Smarandache Curves of Involute-Evolute Curve According to Frenet Frame

: In this paper, the invariants of the Smarandache curves, which consist of Frenet vectors of the involute curve, are calculated in terms of the evolute curve


Introduction and Preliminaries
A regular curve in Minkowski space-time, whose position vector is composed of Frenet frame vectors on another regular curve, is called a Smarandache curve [17].Special Smarandache curves have been studied by some authors.Turgut and Yılmaz's article deals with interesting knowledge of special Smarandache curves in the space E 4  1 .For example, they obtained another orthonormal frame [17].In the light of the reference [17], Ali adapted Smarandache curve to regular curves in the E 3 [2].Ergüt et al. defined the isotropic types of Smarandache curves.Then they examined these kinds of isotropic Smarandache curves according to the Cartan frame in the complex 4-space [6].By using the Darboux frame, Bektaş and Yüce obtained the results about Smarandache curves [4].In another study, they studied the spatial quaternionic curve and the relationship between Frenet frames of the involute curve of the spatial quaternionic curve which are expressed by using the angle between the Darboux vector and binormal vector [15].S ¸enyurt et al. used special curves as a base to create Smarandache curves, and then studied their geometric properties [12][13][14].Al-Dayal and Solouma study some properties of spacelike Smarandache curves regarding Bishop frame of a spacelike curve in Minkowski 3-space [1].There are many studies about Smarandache curves [9,11,16] Huygens discovered an involute-evolute curve while trying to build a more accurate clock.
The involute of a curve is a well-known concept in the Euclidean space [7,8,10].The involute-evolute curve has attracted mathematicians' attention.In [5], authors found the relationships between the Frenet frames of the timelike curve and the spacelike involute curve.In another study, Bishop curvatures of the involute-evolute curve were examined and some important results were found [3].
In this paper, the invariants of the Smarandache curves, which consist of Frenet vectors of the involute curve, are calculated in terms of the evolute curve.
The inner product can be given by where (x 1 , x 2 , x 3 ) ∈ E 3 .Let α ∶ I → E 3 be a unit speed curve with the moving Frenet frame For any unit speed curve α ∶ I → E 3 , the vector W is called Darboux vector defined by If we consider the normalization of the Darboux vector C = 1 ∥W ∥ W , we have where ⟨W, V 3 ⟩ = ∅ .
Theorem 1.1 [10] Let the Frenet frames of α and α * be {V 1 (s), V 2 (s), V 3 (s)} and Theorem 1.3 [7] The distance between corresponding points of the evolute-involute curve in where c is a constant.

Smarandache Curves of Involute-Evolute Curve Couple According to Frenet Frame
In this subsection, special Smarandache curves belonging to involute curves such as V 1 be defined by If equation (1) is taken into account, the above expression is Theorem 2.2 Frenet vectors of Smarandache curve β 1 are given as follows; Here, the coefficients are Proof The derivative of the equation ( 2) is By taking the norm of the above equation, we can write If necessary operations are taken, the tangent vector is In the light of the pieces of information, the principal normal and the binormal vectors are respectively given by 3 Curvature and torsion belonging to Smarandache curve β 1 are, respectively where Proof The first curvature is Taking the derivative of the equation ( 4), we obtain If the expression ( 6) is written in (5), the first curvature is

⋅
If the coefficients are written instead, the desired result is obtained.
To calculate the torsion of the curve β 1 , we differentiate and thus where The torsion is then given by If equation ( 1) is taken into account, the above expression is Theorem 2. 5 The Frenet invariants of the β 2 curve are given as follows; where Proof The theorem is similar to Theorem 2.2 and Theorem 2.3, therefore we omit its proof.◻ be defined by If equation ( 1) is taken into account, the above expression is Theorem 2. 7 The Frenet invariants of the β 3 curve are given as follows; where where

Conclusion
We examined the Smarandache curves formed by the Frenet vectors of the involute curve.Then curvatures and torsions of Smarandache curves are calculated.These invariants (Frenet vectors and curvatures) which depend on the evolute curve are explained.Besides, we illustrate the Smarandache curves formed by taking the helix curve.

Declaration of Ethical Standards
The authors declare that the materials and methods used in their study do not require ethical committee and/or legal special permission.

Authors Contributions
Author [Selin Sivas]: Thought and designed the research/problem, contributed to research method or evaluation of data (%55).

Figure 1 :Example 3 . 2
Figure 1: The black curve is the involute curve of the curve α (c=1).The blue, red, brown and purple curves are Smarandache curves, which consist of Frenet vectors of the involute curve, respectively [7] moving Frenet frame.For an arbitrary curve α ∈ E 3 , with first and second curvature k 1 and k 2 , respectively, the Frenet formulas are given by[7]