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The orthogonal trajectories of the first tangents of the curve are called theinvolutes of $x$. The hyperspheres which have higher order contact with a curve$x$ are known osculating hyperspheres of $x$. The centers of osculatinghyperspheres form a curve which is called generalized evolute of the givencurve $x$ in $n$-dimensional Euclidean space $\mathbb{E}^{n}$. In the presentstudy, 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.

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