A Surface Family with a Common Asymptotic Null Curve in Minkowski 3-Space
This 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.
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 [1–3].
Wang et al.  addressed the constructions of family of surfaces sharing a given geodesic curve in Euclidean 3-space . Inspired by the work of Wang et al. , Bayram et al.  considered surfaces with a shared asymptotic curve in . They derived the parametric representation of a surface family by means of a given curve as an isoparametric and asymptotic curve. In , Abdel-Baky and Al-Ghefari introduced some interesting developable and ruled surfaces as surface families in terms of given asymptotic curves. The study of Li et al. in  was on forming a surface family by means of a given spatial curve and how to be a line of curvature on a surface. They 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, [8–13]). The 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.
Consider the Minkowski 3-space as the ambient space. For our approach, we have utilized the relevant information from [14–17]. Let denote the Minkowski 3-space , i.e., equipped with the metric:in which is the canonical coordinates in . A vector in is referred to as light-like or null when and , space-like if , and time-like if . A light-like or time-like vector in is referred to as causal. We define the norm of in as ; then, is a time-like unit vector if , and space-like unit if . Analogously, a regular curve is time-like, space-like, or null (light-like) if its velocity vector is time-like, space-like, or light-like, respectively. Similarly, we say that a surface is time-like, space-like, or light-like if its tangent planes are time-like, space-like, or light-like, respectively [15, 16]. Given two vectors , , the inner product is the real number and the vector product is defined by
Let be a null curve parameterized by its arc length; that is, the tangent vector is null. Then, there exists a unique Cartan null frame satisfying that [16, 17]
The Frenet–Serret equations associated to such frame are as follows:where is a function on and is constant. Furthermore, if , then is a generalized null cubic . The vector fields and are referred to as the binormal vector field and principal normal vector field of , resp.
2.1. Bertrand Mates
We utilize basic information on Bertrand mates from . Let and be two null curves with Cartan frames ; and ; , respectively. The curve is said to be the Bertrand mate of if there exists a one-to-one correspondence between their points, where and are linearly dependent at their corresponding points. Then, the curve can be written aswhere is the constant distance between their corresponding points. Note that, if is a Bertrand mate of a curve , then the converse is true. Therefore, the formulae link the Cartan frame of with that of its Bertrand mate are 
An isoparametric curve on a surface in is a curve in which there exists a parameter or such that or . Given a parametric curve , we call it an iso-asymptotic of the surface if it is both asymptotic and a parameter curve on .
Lemma 1. A nonlinear regular curve without inflections is an asymptotic on a surface if and only if it satisfies one of the following equivalent conditions: The osculating plane at each noninflection point of meets the surface tangential plane at The binormal at each noninflection point of is orthogonal to the surface tangential plane at
3. 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 , , in which the curve osculating plane is coincident with the surface tangential plane. The expression of the surface over is given bywhere and are functions. If the parameter is considered as the time, the functions and can be considered as directed marching distances of a point unit in the time in the direction and , respectively, and then, the position vector is the initial location of the point. It is readily to check that the tangent vectors of are given by
For convention, we use and . Hence, the normal vector of iswhere
Additionally, since is an isoparametric curve on , there is a parameter in which ; that is,
Thus, when —over —, we obtainThis identifies the curve as an asymptotic curve on the surface. Therefore, we provide the following theorem.
Theorem 1. The given spatial null curve is iso-asymptotic on the surface if and only if
The surfaces defined by equations (7) and (15) are referred to as the family of surfaces with a common asymptotic null curve. Any surface defined by equation (7) and satisfying equation (15) is a member of this family. As in , we also consider the case when the marching-scale functions and can be given into two factors aswhere , , and are functions not identically zero. Therefore, we can derive the following corollary.
Corollary 1. The sufficient and necessary condition of the null curve being an iso-asymptotic null curve on 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 is an iso-asymptotic null curve on the surface as follows: where , , , and are not identically zero functions, .(2)If we pick We can rewrite condition (17), for which is iso-asymptotic null curve on the surface , as follows: in which , , , , , and are functions. The factor-decomposition form is of advantageous. Any set of functions and would meet (18) or (21). Therefore, the designer can adjust the shape of the surface by posing some conditions on the sets of and which guarantee resulting surface which belongs to the iso-asymptotic surface family with the null curve as the asymptotic.
Example 1. Given the null helix,Therefore, we have the Cartan frame as follows:Then, we get a surface family with a common asymptotic null curve given byHence, we can get special members of the family as follows:(1)By choosing and , where , and , then equation (17) is satisfied. Therefore, we attain a surface of the family (Figure 1)(2)If we pick , where and , equation (19) is satisfied. Therefore, we get a surface of the family (Figure 2)(3)By choosing and , where and , then equation (19) is satisfied. Therefore, we get a member of the family (Figure 3)(4)By choosing and , where and , then equation (19) is satisfied. Therefore, we get a member of the family (Figure 4)
4. Ruled Surfaces with a Common Asymptotic Null Curve
Let be a ruled surface with the directrix in which is an isoparametric null curve of ; i.e., there is in which . Hence, the surface is defined aswhere is rulings direction. According to equation (7), we have
Equation (30) forms a system of two equations in two unknown functions: and . Hence, the solutions are
These equations can be viewed as the necessary and sufficient conditions for being a ruled surface with a null directrix .
Now, we utilize the conditions given in Theorem 1 to prove that is asymptotic on . Hence, these conditions become as
Thus, the ruling direction is in the light-like plane at any point on . Additionally, and must not be linearly dependent. Hence,for some real functions and . Utilizing it in the expressions in equation (30), we obtain:
Therefore, the family of isoparametric-ruled surfaces with null asymptotic curve would take the formwhere , for all , and the functions and determine the shape of . We should point out that, in such a model, there are two asymptotic null curves moving through every point on , one of which is itself and the other is a line in the null direction given by equation (33). Every member in the family of isoparametric-ruled surfaces with the shared null asymptotic is determined by the two family parameters: and .
Theorem 2. The necessary and sufficient condition for to be a ruled surface with as a common null asymptotic curve is that and functions and , in which can be constructed by equation (35).
Every member in the family isoparametric-ruled surfaces with the null asymptotic is determined by and , that is, by the direction vector function . In Example 1, for and , with , the corresponding ruled surface is shown in Figure 5, and for and , with , Figure 6 shows the surface with
4.1. 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 , we have
By taking the partial derivatives regarding and , respectively, we obtain
Thus,whereis the distribution parameter of . Since is a space-like vector, we can give the following two cases:
Case (1): since is a nonnull vector, we get the following theorems.
Theorem 3. The noncylindrical ruled surface given in terms of the nonnull vector and (35) is time-like in if and only if one of the following statements hold:(i) is a space-like vector, , or , and (ii) is a time-like vector, and
Theorem 4. The noncylindrical ruled surface given in terms of the nonnull vector and (35) is light-like if and only if .
Case (2): since is a null vector, we obtain the following.
Theorem 5. Let be a noncylindrical ruled surface with parametrization (35) such that is a null vector. Then, is a light-like surface in .
We now give the conditions for the ruled surface 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
Therefore, is cylindrical if and only if
Since , for all , the above equation cannot equal zeros. Hence, there exists no cylindrical ruled surface given by (35).
Corollary 2. There exists no cylindrical and developable ruled surface represented in (35).
4.2. 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. Then, analogous theory to the theory of Bertrand curves can be developed for these surfaces.
Definition 1. Let and be two surfaces based on iso-asymptotic null curves and , respectively. is the Bertrand offset of if and only if , and are the Bertrand mates.
Assume that is a Bertrand offset of with the Serret–Frenet frame given by equation (7). Then, can take the formwhere and 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.
Example 2. In Example 1, according to equation (5) and equation (6), we have and . If we choose and and lower sign of , thenwith the Cartan frame:Therefore, the surface family can be written asIn view of equations (24) and (45), if we take and , and , where , then equation (15) is satisfied. The graphs of and are depicted in Figures 7 and 8.
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.
In the Minkowski 3-space , 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.
All of the data are available within the paper.
Conflicts of Interest
The authors have no conflicts of interest.
The Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, Saudi Arabia, has funded this project, under Grant no. KEP-47-130-42. The authors, therefore, gratefully acknowledge the DSR technical and financial support.
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