A characterization of involutes and evolutes of a given curve in $\mathbb{E}^{n}$

TL;DR

This paper characterizes involutes and evolutes of curves in n-dimensional Euclidean space, providing new insights into their geometric properties and specific results in three and four dimensions.

Contribution

It offers a novel characterization of involute and evolute curves of any order in higher-dimensional Euclidean spaces.

Findings

Characterization of involute curves of order k in $\, ext{E}^n$

Results on involutes and evolutes in $\, ext{E}^3$ and $\, ext{E}^4$

Insights into osculating hyperspheres and generalized evolutes

Abstract

The orthogonal trajectories of the first tangents of the curve are called the involutes of $x$. The hyperspheres which have higher order contact with a curve $x$ are known osculating hyperspheres of $x$. The centers of osculating hyperspheres form a curve which is called generalized evolute of the given curve $x$ in $n$-dimensional Euclidean space $\mathbb{E}^{n}$. In the present study, we give a characterization of involute curves of order $k$ (resp. evolute curves) of the given curve $x$ in $n$-dimensional Euclidean space $\mathbb{E}^{n}$. Further, we obtain some results on these type of curves in $\mathbb{E}^{3}$ and $\mathbb{E}^{4}$, respectively.