Rectifying curves in the $n$-dimensional Euclidean space

TL;DR

This paper characterizes rectifying curves in n-dimensional Euclidean space, providing multiple conditions and constructions, thus extending the understanding of these curves beyond three dimensions.

Contribution

It introduces new characterizations and construction methods for rectifying curves in arbitrary dimensions, generalizing previous 3D results.

Findings

Characterization via curvature conditions

Expressions for position vector components

Construction from hypersphere curves

Abstract

In this article, we study rectifying curves in arbitrary dimensional Euclidean space. A curve is said to be a rectifying curve if, in all points of the curve, the orthogonal complement of its normal vector contains a fixed point. We characterize rectifying curves in the $n$-dimensional Euclidean space in different ways: using conditions on their curvatures, using an expression for the tangential component, the normal component or the binormal components of their position vector, and by constructing them starting from an arclength parameterized curve on the unit hypersphere.