On Pseudohyperbolical Smarandache Curves in Minkowski 3-Space

Abstract

We define pseudohyperbolical Smarandache curves according to the Sabban frame in Minkowski 3-space. We obtain the geodesic curvatures and the expression for the Sabban frame vectors of special pseudohyperbolic Smarandache curves. Finally, we give some examples of such curves.

1. Introduction

In the theory of curves in the Euclidean and Minkowski spaces, one of the interesting problems is the problem of characterization of a regular curve. In the solution of the problem, the curvature functions 𝜅 and 𝜏 of a regular curve have an effective role. It is known that we can determine the shape and size of a regular curve by using its curvatures 𝜅 and 𝜏. Another approach to the solution of the problem is to consider the relationship between the corresponding Frenet vectors of two curves. For instance, Bertrand curves and Mannheim curves arise from this relationship. Another example is the Smarandache curves. They are the objects of Smarandache geometry, that is, a geometry which has at least one Smarandachely denied axiom [1]. An axiom is said to be Smarandachely denied if it behaves in at least two different ways within the same space. Smarandache geometries are connected with the theory of relativity and the parallel universes.

If the position vector of a regular curve 𝛼 is composed by the Frenet frame vectors of another regular curve 𝛽, then the curve 𝛼 is called a Smarandache curve [2]. Special Smarandache curves in Euclidean and Minkowski spaces are studied by some authors [3–8]. The curves lying on a pseudohyperbolic space 𝐻²₀ in Minkowski 3-space 𝔼³₁ are characterized in [9].

In this paper, we define pseudohyperbolical Smarandache curves according to the Sabban frame {𝛼,𝑇,𝜉} in Minkowski 3-space. We obtain the geodesic curvatures and the expressions for the Sabban frame's vectors of special pseudohyperbolical Smarandache curves. In particular, we prove that special 𝑇𝜉-pseudohyperbolical Smarandache curves do not exist. Besides, we give some examples of special pseudohyperbolical Smarandache curves in Minkowski 3-space.

2. Basic Concepts

The Minkowski 3-space ℝ³₁ is the Euclidean 3-space ℝ³ provided with the standard flat metric given by ⟨⋅,⋅⟩=−𝑑𝑥²₁+𝑑𝑥²₂+𝑑𝑥²₃, (1) where (𝑥₁,𝑥₂,𝑥₃) is a rectangular Cartesian coordinate system of ℝ³₁. Since 𝑔 is an indefinite metric, recall that a nonzero vector 𝑥∈ℝ³₁ can have one of the three Lorentzian causal characters: it can be spacelike if ⟨𝑥,𝑥⟩>0, timelike if ⟨𝑥,𝑥⟩<0, and null (lightlike) if ⟨𝑥,𝑥⟩=0. In particular, the norm (length) of a vector 𝑥∈ℝ³₁ is given by ‖𝑥‖=|⟨𝑥,𝑥⟩| and two vectors 𝑥 and 𝑦 are said to be orthogonal if ⟨𝑥,𝑦⟩=0. Next, recall that an arbitrary curve 𝛼=𝛼(𝑠) in 𝔼³₁ can locally be spacelike, timelike, or null (lightlike) if all of its velocity vectors 𝛼^′(𝑠) are, respectively, spacelike, timelike, or null (lightlike) for every 𝑠∈𝐼[10]. If ‖𝛼^′(𝑠)‖≠0 for every 𝑠∈𝐼, then 𝛼 is a regular curve in ℝ³₁. A spacelike (timelike) regular curve 𝛼 is parameterized by pseudo-arclength parameter 𝑠 which is given by 𝛼∶𝐼⊂ℝ→ℝ³₁, and then the tangent vector 𝛼^′(𝑠) along 𝛼 has unit length, that is, ⟨𝛼^′(𝑠),𝛼^′(𝑠)⟩=1(⟨𝛼^′(𝑠),𝛼^′(𝑠)⟩=−1) for all 𝑠∈𝐼, respectively.

For any 𝑥=(𝑥₁,𝑥₂,𝑥₃) and 𝑦=(𝑦₁,𝑦₂,𝑦₃) in the space ℝ³₁, the pseudovector product of 𝑥 and 𝑦 is defined by 𝑥×𝑦=−𝑥₂𝑦₃+𝑥₃𝑦₂,𝑥₃𝑦₁−𝑥₁𝑦₃,𝑥₁𝑦₂−𝑥₂𝑦₁. (2)

Remark 1. Let 𝑥=(𝑥₁,𝑥₂,𝑥₃), 𝑦=(𝑦₁,𝑦₂,𝑦₃), and 𝑧=(𝑧₁,𝑧₂,𝑧₃) be vectors in ℝ³₁. Then, (i)⟨𝑥×𝑦,𝑧⟩=^{𝑥1𝑥2𝑥3𝑦1𝑦2𝑦3}_{𝑧1𝑧2𝑧3},(ii)𝑥×(𝑦×𝑧)=−⟨𝑥,𝑧⟩𝑦+⟨𝑥,𝑦⟩𝑧, (iii)⟨𝑥×𝑦,𝑥×𝑦⟩=−⟨𝑥,𝑥⟩⟨𝑦,𝑦⟩+⟨𝑥,𝑦⟩², where × is the pseudovector product in the space ℝ³₁.

Lemma 2. In the Minkowski 3-space ℝ³₁, the following properties are satisfied [10]:(i)two timelike vectors are never orthogonal;(ii)two null vectors are orthogonal if and only if they are linearly dependent;(iii)timelike vector is never orthogonal to a null vector.

Pseudohyperbolic space in the Minkowski 3-space ℝ³₁ is a quadric defined by 𝐻²₀=𝑥∈ℝ³₁∣−𝑥²₁+𝑥²₂+𝑥²₃=−1. (3)

Let 𝛼∶𝐼⊂ℝ→𝐻²₀ be a regular unit speed curve lying fully in 𝐻²₀ in ℝ³₁. Then its position vector 𝛼 is timelike vector, which implies that tangent vector 𝑇=𝛼^′ is the unit spacelike vector for all 𝑠∈𝐼. Hence we have orthonormal Sabban frame {𝛼(𝑠),𝑇(𝑠),𝜉(𝑠)} along the curve 𝛼, where 𝜉(𝑠)=𝛼(𝑠)×𝑇(𝑠) is the unit spacelike vector. The corresponding Frenet formulae of 𝛼, according to the Sabban frame, read 𝛼^′𝑇^′𝜉^′=01010𝑘_𝑔(𝑠)0−𝑘_𝑔(𝑠)0𝛼𝑇𝜉, (4) where 𝑘_𝑔(𝑠)=det(𝛼(𝑠),𝑇(𝑠),𝑇^′(𝑠)) is the geodesic curvature of 𝛼 on 𝐻²₀ in ℝ³₁ and 𝑠 is arc length parameter of𝑎𝑙𝑝ℎ𝑎. In particular, the following relations hold: 𝛼×𝑇=𝜉,𝑇×𝜉=−𝛼,𝜉×𝛼=𝑇. (5)

3. Pseudohyperbolical Smarandache Curves in Minkowski 3-Space

In this section, we define and investigate pseudohyperbolical Smarandache curves in Minkowski 3-space according to the Sabban frame.

Let 𝛼=𝛼(𝑠) and 𝛽=𝛽(𝑠^∗) be two regular unit speed curves lying fully in pseudohyperbolic space 𝐻²₀ in ℝ³₁ and let {𝛼,𝑇,𝜉} and {𝛽,𝑇_𝛽,𝜉_𝛽} be the moving Sabban frames of these curves, respectively. Then we have the following definitions of pseudohyperbolical Smarandache curves.

Definition 3. Let 𝛼∶𝐼⊂ℝ↦𝐻²₀ be a regular unit speed curve lying fully in 𝐻²₀. Then 𝛼𝜉-pseudohyperbolical Smarandache curve 𝛽∶𝐼⊂ℝ↦𝐻²₀ of 𝛼 is defined by 𝛽𝑠^⋆(𝑠=1)√2(𝑎𝛼(𝑠)+𝑏𝜉(𝑠)), (6) where 𝑎,𝑏∈ℝ₀ and 𝑎²−𝑏²=2.

Definition 4. Let 𝛼∶𝐼⊂ℝ↦𝐻²₀ be a regular unit speed curve lying fully in 𝐻²₀. Then 𝛼𝑇-pseudohyperbolical Smarandache curve 𝛽∶𝐼⊂ℝ↦𝐻²₀ of 𝛼 is defined by 𝛽𝑠^⋆(𝑠=1)√2(𝑎𝛼(𝑠)+𝑏𝑇(𝑠)), (7) where 𝑎,𝑏∈𝑅₀ and 𝑎²−𝑏²=2.

Definition 5. Let 𝛼∶𝐼⊂𝑅↦𝐻²₀ be a regular unit speed curve lying fully in 𝐻²₀. Then 𝛼𝑇𝜉 -pseudohyperbolical Smarandache curve 𝛽∶𝐼⊂𝑅↦𝐻²₀ of 𝛼 is defined by 𝛽𝑠^⋆(𝑠=1)√3(𝑎𝛼(𝑠)+𝑏𝑇(𝑠)+𝑐𝜉(𝑠)), (8) where 𝑎,𝑏,𝑐∈ℝ₀ and 𝑎²−𝑏²−𝑐²=3.

Theorem 6. Let 𝛼∶𝐼⊂ℝ↦𝐻²₀ be a regular unit speed curve lying fully in 𝐻²₀. Then 𝑇𝜉-pseudohyperbolical Smarandache curve 𝛽∶𝐼⊂ℝ↦𝐻²₀ of 𝛼 does not exist.

Proof. Assume that there exists 𝑇𝜉-pseudohyperbolical Smarandache curve of 𝛼. Then it can be written as 𝛽𝑠^⋆(𝑠=1)√2(𝑎𝑇(𝑠)+𝑏𝜉(𝑠)), (9) where 𝑎²+𝑏²=−2, which is a contradiction.

In the theorems which follow, we obtain Sabban frame {𝛽,𝑇_𝛽,𝜉_𝛽} and geodesic curvature 𝜅^𝛽_𝑔 of pseudohyperbolical Smarandache curve 𝛽.

Theorem 7. Let 𝛼∶𝐼⊂ℝ↦𝐻²₀ be a regular unit speed curve lying fully in 𝐻²₀ with the Sabban frame {𝛼,𝑇,𝜉} and the geodesic curvature 𝑘_𝑔. If 𝛽∶𝐼⊂ℝ↦𝐻²₀ is 𝛼𝜉-pseudohyperbolical Smarandache curve of 𝛼, then its frame {𝛽,𝑇_𝛽,𝜉_𝛽} is given by 𝛽𝑇_𝛽𝜉_𝛽=𝑎√20𝑏√2𝜖𝑏0𝜖0√2𝑎0𝜖√2𝛼𝑇𝜉, (10) and the corresponding geodesic curvature 𝑘^𝛽_𝑔 reads 𝑘^𝛽_𝑔=𝑎𝑘_𝑔−𝑏𝑎−𝑏𝑘_𝑔,𝑎,𝑏∈ℝ₀, (11) where 𝑎²−𝑏²=2 and 𝜖=±1.

Proof. Differentiating (6) with respect to 𝑠 and using (4), we obtain 𝛽^′(𝑠)=𝑑𝛽𝑑𝑠^∗𝑑𝑠^∗=𝑑𝑠𝑎−𝑏𝑘_𝑔√2𝑇𝑇,_𝛽𝑑𝑠^∗=𝑑𝑠𝑎−𝑏𝑘_𝑔√2𝑇, (12) where 𝑑𝑠^∗=𝑑𝑠𝑎−𝑏𝑘_𝑔√2. (13) Therefore, the unit spacelike tangent vector of the curve 𝛽 is given by 𝑇_𝛽=𝜖𝑇, (14) where 𝜖=+1 if 𝑎−𝑏𝑘_𝑔>0 for all 𝑠 and 𝜖=−1 if 𝑎−𝑏𝑘_𝑔<0 for all 𝑠.
Differentiating (14) with respect to 𝑠, we find 𝑑𝑇_𝛽𝑑𝑠^∗𝑑𝑠^∗𝑑𝑠=𝜖𝛼+𝑘_𝑔𝜉, (15) and from (13) and (15) we get 𝑇^′_𝛽=√2𝜖𝑎−𝑏𝑘_𝑔𝛼+𝑘_𝑔𝜉. (16)
On the other hand, from (6) and (14) it can be easily seen that 𝜉_𝛽=𝛽×𝑇_𝛽𝑏=𝜖√2𝑎𝛼+𝜖√2𝜉 (17) is a unit spacelike vector.
Consequently, the geodesic curvature 𝑘^𝛽_𝑔 of the curve 𝛽=𝛽(𝑠^∗) is given by 𝑘^𝛽_𝑔=det𝛽,𝑇_𝛽,𝑇^′_𝛽=𝑎𝑘_𝑔−𝑏𝑎−𝑏𝑘_𝑔. (18)

Theorem 8. Let 𝛼∶𝐼⊂ℝ↦𝐻²₀ be a regular unit speed curve lying fully in 𝐻²₀ with the Sabban frame {𝛼,𝑇,𝜉} and the geodesic curvature 𝑘_𝑔. If 𝛽∶𝐼⊂ℝ↦𝐻²₀ is 𝛼𝑇-pseudohyperbolical Smarandache curve of 𝛼, then its frame {𝛽,𝑇_𝛽,𝜉_𝛽} is given by 𝛽𝑇_𝛽𝜉_𝛽=𝑎√2𝑏√20𝑏2+𝑏𝑘_𝑔²𝑎2+𝑏𝑘_𝑔²𝑏𝑘_𝑔2+𝑏𝑘_𝑔²−𝑏²𝑘_𝑔4+2𝑏𝑘_𝑔²−𝑎𝑏𝑘_𝑔4+2𝑏𝑘_𝑔²√22+𝑏𝑘_𝑔²×𝛼𝑇𝜉, (19) and the corresponding geodesic curvature 𝑘^𝛽_𝑔 reads 𝑘^𝛽_𝑔=𝑏²𝑘_𝑔𝜀₁−𝑎𝑏𝑘_𝑔𝜀₂+2𝜀₃2+𝑏²𝑘²_𝑔^5/2,𝑎,𝑏∈ℝ₀, (20) where 𝜀₁=−𝑏³𝑘_𝑔𝑘^′_𝑔+𝑎2+𝑏𝑘_𝑔²,𝜀₂=−𝑎𝑏²𝑘_𝑔𝑘^′_𝑔+𝑏−𝑏𝑘²_𝑔2+𝑏𝑘_𝑔²,𝜀₃=−𝑏³𝑘²_𝑔𝑘^′_𝑔+𝑎𝑘_𝑔+𝑏𝑘^′_𝑔2+𝑏𝑘_𝑔², (21) and 𝑎²−𝑏²=2.

Proof. Differentiating (7) with respect to 𝑠 and using (4), we obtain 𝛽^′(𝑠)=𝑑𝛽𝑑𝑠^∗𝑑𝑠^∗=1𝑑𝑠√2𝑏𝛼+𝑎𝑇+𝑏𝑘_𝑔𝜉,𝑇_𝛽𝑑𝑠^∗=1𝑑𝑠√2𝑏𝛼+𝑎𝑇+𝑏𝑘_𝑔𝜉, (22) where 𝑑𝑠^∗=𝑑𝑠2+𝑏𝑘_𝑔²2. (23)
Therefore, the unit spacelike tangent vector of the curve 𝛽 is given by 𝑇_𝛽=12+𝑏𝑘_𝑔²𝑏𝛼+𝑎𝑇+𝑏𝑘_𝑔𝜉. (24)
Differentiating (24) with respect to 𝑠, it follows that 𝑑𝑇_𝛽𝑑𝑠^∗𝑑𝑠^∗=1𝑑𝑠2+𝑏𝑘_𝑔^23/2𝜀₁𝛼+𝜀₂𝑇+𝜀₃𝜉. (25)
From (4) and (23), we get 𝑇^′_𝛽=√22+𝑏𝑘_𝑔²²𝜀₁𝛼+𝜀₂𝑇+𝜀₃𝜉, (26) where 𝜀₁=−𝑏³𝑘_𝑔𝑘^′_𝑔+𝑎2+𝑏𝑘_𝑔²,𝜀₂=−𝑎𝑏²𝑘_𝑔𝑘^′_𝑔+𝑏−𝑏𝑘²_𝑔2+𝑏𝑘_𝑔²,𝜀₃=−𝑏³𝑘²_𝑔𝑘^′_𝑔+𝑎𝑘_𝑔+𝑏𝑘^′_𝑔2+𝑏𝑘_𝑔². (27) On the other hand, from (7) and (24), it can be easily seen that 𝜉_𝛽=𝛽×𝑇_𝛽𝑏=−²𝑘_𝑔4+2𝑏𝑘_𝑔²𝛼−𝑎𝑏𝑘_𝑔4+2𝑏𝑘_𝑔²√𝑇+22+𝑏𝑘_𝑔²𝜉. (28) Hence 𝜉_𝛽 is a unit spacelike vector.
Therefore, the geodesic curvature 𝑘^𝛽_𝑔 of the curve 𝛽=𝛽(𝑠^∗) is given by 𝑘^𝛽_𝑔=det𝛽,𝑇_𝛽,𝑇^′_𝛽=𝑏²𝑘_𝑔𝜀₁−𝑎𝑏𝑘_𝑔𝜀₂+2𝜀₃2+𝑏²𝑘²_𝑔^5/2. (29)

Theorem 9. Let 𝛼∶𝐼⊂ℝ↦𝐻²₀ be a regular unit speed curve lying fully in 𝐻²₀ with the Sabban frame {𝛼,𝑇,𝜉} and the geodesic curvature 𝑘_𝑔. If 𝛽∶𝐼⊂ℝ↦𝐻²₀ is 𝛼𝑇𝜉-pseudohyperbolical Smarandache curve of 𝛼, then its frame {𝛽,𝑇_𝛽,𝜉_𝛽} is given by^𝛽_{𝑇𝛽𝜉𝛽}=^{𝑎√3𝑏√3𝑐√3𝑏}_{𝑎𝑘𝑔−𝑐}²_−3𝑘²_𝑔+3^{𝑎−𝑐𝑘𝑔}_{𝑎𝑘𝑔−𝑐}²_−3𝑘²_𝑔+3^𝑏𝑘𝑔_{𝑎𝑘𝑔−𝑐}²_−3𝑘²_{𝑔−𝑎+32𝑘−3𝑔+𝑎𝑐3𝑎𝑘𝑔−𝑐2−9𝑘2𝑔+9𝑏𝑐−𝑎𝑏𝑘𝑔3𝑎𝑘𝑔−𝑐2−9𝑘2𝑔+93+𝑐2−𝑎𝑐𝑘𝑔3𝑎𝑘𝑔−𝑐2−9𝑘2𝑔+9}^𝛼_𝑇𝜉 (30)and the corresponding geodesic curvature 𝑘^𝛽_𝑔 reads 𝑘^𝛽_𝑔=−𝑎𝑐−𝑎²−3𝑘_𝑔𝜀₁+𝑏𝑐−𝑎𝑏𝑘_𝑔𝜀₂+3+𝑐²−𝑎𝑐𝑘_𝑔𝜀₃×𝑎𝑘_𝑔−𝑐²−3𝑘²_𝑔+3^5/2−1,𝑎,𝑏,𝑐∈ℝ₀, (31) where 𝜀₁=−𝑎𝑘_𝑔−𝑐𝑎𝑘^′_𝑔−3𝑘^′_𝑔𝑘_𝑔𝑏+𝑎𝑘_𝑔−𝑐²−3𝑘²_𝑔+3𝑎−𝑐𝑘_𝑔,𝜀₂=−𝑎𝑘_𝑔−𝑐𝑎𝑘^′_𝑔−3𝑘^′_𝑔𝑘_𝑔𝑎−𝑐𝑘_𝑔+𝑎𝑘_𝑔−𝑐²−3𝑘²_𝑔+3𝑏−𝑐𝑘^′_𝑔−𝑏𝑘²_𝑔,𝜀₃=−𝑎𝑘_𝑔−𝑐𝑎𝑘^′_𝑔−3𝑘^′_𝑔𝑘_𝑔𝑏𝑘_𝑔+𝑎𝑘_𝑔−𝑐²−3𝑘²_𝑔+3𝑎−𝑐𝑘_𝑔𝑘_𝑔+𝑏𝑘^′_𝑔 (32) and 𝑎²−𝑏²−𝑐²=3.

Proof. Differentiating (8) with respect to 𝑠 and by using (4), we find 𝛽^′(𝑠)=𝑑𝛽𝑑𝑠^∗𝑑𝑠^∗=1𝑑𝑠√3𝑏𝛼+𝑎−𝑐𝑘_𝑔𝑇+𝑏𝑘_𝑔𝜉, (33) and thus 𝑇_𝛽𝑑𝑠^∗=1𝑑𝑠√3𝑏𝛼+𝑎−𝑐𝑘_𝑔𝑇+𝑏𝑘_𝑔𝜉, (34) where 𝑑𝑠^∗=𝑑𝑠𝑎𝑘_𝑔−𝑐²−3𝑘²_𝑔+33. (35)
The geodesic curvature 𝑘_𝑔≠(𝑎𝑐±𝑎²𝑐²−(𝑎²−3)(𝑐²+3))/(𝑎²−3) for all 𝑠, since 𝛽=𝛽(𝑠^∗) is a unit speed regular curve in ℝ³₁.
Therefore, the unit spacelike tangent vector of the curve 𝛽 is given by 𝑇_𝛽=1𝑎𝑘_𝑔−𝑐²−3𝑘²_𝑔+3𝑏𝛼+𝑎−𝑐𝑘_𝑔𝑇+𝑏𝑘_𝑔𝜉. (36)
Differentiating (36) with respect to 𝑠 and from (4) and (35), it follows that 𝑇^′_𝛽=√3𝑎𝑘_𝑔−𝑐²−3𝑘²_𝑔+3²𝜀₁𝛼+𝜀₂𝑇+𝜀₃𝜉, (37) where 𝜀₁=−𝑎𝑘_𝑔−𝑐𝑎𝑘^′_𝑔−3𝑘^′_𝑔𝑘_𝑔𝑏+𝑎𝑘_𝑔−𝑐²−3𝑘²_𝑔+3𝑎−𝑐𝑘_𝑔,𝜀₂=−𝑎𝑘_𝑔−𝑐𝑎𝑘^′_𝑔−3𝑘^′_𝑔𝑘_𝑔𝑎−𝑐𝑘_𝑔+𝑎𝑘_𝑔−𝑐²−3𝑘²_𝑔+3𝑏−𝑐𝑘^′_𝑔−𝑏𝑘²_𝑔,𝜀₃=−𝑎𝑘_𝑔−𝑐𝑎𝑘^′_𝑔−3𝑘^′_𝑔𝑘_𝑔𝑏𝑘_𝑔+𝑎𝑘_𝑔−𝑐²−3𝑘²_𝑔+3𝑎−𝑐𝑘_𝑔𝑘_𝑔+𝑏𝑘^′_𝑔. (38)
On the other hand, from (8) and (36) it can be easily seen that 𝜉_𝛽=𝛽×𝑇_𝛽=−𝑎²−3𝑘_𝑔+𝑎𝑐3𝑎𝑘_𝑔−𝑐²−9𝑘²_𝑔𝛼++9𝑏𝑐−𝑎𝑏𝑘_𝑔3𝑎𝑘_𝑔−𝑐²−9𝑘²_𝑔𝑇++93+𝑐²−𝑎𝑐𝑘_𝑔3𝑎𝑘_𝑔−𝑐²−9𝑘²_𝑔+9𝜉. (39) Hence 𝜉_𝛽 is a unit spacelike vector.
Therefore, the geodesic curvature 𝑘^𝛽_𝑔 of the curve 𝛽=𝛽(𝑠^∗) is given by 𝑘^𝛽_𝑔=det𝛽,𝑇_𝛽,𝑇^′_𝛽=−𝑎𝑐−𝑎²−3𝑘_𝑔𝜀₁+𝑏𝑐−𝑎𝑏𝑘_𝑔𝜀₂+3+𝑐²−𝑎𝑐𝑘_𝑔𝜀₃×𝑎𝑘_𝑔−𝑐²−3𝑘²_𝑔+3^5/2−1. (40)

Corollary 10. If 𝛼∶𝐼⊂ℝ↦𝐻²₀ is a geodesic curve on 𝐻²₀ in Minkowski 3-space 𝔼³₁, then(1)𝛼𝑇-pseudohyperbolic Smarandache curve is also geodesic on 𝐻²₀;(2)𝛼𝜉-pseudohyperbolic and 𝛼𝑇𝜉-pseudohyperbolic Smarandache curves have constant geodesic curvatures on 𝐻²₀.

4. Examples

Example 1. Let 𝛼 be a unit speed curve lying in pseudohyperbolic space 𝐻²₀(1) in the Minkowski 3-space 𝔼³₁ with parameter equation (see Figure 1) 𝛼(𝑠)=𝑠²2𝑠+1,²2,𝑠. (41) The orthonormal Sabban frame {𝛼(𝑠),𝑇(𝑠),𝜉(𝑠)} along the curve 𝛼 is given by 𝛼(𝑠)=𝑠²2𝑠+1,²2,𝑇(𝑠),𝑠=𝛼^′(𝑠)=(),𝜉(𝑠)(𝑠)(𝑠)=𝑠𝑠,𝑠,1=𝛼×𝑇²2,𝑠²2−1,𝑠. (42) In particular, the geodesic curvature 𝑘_𝑔 of the curve 𝛼 has the form 𝑘_𝑔(𝑠)=−1. (43)

Figure 1 

The curve 𝛼 on 𝐻20(1).

Case 1. If we take 𝑎=2, 𝑏=√2, then from (6) the 𝛼𝜉-pseudohyperbolical Smarandache curve 𝛽 is given by (see Figure 2) 𝛽𝑠^∗(𝑠=1)2√2+1𝑠²+2√√2,2+1𝑠²−2,2√2+2𝑠. (44) From Theorem 7, its frame {𝛽,𝑇_𝛽,𝜉_𝛽} is given by 𝛽𝑇_𝛽𝜉_𝛽=√√201010−102𝛼𝑇𝜉, (45) and the corresponding geodesic curvature 𝑘^𝛽_𝑔 reads 𝑘^𝛽_𝑔=−1. (46)

Figure 2 

The 𝛼𝜉-pseudohyperbolical Smarandache curve 𝛽 and the curve 𝛼 on 𝐻20(1).

Case 2. If we take 𝑎=√3, 𝑏=−1, then from (7) the 𝛼𝑇-pseudohyperbolical Smarandache curve is given by (see Figure 3) 𝛽𝑠^∗(𝑠)=√24√3𝑠²−2𝑠+2√3,√3𝑠²−2𝑠,2√3𝑠−2. (47) According to Theorem 8, its frame {𝛽,𝑇_𝛽,𝜉_𝛽} is given by 𝛽𝑇_𝛽𝜉_𝛽=√62−√220−√331√33√66−√22√63𝛼𝑇𝜉, (48) and the corresponding geodesic curvature 𝑘^𝛽_𝑔 reads 𝑘^𝛽_𝑔=−1. (49)

Figure 3 

The 𝛼𝑇-pseudohyperbolical Smarandache curve 𝛽 and the curve 𝛼 on 𝐻20(1).

Case 3. If 𝑎=3, 𝑏=√3, and 𝑐=√3, then from (7) the 𝛼𝑇𝜉-pseudohyperbolical Smarandache curve 𝛽 is given by (see Figure 4) 𝛽𝑠^∗(𝑠=1)2√3+1𝑠²+2𝑠+2√√3,3+1𝑠²+2𝑠−2,2√3+2𝑠+2. (50) By using Theorem 9, it follows that the frame {𝛽,𝑇_𝛽,𝜉_𝛽} is given by (see Figure 4) 𝛽𝑇_𝛽𝜉_𝛽=√√3113−1211−√32√3+121√3+12𝛼𝑇𝜉, (51) and the corresponding geodesic curvature 𝑘^𝛽_𝑔 (Figure 4) reads 𝑘^𝛽_𝑔=−62+√33+√3^1/2. (52)

Figure 4 

The 𝛼𝑇𝜉-pseudohyperbolical Smarandache curve 𝛽 and the curve 𝛼 on 𝐻20(1).

Also, Pseudohyperbolical Smarandache curves of 𝛼 and the curve𝛼 on 𝐻²₀ (1) with Figure 5 are shown.

Figure 5 

Pseudohyperbolical Smarandache curves of 𝛼 and the curve 𝛼 on 𝐻20(1).

Acknowledgment

The authors would like to thank the referees for their helpful suggestions and comments which significantly improved the first version of the paper.