A Surface Family with a Common Asymptotic Null Curve in Minkowski 3-Space E 31

*is approach is on constructing a surface family with a common asymptotic null curve. It has provided the necessary and sufficient condition for the curve to be an asymptotic null curve and extended the study to ruled and developable surfaces. Subsequently, the study has examined the Bertrand offsets of a surface family with a common asymptotic null curve. Lastly, we support the results of this approach by some examples.


Introduction
In differential geometry of surfaces, an asymptotic direction is a curve with zero normal curvature. At a point on an asymptotic curve, we take a plane having both the curve's tangent and the normal of a surface. At such point, the trace curve of intersection of the surface and the plane has vanished curvature. Asymptotic directions exactly occur in case of negative (or zero) Gaussian curvature. Exactly speaking, there we will have two asymptotic directions bisected by the principal directions [1][2][3].
Wang et al. [4] addressed the constructions of family of surfaces sharing a given geodesic curve in Euclidean 3-space E 3 . Inspired by the work of Wang et al. [4], Bayram et al. [5] considered surfaces with a shared asymptotic curve in E 3 . ey derived the parametric representation of a surface family by means of a given curve as an isoparametric and asymptotic curve. In [6], Abdel-Baky and Al-Ghefari introduced some interesting developable and ruled surfaces as surface families in terms of given asymptotic curves. e study of Li et al. in [7] was on forming a surface family by means of a given spatial curve and how to be a line of curvature on a surface. ey provided three kinds of marching-scale functions and the necessary and sufficient conditions on them to meet both isoparametric and line of curvature requirements.
A lot of works dealing with family of surfaces having a common special curve in both Euclidean space and Minkowski space have been published (see, for instance, [8][9][10][11][12][13]). e main interest of this work is to construct a surface family from a given asymptotic null curve. Hence, the sufficient and necessary conditions for the given curve to be the asymptotic null curve are given in details. As an application, some representative curves are selected to form their corresponding surfaces that have such curves as asymptotic null curves. We extend the study to ruled and developable surfaces. Finally, some related examples to these surfaces are illustrated.

Preliminaries
Consider the Minkowski 3-space E 3 1 as the ambient space. For our approach, we have utilized the relevant information from [14][15][16][17]. Let E 3 1 denote the Minkowski 3-space E 3 1 , i.e., R 3 equipped with the metric: in which (x 1 , x 2 , x 3 ) is the canonical coordinates in R 3 . A vector x in E 3 1 is referred to as light-like or null when 〈x, x〉 � 0 and x ≠ 0, space-like if <x, x ≪ 0, and time-like if 〈x, x〉 < 0. A light-like or time-like vector in E 3 1 is referred to as causal. We define the norm of x in E 3 1 as ‖x‖ � ������ � |〈x, x〉| √ ; then, x is a time-like unit vector if 〈x, x〉 � − 1, and spacelike unit if 〈x, x〉 � 1. Analogously, a regular curve α: I ⟶ E 3 1 is time-like, space-like, or null (light-like) if its velocity vector α ′ is time-like, space-like, or light-like, respectively. Similarly, we say that a surface is time-like, spacelike, or light-like if its tangent planes are time-like, spacelike, or light-like, respectively [15,16]. Given two vectors x, y ∈ E 3 1 , the inner product is the real number 〈x, y〉 � − x 1 y 1 + x 2 y 2 + x 3 y 3 and the vector product is defined by Let α � α(s) be a null curve parameterized by its arc length; that is, the tangent vector α ′ � A is null. en, there exists a unique Cartan null frame A, B, C { } satisfying that [16,17] 〈A, A〉 � 〈B, B〉 � 〈C, A〉 � 〈C, B〉 � 0, e Frenet-Serret equations associated to such frame are as follows: where κ(s) ≠ 0 is a function on α(s) and ω is constant. Furthermore, if ω � 0, then α � α(s) is a generalized null cubic [13]. e vector fields B and C are referred to as the binormal vector field and principal normal vector field of α(s), resp.

Bertrand Mates.
We utilize basic information on Bertrand mates from [17]. Let α(s) and α * (s * ) be two null curves with Cartan frames α(s) { ; A, B, C} and α * (s * ) { ; A * , B * , C * }, respectively. e curve α * (s * ) is said to be the Bertrand mate of α(s) if there exists a one-to-one correspondence between their points, where C and C * are linearly dependent at their corresponding points. en, the curve α * (s * ) can be written as where f � ∓ω − 1 is the constant distance between their corresponding points. Note that, if α * is a Bertrand mate of a curve α, then the converse is true. erefore, the formulae link the Cartan frame of α with that of its Bertrand mate α * are [17] A * An isoparametric curve α � on a surface P(s, t) in E 3 1 is a curve in which there exists a parameter s 0 or t 0 such that α(s) � P(s, t 0 ) or α(t) � P(s 0 , t). Given a parametric curve α � α(s), we call it an iso-asymptotic of the surface P(s, t) if it is both asymptotic and a parameter curve on P(s, t).

Surfaces with Common Asymptotic Null Curve
In this section, we present a new approach for constructing a surface family with a shared asymptotic null curve α � α(s), 0 ≤ s ≤ L, in which the curve osculating plane Sp � A, C { } is coincident with the surface tangential plane. e expression of the surface over α(s) is given by where u(s, t) and v(s, t) are C 1 functions. If the parameter t is considered as the time, the functions u(s, t) and v(s, t) can be considered as directed marching distances of a point unit in the time t in the direction A and C, respectively, and then, the position vector α(s) is the initial location of the point. It is readily to check that the tangent vectors of M are given by For convention, we use P s � zP/zs and P t � zP/zt. Hence, the normal vector of M is where 2 Mathematical Problems in Engineering where is identifies the curve α(s) as an asymptotic curve on the surface. erefore, we provide the following theorem.

Theorem 1. e given spatial null curve α(s) is iso-asymptotic on the surface P(s, t) if and only if
e surfaces defined by equations (7) and (15) are referred to as the family of surfaces with a common asymptotic null curve. Any surface P(s, t) defined by equation (7) and satisfying equation (15) is a member of this family. As in [4], we also consider the case when the marching-scale functions u(s, t) and v(s, t) can be given into two factors as where l(s), m(s), U(t), and v(t) are C 1 functions not identically zero. erefore, we can derive the following corollary.

Corollary 1.
e sufficient and necessary condition of the null curve α(s) being an iso-asymptotic null curve on P(s, t) is It follows from Corollary 1 that, to attain a surface family, with a shared null asymptotic curve, we first pick the marching-scale functions as in equation (17) and then apply them to equation (7). For more convenience, although the marching-scale functions can be given in restricted forms, they can define a large class of surface family with a shared asymptotic null curve as follows: (1) If we take We can write the sufficient condition for which the curve α(s) is an iso-asymptotic null curve on the surface P(s, t) as follows: where l(s), m(s), U(t), and V(t) are not identically zero C 1 functions, a ij ∈ R (i � 1, 2; j � 1, 2, . . . , p).
We can rewrite condition (17), for which α � α(s) is iso-asymptotic null curve on the surface P(s, t), as follows: en, we get a surface family with a common asymptotic null curve given by P(s, t) � (s, sin s, cos s) +(u(s, t)), 0, v(s, t)) 1 cos s − sin s Hence, we can get special members of the family as follows: (1) By choosing u(s, t) � t and v(s, t) � − t, t 0 � 0, where β, c ∈ R, c ≠ 0, and − 2 ≤ t ≤ 2, then equation (17) is satisfied. erefore, we attain a surface of the family (Figure 1) P(s, t; β, c) � (s, sin s, cos s)

Ruled Surfaces with a Common Asymptotic Null Curve
Let P(s, t) be a ruled surface with the directrix α(s) in which α(s) is an isoparametric null curve of P(s, t); i.e., there is t 0 in which P(s, t 0 ) � α(s). Hence, the surface is defined as where d(s) is rulings direction. According to equation (7), Equation (30) forms a system of two equations in two unknown functions: u(s, t) and v(s, t). Hence, the solutions are ese equations can be viewed as the necessary and sufficient conditions for P(s, t) being a ruled surface with a null directrix α(s). Now, we utilize the conditions given in eorem 1 to prove that α(s) is asymptotic on P(s, t). Hence, these conditions become as 〈d, C〉 ≠ 0.
(32) us, the ruling direction d(s) is in the light-like plane Sp A(s), C(s) { } at any point on α(s). Additionally, d(s) and A(s) must not be linearly dependent. Hence, for some real functions β(s) and c(s) ≠ 0. Utilizing it in the expressions in equation (30), we obtain: erefore, the family of isoparametric-ruled surfaces with null asymptotic curve α(s) would take the form

Theorem 2.
e necessary and sufficient condition for P(s, t) to be a ruled surface with α(s) as a common null asymptotic curve is that ∃t 0 ∈ [0, T] and functions β(s) and c(s) ≠ 0, in which P(s, t) can be constructed by equation (35).
Every member in the family isoparametric-ruled surfaces with the null asymptotic α(s) is determined by c(s) and β(s), that is, by the direction vector function d(s). In Example 1, for u(s, t) � t 2 and v(s, t) � t, with − ·6 ≤ t ≤ ·6, the corresponding ruled surface is shown in Figure 5, and for u(s, t) � 0 and v(s, t) � t, with − 2 ≤ t ≤ ·2, Figure 6 shows the surface with u(s, t) � 0 v(s, t) � 0 − 2 ≤ t ≤ 0.2.

Classification of Ruled Surfaces with a Common Asymptotic Null Curve.
In this section, we classify the ruled surfaces with a common asymptotic null curve as time-like, light-like, and space-like surfaces. For this purpose, from equation (33), for all 0 ≤ s ≤ L, we have By taking the partial derivatives regarding s and t, respectively, we obtain (37) us,

Mathematical Problems in Engineering
where is the distribution parameter of M. Since d is a space-like vector, we can give the following two cases: Case (1): since d ′ is a nonnull vector, we get the following theorems.  Case (2): since d ′ is a null vector, we obtain the following. We now give the conditions for the ruled surface M defined by (35) to be cylindrical. To do that, we take the derivative of equation (33) regarding s and use equation (4); hence, we obtain  Since c(s) ≠ 0, for all 0 ≤ s ≤ L, the above equation cannot equal zeros. Hence, there exists no cylindrical ruled surface given by (35).

Bertrand Offsets for Surfaces with a Common Asymptotic
Null Curve. Here, we examine the Bertrand offsets of a surface family with a shared asymptotic null curve. en, analogous theory to the theory of Bertrand curves can be developed for these surfaces. Assume that M * is a Bertrand offset of M with the Serret-Frenet frame α(s); A, B, C { } given by equation (7). en, M * can take the form where u * (s * , t) and v(s * , t) have the same meaning as in equation (7). Now, we provide a representative example to illustrate such method and verify the correctness of the derived formulae.
Note that if we choose a different combination of characteristic curve, or even a number of curves, we would get and produce such series of surfaces.

Conclusion
In the Minkowski 3-space E 3 1 , there are many paper dealing with the problem of forming a family of surfaces from a given asymptotic curve [9,10,12,13]. Here, we provide a different approach for building a surface family in which its members are sharing a given asymptotic null curve as isoparametric. Given a null space curve, we derive the characterization for the given curve to be asymptotic and for the resulting surface to be ruled. Finally, we analyzed the case that surfaces family having a Bertrand offset of a given curve as asymptotic null. Hopefully, the results of this study would be applicable for physicists and those of interest in general relativity theory.

Data Availability
All of the data are available within the paper.

Conflicts of Interest
e authors have no conflicts of interest.