Abstract
In this work, we consider the Darboux frame of a curve lying on an arbitrary regular surface and we construct ruled surfaces having a base curve which is a -direction curve. Subsequently, a detailed study of these surfaces is made in the case where the directing vector of their generatrices is a vector of the Darboux frame, a Darboux vector field. Finally, we give some examples for special curves such as the asymptotic line, geodesic curve, and principal line, with illustrations of the different cases studied.
1. Introduction
Ruled surfaces are well known as one of the most important surface families in the differential geometry of the surfaces. An important fact about these types of surfaces is that any ruled surface can always result from a continuous movement of a straight line along a curve.
Among the associative curves are the direction curves introduced by Choi and Kim [1] as integral curves of some vector fields generated by Frenet vectors of a given curve. This type of curves is included in several works. In terms of the Frenet frame, the direction curves are studied in [1, 2]; this study has been extended to the alternative frame [3], Darboux frame [4], and Bishop frame [5].
Examen the ruled surfaces constructed by means of direction curves has recently attracted the attention of many differential geometers. In terms of the Frenet frame, Güven [6] defined two ruled surfaces such as normal and binormal surfaces by considering their base curves as the -direction curve. He get some results about the developability and minimality of these surfaces and the conditions for which their base curve is an asymptotic line, a geodesic curve, or a principal line. In the same way, this two ruled surfaces were also defined with a base curve and adjoint of the base curve in [7]. Moreover, in [8], the authors improve the theory of the ruled surfaces in terms of principal-direction curves of a given curve; they obtained a new representation of these ruled surfaces by slant helices and principal elements of the ruled surface such as the pitch and angle of pitch. After that, in [9] and in terms of the Darboux frame, the authors used the direction curve to define a new ruled surface called the relatively osculating developable surface. Then, they have obtained some results about the existence, uniqueness, and singularity of such surface.
In this work, we consider the Darboux frame of a curve lying on an arbitrary regular surface and we construct ruled surfaces whose base curve is a -direction curve. We give some results about the developability, minimality, and condition for which the striction curve is the base curve for the particular cases where the directing vector of the ruled surface is a vector of the Darboux frame, a Darboux vector field. Finally, we give some examples for special curves such as the asymptotic line, geodesic curve, and principal line, with illustrations of the different cases studied.
2. Preliminaries
In this section, we recall some basic concepts and properties on classical differential geometry of curves lying on a regular surface and of ruled surfaces, in the Euclidean 3-space. (i)We denote by the Euclidean 3-space, with the usual metricwhere and are two vectors of
Let be a regular surface and be a unit speed curve on the surface . The Darboux frame along the curve is an orthonormal frame where is the unit tangent, is the unit normal on the surface , and Then, the Darboux equations are given by the following relations: or where , and are the normal Darboux vectors field, the rectifying Darboux vector field, and the osculator Darboux vectors field, respectively, and are defined by where , and are the geodesic curvature, the normal curvature, and the geodesic torsion of the curve , respectively. (i)A ruled surface [10] is generated by a one-parameter family of straight lines and has a parametric representation:where is called the base curve of the ruled surface and the unit vectors representing the direction of straight lines. If is constant, then, the ruled surface is cylindrical; otherwise, the surface is said to be noncylindrical.
Definition 1 [10]. For a curve lying on a regular surface, the following are well known: (1) is an asymptotic line if and only if the normal curvature vanishes(2) is a geodesic curve if and only if the geodesic curvature vanishes(3) is a principal line if and only if the geodesic torsion vanishes
Definition 2 [4]. Let be a unit speed curve on the regular surface and the Darboux frame along the curve The integral curve of the vector field is called -direction curve of , in other words
We have
Then, the point is said to be singular if
If there exists a common perpendicular to two constructive rulings in the ruled surface, then, the foot of the common perpendicular on the main rulings is called a central point. The locus of the central point is called a striction curve [10].
The parametrization of the striction curve on the ruled surface is given by
Definition 3. A ruled surface is called developable if
Letting and be the first and the second fundamental forms from the ruled surface , respectively, we have where
The mean curvature of a ruled surface is given as follows
Definition 4. A ruled surface is said to be minimal if its mean curvature vanishes identically.
3. Ruled Surfaces Defined by -Direction Curves
Letting be a unit speed curve lying on a regular surface , the Darboux frames of are the geodesic curvature, the normal curvature, the geodesic torsion of and a -direction curve, respectively.
We consider the following ruled surface where is the base curve and the unit director vector of the straight line.
3.1. General Study
In this section, we propose to give some properties of the ruled surface
Differentiating (12) with respect to and , we get
Then, by using (10), we obtain the components of the first fundamental form and as follows:
Consequently, the condition of regularity of the ruled surface (12) is
Differentiating (13) with respect to and , we get which allows, using formula (10), calculating the two components of the second fundamental form and respectively, and we obtain
Therefore, by using (11), the mean curvature of the ruled surface (12) is given as follows
On the other hand, the striction curve of the surface (12) is
3.2. Case Where (Resp. )
Let us consider the ruled surfaces defined by
By using (2), we have
If we substitute (21) in (15), we find
Likewise, we have
If we substitute (23) in (15), we find
On the other hand,
If we substitute (25) in (15), we find
Corollary 5. The ruled surfaces and are regulars along the curve , while is singular along
We denote by (resp. ) the striction curve of and from (19), we have
The result follows.
Corollary 6. (1) is the striction curve of the surface if and only if is a geodesic curve(2) is the striction curve of the surface (3) is the striction curve of the surface if and only if is a principal line
On the other hand, by using (2), we have
Corollary 7. (1)The ruled surface (resp. ) is developable if and only if is an asymptotic line(2)The ruled surface is developable
Differentiating (2), we obtain
By using (2) and (29), we have
If we substitute (30) in (18), we find
Likewise, we have
If we substitute (32) in (18), we find
Corollary 8. The ruled surface is minimal along its base curve if and only if is a principal line (resp. a geodesic curve).
3.3. Case Where Is a Darboux Vector Field
3.3.1. Case Where
The ruled surface becomes where with and
Using (2), we get
Hence, where
We obtain
If we substitute (38) in (15), we find
The following corollaries follow:
Corollary 9. The ruled surface is regular.
Corollary 10. In the noncylindrical case (i.e., ), the curve is the striction curve of the surface .
On the other hand, we have
Corollary 11. The ruled surface is developable if and only if it is cylindrical.
According to (36), we have
Therefore, by using (2) and (29), we obtain
Then,
If we substitute (42) in (18), we find
Hence, we have the following corollary:
Corollary 12. The ruled surface is not minimal.
3.3.2. Case Where
The ruled surface becomes where with and
Differentiating we obtain
Thus, where
We obtain
If we substitute (48) in (15), we get
On the other hand, the striction curve of is
Hence, we have the following corollary:
Corollary 13. In the noncylindrical case (i.e., ), we have the following: (1) is singular along its striction curve(2) is the striction curve of the surface if and only if is a principal line
We have Hence, the result is as follows.
Corollary 14. The ruled surface is developable.
From (46), we have
Then, by using (2) and (29), we obtain
Hence,
If we substitute (53) in (18), we obtain
Corollary 15. The surface is not minimal.
3.3.3. Case Where
The ruled surface becomes
where with and
By using (2), we get
Thus, where It follows that
If we substitute (59) in (15), we obtain
On the other hand, the striction curve of is
Hence, we have the following corollary:
Corollary 16. In the noncylindrical case (i.e., ), we have the following: (1) is singular along its striction curve(2) is the striction curve of the surface if and only if is a geodesic curve
We have Hence, we have the following result.
Corollary 17. The ruled surface is developable.
From (57), we have
Then, by using (2) and (29), we obtain
Then,
If we substitute (64) in (18), we obtain
Corollary 18. The ruled surface is not minimal.
4. Examples
In this section, we reinforce the previous study by the given four examples. The first one corresponds to the general case, and the three others represent the particular cases where the initial curve is an asymptotic line, a geodesic curve, and a line of curvature.
In the examples which follow, the same notations as in the preceding paragraphs are retained. We denote by the initial curve lying on a surface defined by the Darboux frame of , the normal curvature, the geodesic curvature, and the geodesic torsion of the curve respectively; by the -direction curve; and by (resp., ) the unit rectifying (resp., osculator, normal) vector field. We give for each example the illustrations of the ruled surfaces denoted by ,
Example 1. Let be a curve lying on the surface given by the following parametrization: which can be seen in Figure 1. The Darboux frame of is Then, It follows that The -direction curve is given by where , are integration constants. Consequently, the surfaces , , , , , are given as illustrate in Figures 2–7.
Example 2. Let be the curve lying on the surface given by the following parametrization: which can be seen in Figure 8. The Darboux frame of is Then, , and and Let , where are integration costants. Consequently, the surfaces , , , , , are given as illustrated in Figures 9–13.
Example 3. We consider the curve , lying on the surface defined by which can be seen in Figure 14. The Darboux frame of is Then, , , and and Let where are integration costants. Consequently, the surfaces , , , , , are given as illustrated in Figures 15–19.
Example 4. We consider the curve lying on the surface given by the following parametrization: which can be seen in Figure 20. The Darboux frame of is Then, , , and and Let where are integration costants. Consequently, the surfaces , , , , , are given as illustrated in Figures 21–24.
Data Availability
No data were used to support the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.