Generalized involute and evolute curves of equiform spacelike curves with a timelike equiform principal normal in E13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$E_{1}^{3}$\end{document}

Equiform geometry is considered as a generalization of the other geometries. In this paper, involute and evolute curves are studied in the case of the curve α is an equiform spacelike with a timelike equiform principal normal vector N. Furthermore, the equiform frames of the involute and evolute curves are obtained. Also, the equiform curvatures of the involute and evolute curves are obtained in Minkowski 3-space.

In this paper, we firstly introduce the equiform parameter, the equiform frame, and the equiform formulas in the case of equiform spacelike curves with a timelike equiform principal normal in Minkowski space E 3 1 . Secondly, we introduce the involute and the evolute of the equiform spacelike curve with a timelike equiform principal normal. Further, the equiform frames for the involute and the evolute curves are obtained. Also, the equiform curvatures of the involute and the evolute curves are obtained.
A differentiable map α : I ⊂ R → E 3 1 is called smooth curve in Minkowski 3-space, where I is an open interval. Suppose that {t(s), n(s), b(s)} be the orthonormal Frenet frame along the curve α(s), where t(s), n(s), and b(s) are the tangent, the principal normal, and the binormal vectors of the curve α, respectively.
Any curve α in Minkowski 3-space can be one of the following cases and below the corresponding Frenet formulas: (1) α is a spacelike curve with (i) a spacelike principal normal, then Frenet formulas are given by: (ii) a timelike principal normal, then Frenet formulas are given by: (iii) a null (lightlike) principal normal, then Frenet formulas are given by: (2) α is a timelike curve, then Frenet formulas are given by: g(t, n) = g(t, b) = 0. [13] (3) α is a lightlike curve, then Frenet formulas are given by: The equiform geometry has minor importance related to usual one, and the curves that appear here in equiform geometry can be seen as a generalization of well-known curves from other geometries.
Let γ (s) = t(s) be the spherical tangent indicatrix of the curve α and σ be an arc length parameter of γ . We can make a reparameterization of α by the parameter σ , α = α(σ ) : I → E 3 1 , the parameter σ is called the equiform parameter, of the curve α(σ ). Let σ be the arc length parameter of spherical tangent indicatrix ζ , then we have: By integration with respect to s, we have: where ρ is the radius of curvature of α [11]. Let T, N, and B be the orthogonal equiform frame along the curve α(σ ) in Minkowski 3-space, where T, N, and B are the equiform-tangent, the equiform-normal, and the equiform-binomial vectors of the curve α(σ ), respectively. They are given by T = dα dσ = ρt, N = ρn, b = ρb [10,11].
The function K 1 : I → R defined by K 1 = dρ ds is called the first equiform curvature of α(σ ), and the function K 2 : I → R defined by K 2 = τ κ is the called second equiform curvature of α(σ ).
If α(σ ) is an equiform spacelike with a timelike equiform principal normal vector, then the equiform formulas are given in [10] by: Lemma 1 Suppose that a curve α is an equiform spacelike with a timelike equiform principal normal N. If α(σ ) is parameterized by the equiform parameter σ , then: Lemma 2 If a curve α(σ * ) is an equiform spacelike with a timelike equiform principal normal N and σ * is not necessary the equiform parameter of the curve α, then: Suppose that a curve α is an equiform spacelike with a timelike equiform principal normal N. Then, the equiform curvatures are given by:

Definition 2 A curve α(σ )
is an ordinary helix if the second equiform curvature K 2 = 0, and it is a general helix if K 2 is constant.

The involute of an equiform spacelike curve with a timelike equiform principal normal
In this section, we study the involute curve of the equiform spacelike curve with a timelike equiform principal normal vector N in E 3 1 . Also, the equiform frame of the involute curve is introduced. Furthermore, the equiform curvatures of the involute curve are obtained.
where c is constant.

Theorem 2
Let α(σ ) be an equiform spacelike curve with a timelike equiform principal normal and suppose that a curve β is an involute of the curve α, then: Proof By taking the derivative of Eq. (3) with respect to σ , we have: Then, Let us assume that: By taking the derivative of Eq. (4) with respect to σ , we have: Hence, we have: Since N * = − B * ∧T * ρ * , then we obtain:

Theorem 3 Let β(σ ) be an involute of the curve α(σ )
, and K * 1 , K * 2 be the first and second equiform curvatures of the curve β, respectively. Then, K * 1 and K * 2 are given respectively by: .
Proof The proofs come forward from the equation of K * 2 .

The evolute of equiform spacelike curve with a timelike equiform principal normal
In this section, the evolute curves of the equiform spacelike curve with a timelike equiform principal normal N are studied in E 3 1 . Moreover, the equiform frame of the evolute curve is introduced. Furthermore, the equiform curvatures of the evolute are computed.
Proof By similar proof of Theorem 2, we obtain the required.

Corollary 4
If the curve α is a equiform spacelike curve with a timelike equiform principal normal, then its evolutes are equiform timelike curves.
Proof The proof comes forward from Theorem 5.