Geometry of Admissible Curves of Constant-Ratio in Pseudo-Galilean Space

. An admissible curve of a pseudo-Galilean space is said to be of constant-ratio if the ratio of the length of the tangent and normal components of its position vector function is a constant. In this paper, we investigate and characterize a spacelike admissible curve of constant-ratio in terms of its curvature functions in the pseudo-Galilean space G 13 . Also, we study some special curves of constant-ratio such as T -constant and N -constant types of these curves. Finally, we give some computational examples for constructing the meant curves to demonstrate our theoretical results


Introduction
According to the space curve theory, it is well known that, a curve α(s) in E 3 lies on a sphere if its position vector lies on its normal plane at each point.If the position vector α lies on its rectifying plane then α(s) is called a rectifying curve [1].Rectifying curves are characterized by the simple equation: where λ(s) and µ(s) are smooth functions and T (s) and B(s) are tangent and binormal vector fields of α, respectively.In [2] the author provided that a twisted curve is congruent to a non constant linear function of s.On the other hand, in the Minkowski 3-space E 3 1 , the rectifying curves were investigated in [3,4].Besides, in [4] a characterization of the spacelike, the timelike and the null rectifying curves in the Minkowski 3-space in terms of centrodes were given.The characterization of rectifying curves in three dimensional compact Lee groups as well as in dual spaces were given in [5], [6], respectively.
For the study of constant-ratio curves, the authors gave the necessary and sufficient conditions for curves in Euclidean and Minkowski spaces to become T -constant or N-constant [7][8][9][10].In analogy with the Euclidean 3-dimensional case, our main goal in this work is to define the spacelike admissible curves of constant-ratio in the pseudo Galilean 3-space as a curve whose position vector always lies in the orthogonal complement N ⊥ of its principal normal vector field N. Consequently, N ⊥ is given by where <•,•> denotes the inner product in G 1 3 .Hence N ⊥ is a 2-dimensional plane of G 1 3 , spanned by the tangent and binormal vector fields T and B, respectively.Therefore, the position vector with respect to some chosen origin of a considered curve α in G 1  3 , satisfies the parametric equation: for some differential functions m i (s), 0 ≤ i ≤ 2, where s is arc-length parameter.Then, we give the necessary and sufficient conditions for the curve α in G 1 3 to be a constant-ratio curve.

Pseudo-Galilean geometry
In this section, we introduce the basic concepts, familiar definitions and notations on pseudo-Galilean space which are needed throughout this study.The pseudo-Galilean geometry is one of the real Cayley-Klein geometries of projective signature (0,0,+,-).The absolute of the pseudo-Galilean geometry is an ordered triple {w , f , I} where w is the ideal (absolute) plane, f is a line in w and I is the fixed hyperbolic involution of points of f , for more details, we refer to [11,12].The geometry of the pseudo-Galilean space is similar (but not the same) to the Galilean space which was presented in [11].The inner and cross product of two vectors x = (x 1 , y 1 , z 1 ) and y = (x 2 , y 2 , z 2 ) in G 1 3 are, respectively defined as follows: Also the norm of a vector x = (x, y , z) is given by The group of motions of the pseudo-Galilean G 1 3 is a six-parameter group given (in affine coordinates) by According to the motion group in pseudo-Galilean space, a vector x(x, y , z) is said to be non isotropic if x = 0.All unit non-isotropic vectors are of the form (1, y , z).For isotropic vectors, x = 0 holds.There are four types of isotropic vectors: spacelike (y 2 − z 2 ) > 0, timelike (y 2 − z 2 ) < 0, and two types of lightlike (y = ±z) vectors.A non-lightlike isotropic vector is a unit vector if y 2 −z 2 = ±1.
Also, If α is taken as follows: with the condition then the arc-length parameter s is defined by Here, we assume that ds = dx and s = x as the arc-length of the curve α [12].The vector is called the tangent unit vector of α.Also, the unit vector field is given by and the binormal vector is expressed as and it is orthogonal in pseudo-Galilean sense to the osculating plane of α spanned by the vectors α (s) and α (s).The curve α given by Eq. (2.2) is a spacelike (resp.timelike) if N(s) is a timelike (resp.spacelike) vector.The principal normal vector or simply normal is spacelike if ε = +1 and timelike if In each point of an admissible curve in G 1 3 , the associated orthonormal (in pseudo-Galilean sense) trihedron {T (s), N(s), B(s)} can be defined.This trihedron is called pseudo-Galilean Frenet trihedron.
For the pseudo-Galilean Frenet trihedron of an admissible curve α, the Frenet equations are defined as: where κ and τ are the pseudo-Galilean curvatures of α defined as follows: and the pseudo-Galilean torsion can be written in the form The Serret-Frenet equations (2.8) can be written in matrix form as The Pseudo-Galilean sphere with radius r is defined by 3 be an arbitrary spacelike admissible curve.In the light of which introduced in [13][14][15], we consider the following theorem.
Theorem 3.1.The position vector of α with curvatures κ(s) and τ (s) = 0, and with respect to the Frenet frame in the pseudo-Galilean space G 1  3 , it can be written as where c o , c 1 and c 2 are arbitrary constants.
Proof.Let α be an arbitrary spacelike curve in the pseudo-Galilean space G 1 3 , then we may express its position vector as Differentiating this equation with respect to the arc-length parameter s and using the Serret-Frenet equations (2.8), we obtain From Eqs. (3.2), we have It is useful to change the variable s to the variable t = τ (s)ds.Therefore all functions of s will transform to the functions of t.Here, we will use dot to denote the derivative with respect to t (where the prime denotes the derivative with respect to s).Also, From Eq. (3.2), we get it leads to The general solution of this equation is given by Differentiating Eq. (3.10) with respect to s and substituting in Eq. (3.10), we find Similarly, taking the differentiation of Eq. (3.13) and equalize with Eq. (2.8), we obtain Hence, it completes the proof.
Theorem 3.3.The position vector α(s) of a spacelike admissible curve with curvature κ(s) and torsion τ (s) in the pseudo-Galilean space G 1 3 is computed from the intrinsic representation form with tangent, principal normal and binormal vectors respectively, are given by Now, for each given α : , there is a natural orthogonal decomposition of the position vector α at each point on α; namely, where α T and α N denote the tangential and normal components of α at the point, respectively.Let α T and α N denote the length of α T and α N , respectively.In what follows we introduce the notion of constant-ratio curves.So, similar to the Euclidean case [16], we consider the following definitions [17].
Definition 3.1.A curve α of the pseudo-Galilean space G 1 3 is said to be of constant-r ati o curve if the ratio α T : α N is constant on α(I).
Clearly, for a constant-ratio curve in G 1  3 , we have for some constant c 3 .
3 be an admissible curve in G 1 3 .If α N is constant, then α is called a N-constant curve.For a N-constant curve α, either α N = 0 or α N = µ for some nonzero smooth function µ.Further, a N-constant curve α is called of first kind if α N = 0, otherwise it is of second kind.
For N-constant curve α in G 1  3 , we can write where c 4 is constant.
In what follows, we characterize the admissible curves in terms of their curvature functions m i (s) and give the necessary and sufficient conditions for these curves to be T -constant or N-constant curves.
Theorem 3.4.Let α : Proof.Let α : I ⊂ R → G 1 3 be a spacelike curve given with the invariant parameter s.Then, we have where c o is an arbitrary constant.Also, from Eq. (3.16), the curvature functions m i (s), 0 By using Eqs.(3.2) with Eq. (3.18), we obtain thus, the result is clear.
So, we have two cases to be discussed: Using Eqs.(3.2), we get Also, from Eqs. (3.2), we obtain It means that the curve α is a straight line in G 1 3 .
Theorem 3.5.Let α : I ⊂ R → G 1 3 be a spacelike curve in G 1 3 .If α is N-constant curve of second kind, then the position vector α has the parametrization: where u(s) = τ (s)ds + c 5 , c 5 is integral constant.
Proof.From Eq. (3.3), we have Besides, from of Eq. (3.2) and Eq.(3.17), we obtain where c 4 = 0 is a real constant.The solution of this equation is given by If we substitute Eq. (3.3) in Eq. (3.2), we can get hence, in light of Eqs.(3.3), (3.20) and (3.21), we obtain the required result.
Theorem 3.6.Let α be a spacelike curve in G 1 3 with its pseudo-Galilean trihedron {T (s), N(s), B(s)}.If the curve α lies on a pseudo-Galilean sphere S 2 ± , then it is N-constant curve of second kind and the center of a pseudo-Galilean sphere of α at the point c(s) is given by ± be a sphere in G 1 3 , then S 2 ± is given by where r is the radius of the pseudo-Galilean sphere and it is a constant.Let c be the center of the pseudo-Galilean sphere, then we have Differentiating this equation with respect to s, we get more differentiation yields From Eq. (2.8), we find and since c(s) − α(s) ∈ Sp{T (s), N(s), B(s)}, then we can write Now, from Eq. (3.23) and (3.24), we find .
Also, from Eq. Proof.Let α be a N-constant curve in G 1 3 , then we have Thus, the proof is completed.
Theorem 3.9.Let α be a spacelike curve in G 1 3 .If α is a circle then α is N-constant curve of second kind.
Proof.If α is a circle, then we have κ(s) = const and τ (s) = 0. Also, from Theorem 3.4, one can write which leads to m 2 2 (s) − m 2 1 (s) = const.thus, it completes the proof.

Examples
In this section, we give some examples to illustrate our main results.From Serret-Frenet equations, one can obtain τ (s) = −2 s .Moreover, the curvature functions m i (s) are So, from Eq. (3.16), we get Under the above considerations, α is of constant-ratio and the ratio is equal .Also, since α N (s) where κ(s) = a = const and τ (s) = s.
Since the curve has a constant curvature and non-constant torsion, so it is a Salkowski curve.
From Theorem 3.4, we have the curvature functions:

Conclusion
In the three-dimensional pseudo-Galilean space, spacelike admissible curves of constant-ratio and some special curves such as T -constant and N-constant curves have been studied.Furthermore, the spherical images of these curves have been studied.Some interesting results of N − constant curves have been obtained.Finally, as an application for this work, two examples are given and plotted to confirm our main results.
It follows that γ is N-constant curve but not constant-ratio curve, see Fig(1b).
then the curve α is a constant-ratio curve but not N-constant curve, see Fig(1a).