Framed slant helices in Euclidean 3-space

In this paper, we define framed slant helices and give a necessary and sufficient condition for them in three-dimensional Euclidean space. Then, we introduce the spherical images of a framed curve. Also, we examine the relations between a framed slant helix and its spherical images. Moreover, we give an example of a framed slant helix and its spherical images with figures.


Introduction
Let γ be a regular curve with the Frenet apparatus {T, N, B, κ, τ } in three-dimensional Euclidean space R 3 . We know that the curve γ is a general helix if the tangent vector of γ makes a constant angle with a fixed straight line. In 1802, a classical characterization of general helices was given by M.A. Lancert and was proved first by B. de Saint Venant in 1845: "The curve γ is a general helix if and only if its curvatures ratio is constant. " (see [1,2]). On the other hand, the curve γ is a slant helix if its normal vector makes a constant angle with a fixed straight line. It is well known that slant helices have the following characterization: "The curve γ is a slant helix in R 3 if and only if the geodesic curvature of the principal normal indicatrix (N) of the curve is a constant function. " That is σ N (s) = κ 2 (κ 2 + τ 2 ) 3 2 τ κ is a constant function (see [3]). To date, general helices and slant helices have been introduced in different spaces and characterizations obtained of these curves by many researchers (see [4][5][6][7][8][9]). Recently, Shun'ichi Honda and Masatomo Takahashi investigated framed curves in Euclidean space (see [10]). Moreover, Yongqiao Wang et al. defined framed helices and gave a characterization for a framed curve to be a framed helix (see [11]). Tuncer et al. introduced pedal and contrapedal curves of fronts by using the Legendrian Frenet frame in the Euclidean plane and also examined singularities of pedal and contrapedal curves of fronts in [12]. Yazıcı et al. investigated framed rectifying curves via the dilation of framed curves on S 2 in R 3 , [13]. Yıldız studied the evolution of framed curves in R 3 , [14].
In this paper, we define framed slant helices and give a characterization to be a framed slant helix of a framed curve in three-dimensional Euclidean space. We then define framed spherical images of a framed curve. Also, we obtain some relations between a framed slant helix and its spherical images. Finally, we present an illustrated example to support the theory.

Basic materials
In this section, we outline the definitions of framed curves and framed helices in threedimensional Euclidean space (see for details [10,11]).
Let γ : I ⊂ R → R 3 be a curve with singular points. The set 2 , which is defined by is a three-dimensional smooth manifold. For a given μ = (μ 1 , μ 2 ) ∈ 2 , we can define a unit vector of R 3 , as follows: This means that v is orthogonal to μ 1 and μ 2 .

Definition 2.1
We say that (γ , μ) : I → R 3 × 2 is a framed curve if γ (s), μ i (s) = 0 for all s ∈ I and i = 1, 2. We also say that γ : I → R 3 is a framed base curve if there exists μ : I → 2 such that (γ , μ) is a framed curve.

Definition 2.3
Let (γ , μ 1 , μ 2 ) : I → R 3 × 2 be a framed curve with p(s) > 0. The γ is called a framed helix if its generalized tangent vector v makes a constant angle with a fixed unit vector ζ . That is where φ is a constant angle (see [11]).

Framed slant helices in R 3
In this section we define a framed slant helix and its axis in three-dimensional Euclidean space R 3 . Also, we give a characterization of a framed slant helix.
Definition 3.1 Let (γ , μ 1 , μ 2 ) : I → R 3 × 2 be a framed curve. Then, γ is called a framed slant helix if its generalized principal normal vector μ 1 makes a constant angle with a fixed unit vector ζ . That is where ϕ = π 2 is a constant angle between ζ and μ 1 (s).
Then, the framed harmonic curvature of the framed curve (γ , μ) is defined by If the curve γ is a framed slant helix, then the axis of γ is where H is the framed harmonic curvature function of the curve γ and ϕ = π 2 is a constant angle.
Also, we know from Definition 3.1 that By differentiating equation (2), we obtain and using the Frenet formulas for the adapted frame of the curve γ given in equation (1), Again, differentiating equation (3) and using equation (1), we obtain Then, if we substitute equation (4) into equation (3), we obtain Consequently, using equations (2), (4) and (5) the axis of the framed slant helix γ is given by which completes the proof. is a constant function, where H is the framed harmonic curvature function of the curve γ .
Proof If the axis of the framed slant helix γ is ζ , we have from Proposition 3.1: As ζ is a unit vector we can readily see that where ϕ is the constant angle between ζ and μ 1 . Conversely, if σ is a constant function then the result is obvious. This completes the proof.

Spherical images of framed slant helices in R 3
In this section, first we define the spherical indicatrices of a framed curve and we investigate the relations between framed slant helices and their spherical indicatrices. The adapted frame apparatus of β is given by the notation {v β , μ 1 β , μ 2 β }. Clearly, there exists a smooth mapping α β : I → R such that:

v-Indicatrices of framed slant helices
then, differentiating the last equation according to parameter s and using equation (1), we obtain From the norm of the last equation, we obtain α β (s) = p(s).
From the norm of the above equation, we obtain p β (s) = p(s) 1 + H 2 (s).
Hence, we obtain the following equation Then, using equations (6) and (8), we obtain From the norm of the derivative of the last equation, we obtain the following equation: Then, we can readily see that p(s)(1+H 2 (s)) 3 2 = σ is a constant function since γ is a framed slant helix. In other words, using Theorem 2.3 we can readily see that β is a framed helix.
Conversely, if we assume that β is a framed helix then it is clear that γ is a framed slant helix. This completes the proof.
where H is the framed harmonic curvature function of the curve γ .
where H is the framed harmonic curvature function of the curve γ .
Proof It is obvious from equations (7) and (10). The adapted frame apparatus of η is given by the notation {v η , μ 1 η , μ 2 η }. Clearly, there exists a smooth mapping α η : I → R such that: Proof Let γ be a framed slant helix in R 3 and η a framed μ 1 -indicatrix of γ . From Definition 4.2, we have

μ 1 -Indicatrices of framed slant helices
Then, differentiating equation (11) according to the parameter s and using equation (1), we obtain From the norm of the last equality, we obtain Hence, we obtain the following equation: v η (s) = -1 If we differentiate the last equation and use equation (1), we obtain Since γ is a framed slant helix σ (s) is a constant function. Hence, we can obtain Then, the norm of the last equation gives us p η (s) = p(s) 1 + H 2 (s) 1 + σ 2 and so Then, using equations (12) and (14), we obtain From the norm of the derivative of μ 2 (s η ), we obtain q η (s) = 0.
Hence, γ is a plane curve. This completes the proof.  1 + H 2 (s) μ 2 (s), where H is the framed harmonic curvature function of the curve γ .

μ 2 -Indicatrices of framed slant helices
Hence, we have Then, using equations (16) and (18), we have From the norm of the derivative of the last equation, we obtain the following equation: Then, we can readily see that q δ (s) p δ (s) = H (s) p(s)(1+H 2 (s)) 3 2 = σ is a constant function since γ is a framed slant helix. In other words, using Theorem 2.3 we can see readily that δ is a framed helix.
Proof It is obvious from equations (17) and (20). The curve γ has a singular point at t = 0, so that it is not a Frenet curve. On the other hand, the curve γ is a framed curve with the mapping (γ , μ 1 , μ 2 ) : (-2π, 2π) ⊂ R → R 3 × 2 .
The adapted frame vectors of the framed curve (γ , μ 1 , μ 2 ) are given by