The notions of evolutes and involutes were studied by C. Huygens in his work [7] and studied in differential geometry and singularity theory of planar curves [1]. The evolute of a regular curve in the Euclidean plane is given by not only the locus of all its centres of the curvature, but also the envelope of normal lines of the regular curve, namely, the locus of singular loci of parallel curves. On the other hand, the involute of a regular curve is to replace the taut string by a line segment that is tangent to the curve on one end, while the other end traces out the involute. In ([3], [4]) T. Fukunaga and M. Takahashi defined the evolutes and the involutes of fronts in the plane without inflection points and gave properties of them. Meanwhile, E. Özyılmaz and S. Yılmaz studied the involute-evolute of W-curves in Euclidean 4-space [13], see also, [16]. Recently, B. Divjak and Ž. M. Šipuš, considered the isotropic involutes (of order k) and the isotropic evolutes in n-dimensional isotropic space In(1) [2, 10].

This paper is organized as follows: Section 2 gives some basic concepts of Frenet curves in Euclidean spaces. Section 3 gives some basic concepts of the involute curves of order k in . Section 4 explains some geometric properties about the involute curves of order k in , where k = 1, 2. Section 5 tells about the involute curves of order k in , where k = 1, 2, 3. Further these sections provides some properties and results of these type of curves. In the final section we consider generalized evolute curves in . Moreover, we present some results of generalized evolute curves in and , respectively.

Let be a regular curve in , (i.e., ||x′(t)|| ≠ 0). Then x is called a Frenet curve of osculating order d, (2 ≤ d ≤ n) if x′(t), x″(t), …, x^(d)(t) are linearly independent and x′(t), x″(t),…, x^(d+1)(t) linearly dependent for all t in I. For the case d = n, the Frenet curve x is called generic curve in [17]. From now on we assume that x is a generic curve in . To each generic curve x one can associates an orthonormal d-frame V1=x′(t)‖x′(t)‖, V₂, V₃ …, V_n along x, the Frenet n-frame, and n − 1 functions κ₁, κ₂, …, κ_n−1:I –→ ℝ, the Frenet curvature, such that

where, v = ||x′(t)|| is the speed of the curve x. In fact, to obtain V₁, V₂, V₃ …, V_n, it is sufficient to apply the Gram-Schmidt orthonormalization process to x′(t), x″(t), …, x⁽ⁿ⁾(t). Moreover, the functions κ₁, κ₂, …, κ_n−1 are easily obtained as by-product during this calculation.

More precisely, V₁, V₂, V₃ …, V_n and κ₁, κ₂, …, κ_n−1 are determined by the following formulas:

and

respectively, where δ ∈ {1, 2, 3, …, n − 1} (see, [5]).

The osculating hyperplane of a generic curve x at t is the subspace generated by {V₁, V₂, V₃ …, V_n} that passes through x(t). The unit vector V_n(t) is called binormal vector of x at t. The normal hyperplane of x at t is defined to be the one generated by {V₂, V₃ …, V_n} passing through x(t) [14].

A Frenet curve of rank d for which the first Frenet curvature κ₁ is constant is called a Salkowski curve [15] (or T.C-curve [8]). Further, a Frenet curve for which κ₁, κ₂, …, κ_n−1 are constant is called (circular ) helix or W-curve [9]. Meanwhile, a Frenet curve with constant curvature ratios κ2κ1,κ3κ2,κ4κ3,…,κn-1κn-2 is called a ccr-curve (see, [12], [11]). A ccr-curve in is known as generalized helix.

Given a generic curve x in , the Frenet 4-frame, V₁, V₂, V₃, V₄ and the Frenet curvatures κ₁, κ₂, κ₃ are given by

and

respectively, where ∧ is the exterior product in [5].

Definition 3.1

Let x = x(s) be a regular generic curve in given with the arclength parameter s (i.e., ||x′(s)|| = 1). Then the curves which are orthogonal to the system of k-dimensional osculating hyperplanes of x, are called the involutes of order k [2] (or, k^th involute [6]) of the curve x. For simplicity, we call the involutes of order 1, simply the involutes of the given curve.

In order to find the parametrization of involutes x of order k of the curve x, we put

where λ_α is a differentiable function and s is the parameter of χ̄ which is not necessarily an arclength parameter. The differentiation of the equation (3.1) and the Frenet formulae (2.1) are given in the following equation

Furthermore, the involutes χ̄ of order k of the curve x are determined by

This condition is satisfied if and only if

where 2 ≤ α ≤ n − 1 [2].

In the sequel we characterize the involutes of generic curves in and .

In the present section we consider involutes of order 1 and of order 2 of curves in Euclidean 3-space , respectively.

4.1. Involutes of Order 1 in