A Note on Parametric Surfaces in Minkowski 3-Space

With the help of the Frenet frame of a given pseudo null curve, a family of parametric surfaces is expressed as a linear combination of this frame. The necessary and sufficient conditions are examined for that curve to be an isoparametric and asymptotic on the parametric surface. It is shown that there is not any cylindrical and developable ruled surface as a parametric surface. Also, some interesting examples are illustrated about these surfaces.


Introduction
The Minkowski 3-space E 3 1 is the Euclidean 3-space E 3 provided with the standard flat metric given by where ( 1 , 2 , 3 ) is a rectangular coordinate system of E 3 1 . Since is an indefinite metric, recall that a vector V ∈ E 3 1 can have one of three Lorentzian causal characters: it can be spacelike if (V, V) > 0 or V = 0, timelike if (V, V) < 0, and null (lightlike) if (V, V) = 0 and V ̸ = 0. In particular, the norm (length) of a vector V is given by ‖V‖ = √| (V, V)| and two vectors ⃗ V and ⃗ are said to be orthogonal, if (V, ) = 0. Next, recall that an arbitrary curve = ( ), in E 3 1 , can locally be spacelike, timelike, or null (lightlike), if all of its velocity vectors ( ) are, respectively, spacelike, timelike, or null (lightlike) [1]. In Minkowski 3-space, a spacelike curve whose principal normal and binormal are null vectors is called pseudo null curve [2].
For a pseudo null curve, the first curvature can take only two values: = 0 when is a straight line, or = 1 in all other cases [2,3]. Also, we have In the differential geometry of surfaces, an asymptotic curve is formally defined as a curve on a regular surface such that the normal curvature is zero in the asymptotic direction. Asymptotic directions can only occur when the Gaussian curvature on surface is negative or zero along the asymptotic curve [1,4,5].
Asymptotic curves or asymptotics have been the subject in differential geometry, in architectural CAD, and in molecular design (see [6]). There are recent works about asymptotics: Wang et al. [7] introduced the concept of surface pencil with a common isogeodesic curve. Bayram et al. [8] obtained the parametric representation for a surface pencil from a given curve as an isoparametric and asymptotic curve 2 The Scientific World Journal in Minkowski 3-space E 3 1 . Abdel-Baky and Al-Ghefari in [9] demonstrated some interesting ruled and developable surfaces as a surface pencil from a given asymptotic curve. Also, Saffak et al. [10] expressed a family of surfaces from a given spacelike or timelike asymptotic curve using the Frenet trihedron frame of the curve in Minkowski 3-space E 3 1 . The goal of the study is to construct the parametric representation of surface from a given pseudo null curve and derive the necessary and sufficient conditions for the given pseudo null curve to be an isoparametric and asymptotic on the parametric surface. The family of parametric surfaces with common pseudo null asymptotic curve is defined. Also, it is shown that there is not any cylindrical and developable ruled surface as a parametric surface and some interesting examples about these surfaces are illustrated.
In this paper we will assume that pseudo null base curve has the first curvature ( ) = 1; that is, that the curve is not a straight line.

Surfaces with Common Asymptotic
Curve in E If the parameter is seen as the time, the functions ( , ), ( , ), and ( , ) can then be viewed as directed marching distances of a point unit in the time in the direction , , and , respectively, and the position vector is seen as the initial location of this point.
By taking the derivative of (7) with respect to and using the Frenet equations (2), we get The normal ( , ) of the surface is given by and since = = ( − + 1) + ( + + ) + ( − ) , using (5), the normal vector can be written as Let the curve = ( ) on the ruled surface , given by (6), be an isoparametric. Then there should exist a parameter = 0 such that ( ) = ( , 0 ) where And from (11) we obtain According to [11], the curve on the surface is asymptotic if and only if the binormal ( ) of the curve and the normal ( , 0 ) of the surface at any point on the curve are parallel to each other. Thus for all ∈ [0, ] if and only if Therefore, we can give the necessary and sufficient conditions for the surface to have the pseudo null curve as an isoparametric and asymptotic with the following theorem.

Theorem 1. Let be a surface having a pseudo null base curve
in the 3-dimensional Minkowski space with parametrization (6). The curve is an isoparametric and asymptotic curve on the surface if and only if the following conditions are satisfied: where ( , ), ( , ), and ( , ) are 1 functions.
We call the set of surfaces defined by (6) and (15) the family of surfaces with common isoasymptotic, since the common isoasymptotic is also an isoparametric curve on these surfaces. Any surface defined by (6) and satisfying conditions (12) and (15) is a member of the family.
By simplifying, condition (18) can be represented as where e satisfy the conditions (7).
As a curve , consider the pseudo null curve (see Figure 1) Then we get the Frenet vectors as follows: where , ∈ R, ̸ = 0, then the surfaces family with the common isoasymptotic is given by ) .

Ruled Surface with Common Asymptotic Curve
Let = ( ) be a pseudo null curve in the 3-dimensional Minkowski space. Suppose ( , ) is a ruled surface with the directrix ( ) which is also an isoparametric curve of ( , ).
In that case, there exists a parameter = 0 such that ( , 0 ) = ( ) for all ∈ [0, ], then for ∈ [0, ], ∈ [0, ], and 0 ∈ [0, ] the surface ( , ) can be expressed as where ( ) denotes the direction of the rulings. Also, from (6) and (7), we get It follows that, at any point on the curve ( ), the ruling direction ( ) must be in the plane formed by ( ) and ( ). Moreover, the ruling direction ( ) and the vector ( ) must not be parallel. Thus, for some real functions ( ) and V( ), we can write  Thus, the isoasymptotic ruled surface with the common asymptotic directrix ( ) is given by where the real functions ( ) and V( ) control the shape of the ruled surface, and V( ) ̸ = 0 for all ∈ [0, ]. On the other hand, there exist two asymptotic curves passing through every point on the curve ( ): one is ( ) itself and the other is a straight line in the direction ( ) as given in (35). Every member of the isoasymptotic ruled surface is decided by two parameters ( ) and V( ), that is, by the direction vector function ( ).
From (4) and (35), for all ∈ [0, ], we have and we can give the following cases.

Corollary 3.
There is not any cylindrical and developable ruled surface as defined by (37).
Example 4. Let be a ruled surface whose asymptotic curve is the pseudo null curve in Example 2.
If the controlling functions of the ruled surface are The Scientific World Journal then the corresponding cylindrical surface is shown in Figure 4.
If the controlling functions of the ruled surface for all are ( ) = , then the corresponding noncylindrical surface is shown in Figure 5.

Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.