$-rectifying curve in $\mathbb{E}^n$ as a curve $\gamma$ in $\mathbb{E}^n$ parametrized by its arc-length $s$ such that its $f$-position vector field $\gamma_f$, defined by $\gamma_f(s) = \int f(s) d\gamma$, always lies in its rectifying space in $\mathbb{E}^n$, where $f$ is a nowhere vanishing real-valued integrable function in parameter $s$. The main purpose is to characterize and classify such curves in $\mathbb{E}^n$.

Keywords

• euclidean space
• frenet-serret formulae
• higher curvatures
• rectifying curve
• $f$-position vector field
• $f$-rectifying curve