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

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.