Abstract
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.
1. Introduction
The Minkowski -space is the Euclidean -space provided with the standard flat metric given by where is a rectangular coordinate system of . Since is an indefinite metric, recall that a vector can have one of three Lorentzian causal characters: it can be spacelike if or , timelike if , and null (lightlike) if and . In particular, the norm (length) of a vector is given by and two vectors and are said to be orthogonal, if . Next, recall that an arbitrary curve , in , 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].
If is a pseudo null curve, the Frenet formulas have the form [2, 3] where For a pseudo null curve, the first curvature can take only two values: when is a straight line, or 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 in Minkowski -space . 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 -space .
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 ; that is, that the curve is not a straight line.
2. Surfaces with Common Asymptotic Curve in
Let be a parametric surface on a pseudo null curve in the -dimensional Minkowski space with parametrization where , , and are functions and is given by 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 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 such that 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 of the surface at any point on the curve are parallel to each other. Thus for all 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 -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 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.
As mentioned in [7], the marching-scale functions , , and can be decomposed into two factors.
Case 1. If we choose where , , , , , and are functions, and , , and are not identically zero, then, from Theorem 1, we can simply express that the necessary and sufficient condition of the curve being an isoparametric and asymptotic curve on the surface is For the case when the marching-scale functions , , and depend only on the parameter , if we choose , then the corresponding family of surfaces with the common isoasymptotic becomes By simplifying, condition (18) can be represented as
Case 2. If we choose where , , , , , , , , and are functions, then, from Theorem 1 and (18), we can simply express that the necessary and sufficient condition of the curve being an isoparametric and asymptotic curve on the surface is
The choices given above give an advantage: any set of functions , , and would satisfy (18) or (22). Thus we can select different sets of functions , , and to adjust the shape of the surface until they are gratified with the design, and the resulting surface is guaranteed to belong to the surface family with the pseudo null curve as the common asymptotic.
Example 2. Let be a parametric surface on a pseudo null curve in the -dimensional Minkowski space parameterized by
where satisfy the conditions (7).
As a curve , consider the pseudo null curve (see Figure 1)
Then we get the Frenet vectors as follows:
Moreover, the curvatures and of have the form
If we choose
where , , then the surfaces family with the common isoasymptotic is given byFor and we obtain a member of the surface (see Figure 2) aswhere and .
For and we obtain a member of the surface (see Figure 3) aswhere and .
3. Ruled Surface with Common Asymptotic Curve
Let be a pseudo null curve in the -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 such that for all , then for , , and the surface can be expressed as where denotes the direction of the rulings.
Also, from (6) and (7), we get and we get a system of three equations with three unknown functions , , and as follows: The above equations in (33) are just the necessary and sufficient conditions for which is a ruled surface with a directrix .
If the curve is also asymptotic on the surface , by using the conditions given in (18), then for all these conditions become 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 , we can write where for all . Substituting it into the expressions in (33)–(35), we have where for all .
Thus, the isoasymptotic ruled surface with the common asymptotic directrix is given by where the real functions and control the shape of the ruled surface, and for all . 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 , that is, by the direction vector function .
From (4) and (35), for all , we have and we can give the following cases.
Case 3. If for all , then and the direction is a null vector given by
where for all . By taking the derivative of (39) with respect to and using (2), we get
and does not equal zero vectors because of for all . Thus, there is not any cylindrical ruled surface as defined by (37).
Also, from (4) and (40), we have
and since for all , the ruled surface is not developable.
Case 4. If for all , then and the direction is a spacelike vector. By taking the derivative of (35) with respect to and using (2), we get
and this equation does not equal zero vectors because of and for all . Thus, there is not any cylindrical ruled surface as defined by (37).
Also, from (4)-(5) and (35), we have
and since and for all , the ruled surface is not developable.
Therefore, from Cases 3 and 4, we can give the following corollary.
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
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.
Acknowledgment
The author would like to thank the anonymous referee for his/her helpful suggestions and comments which improved significantly the presentation of the paper.