Citation: Mehmet Önder. Non-null slant ruled surfaces[J]. AIMS Mathematics, 2019, 4(3): 384-396. doi: 10.3934/math.2019.3.384
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In the study of curve theory, the curves whose curvatures satisfy some special conditions have an important role. The well-known of such curves is general helix defined by the classical definition that the tangent lines of the curve make a constant angle with a fixed straight line [5]. In 1802, M.A. Lancret stated a result on the helices which was first proved by B. de Saint Venant in 1845 [25]. Venant showed that a curve is a general helix if and only if the ratio of the curvatures and of the curve is constant, i.e., is constant at all points of the curve. Helices have been studied not only in Euclidean spaces but also in Lorentzian spaces by some mathematicians and different characterizations of these curves have been obtained according to the properties of the spaces [7,8,13,18].
Recently, Izumiya and Takeuchi have introduced a new curve called slant helix which is defined by the property that the normal lines of the curve make a constant angle with a fixed direction in the Euclidean 3-space [9]. Later, the spherical images, the tangent indicatrix and the binormal indicatrix of a slant helix have been studied by Kula and Yaylı and they have obtained that the spherical images of a slant helix are spherical helices [14]. The position vector of a slant helix in has been studied by Ali [4]. Then the corresponding characterizations for the position vector of a timelike slant helix in Minkowski 3-space have been given by Ali and Turgut [3]. Moreover, Ali and Lopez have also given some new characterizations of slant helices in [2].
Analogue to the curves, ruled surfaces have orthonormal frames along their striction curves. So, the notions "helix" or "slant helix" can be considered for ruled surfaces. Before, Abdel-Baky considered the notion of "slant" for ruled surfaces by means of dual vector analysis [1]. Later, by considering the orthonormal frame along the striction curve of a ruled surface, Önder has defined slant ruled surfaces in the real Euclidean 3-space [24]. Moreover, Kaya and Önder have studied the position vectors and some differential equation characterizations for slant ruled surfaces in [10,11,23]. They have also studied this subject for null scrolls and defined slant null scrolls in [22].
In this paper, we define non-null slant ruled surfaces by considering the Frenet vectors of timelike and spacelike ruled surfaces in . We give the conditions for a non-null ruled surface to be a slant ruled surface.
Let be a Minkowski 3-space with natural Lorentz metric , where is a rectangular coordinate system of . Since this metric is not positive definite, for an arbitrary vector in we have (ⅰ) and , (ⅱ) (ⅲ) and . Then we have three types of vectors: spacelike, timelike or null (lightlike) if (ⅰ), (ⅱ) or (ⅲ) holds, respectively [16]. Similarly, an arbitrary curve can locally be spacelike, timelike or null (lightlike), if all of its velocity vectors satisfy (ⅰ), (ⅱ) or (ⅲ), respectively. For the vectors and in , the vector product of and is defined by
Analogue to the curves, a surface can be timelike or spacelike in . The Lorentzian character of a surface in is determined by the induced metric on the surface. The surface is called timelike (spacelike), if the induced metric on the surface is a Lorentz metric (positive definite Riemannian metric) [6].
Let now be an open interval in the real line . Let be a curve in defined on and be a unit direction vector of an oriented line in . Then we have following parametrization for a ruled surface ,
(2.1) |
The straight lines of surface are called rulings and the curve is called base curve or generating curve. In particular, if the direction of is constant, then ruled surface is said to be cylindrical, and non-cylindrical otherwise.
The function defined by
is called distribution parameter (or drall) of ruled surface. Then, is called developable surface if and only if [15,19,21]. Then at all points of same ruling, the tangent planes are identical, i.e., tangent plane contacts the surface along a ruling. If , then the tangent planes of are distinct at all points of same ruling which is called nontorsal [19,21].
Let consider the unit normal vector of defined by . So, at the points of a nontorsal ruling we have
which is called central tangent. The point at which the vectors and are orthogonal is called the striction point (or central point) and the set of striction points of all rulings is called striction curve which has parametric representation
(2.2) |
It is clear that base curve is same with striction curve if and only if .
Since the vectors and are orthogonal, we can define an orthonormal frame on surface. For this purpose, let write . The unit vector is called central normal and the orthonormal frame at central point is called Frenet frame of .
According to the Lorentzian casual characters of ruling and central normal, the Lorentzian character of surface is classified as follows:
ⅰ) If the central normal vector is spacelike and is timelike, then the ruled surface is said to be of type .
ⅱ) If the central normal vector and the ruling are both spacelike, then the ruled surface is said to be of type .
ⅲ) If the central normal vector is timelike, then the ruled surface is said to be of type [19,21].
The ruled surfaces of type and are clearly timelike and ruled surface of type is spacelike. By using these classifications and taking striction curve as base curve, parametrization of ruled surface can be given as follows,
(2.3) |
where and is arc length of striction curve.
For the derivatives of vectors of Frenet frame of ruled surface with respect to arc length of striction curve we have followings:
ⅰ) If the ruled surface is a timelike ruled surface then we have
(2.4) |
and [19].
ⅱ)If the ruled surface is spacelike ruled surface then we have
(2.5) |
and [21].
In the equations (2.4) and (2.5), , and , are arc lengths of spherical curves generated by unit vectors and , respectively.
Theorem 2.1. ([17]) Let the striction curve of ruled surface be a unit speed curve and has the same Lorentzian casual character with the ruling and let also be the base curve of the surface. Then is developable if and only if the unit tangent of the striction curve is the same with the ruling along the curve.
In this section, we introduce the definition and characterizations of non-null -slant ruled surfaces in . First, we give the following definition.
Definition 3.1. Let be a non-null ruled surface in given by the parametrization
(3.1) |
where is striction curve of and is arc length parameter of . Let the Frenet frame and non-zero invariants of be and , respectively. Then, is called a -slant ruled surface if the ruling makes a constant angle with a fixed non-null unit direction in the space, i.e.,
(3.2) |
Then we give following characterizations for -slant ruled surfaces in . Whenever we talk about we will mean that the surface has properties as assumed in Definition 3.1.
Theorem 3.1. The ruled surface is a -slant ruled surface if and only if the function is constant and given by
(3.3) |
where are non-zero constants.
Proof. Let be a -slant ruled surface in . Then denoting by the unit vector of fixed direction, satisfies
(3.4) |
Differentiating (3.4) with respect to gives . Therefore, lies on the plane spanned by the vectors and , i.e.,
(3.5) |
where and are real constants. By differentiating (3.5) with respect to it follows
(3.6) |
and then we have that the function
is constant.
Conversely, given a non-null ruled surface , let the equation (3.3) is satisfied. We define
(3.7) |
where are non-zero constants. Differentiating (3.7) and using (3.3) it follows , i.e., is a constant vector. On the other hand . Then is a -slant ruled surface in .
Theorem 3.2. Non-null ruled surface is a -slant ruled surface if and only if .
Proof. From the Frenet formulae in (2.4) and (2.5) we have
(3.8) |
Let now be a -slant ruled surface in . By Theorem 3.1 we have is constant. Then from (3.8) it follows that .
Conversely, if , since the curvatures are non-zero, from (3.8) it is obtained that is constant and Theorem 3.1 gives that is a -slant ruled surface in .
Theorem 3.3. Non-null ruled surface is a -slant ruled surface if and only if .
Proof. From the Frenet formulae in (2.4) and (2.5) it follows
(3.9) |
Let now be a -slant ruled surface in . By Theorem 3.1, we have is constant. Then from (3.9) it follows that .
Conversely, if , since the curvature is non-zero, from (3.9) it is obtained that is constant and Theorem 3.1 gives that is a -slant ruled surface in .
Theorem 3.4. Non-null ruled surface is a -slant ruled surface if and only if
(3.10) |
holds where
Proof. Assume that is a timelike -slant ruled surface. From (2.4) we get followings
(3.11) |
(3.12) |
Since is a -slant ruled surface, is constant and by differentiation we have
(3.13) |
and from (2.4)
(3.14) |
Substituting (3.13) and (3.14) in (3.12) gives
(3.15) |
Using the second equation of (2.4), (3.10) is obtained from (3.15).
Conversely, let us assume that (3.10) holds. Differentiating (3.14) gives
(3.16) |
and so,
(3.17) |
Substituting (3.10) in (3.17) it follows
(3.18) |
Now, writing (3.11) in (3.18) and using (2.4) we have
(3.19) |
On the other hand, from (2.4) it is obtained
(3.20) |
Substituting (3.20) in (3.19) we have
(3.21) |
Integrating (3.21), we get that is constant and by Theorem 3.1, is a -slant ruled surface.
If is a spacelike ruled surface, then by the similar way it is obtained that is a -slant ruled surface if and only if (3.10) holds for .
Theorem 3.5. Let be a developable non-null ruled surface in . Then is a -slant ruled surface if and only if the striction line is a general helix in .
Proof. Since is a developable non-null ruled surface in , from Theorem 2.1 we have where is the unit tangent of . Then from Definition 3.1, it is clear that is a -slant ruled surface if and only if the striction line is a general helix in .
In this section, we introduce the definition and characterizations of non-null -slant ruled surfaces in . First, we give the following definition.
Definition 4.1. Let be a non-null ruled surface in given by the parametrization
(4.1) |
where is striction curve of and is arc length parameter of . Let the Frenet frame and non-zero invariants of be and , respectively. Then, is called a -slant ruled surface if the central normal vector makes a constant angle with a fixed non-zero unit direction in the space, i.e.,
(4.2) |
Then, under the assumptions given in Definition 4.1, we can give the following theorems characterizing non-null -slant ruled surfaces.
Theorem 4.1. is a non-null -slant ruled surface if and only if the function
(4.3) |
is constant.
Proof. Assume that is a non-null -slant ruled surface and let be timelike. Let be a fixed non-zero constant vector such that . Then for the vector we have
(4.4) |
where and are smooth functions of arc length parameter . Since is constant, differentiation of (4.4) gives
(4.5) |
From the second equation of system (4.5) we have
(4.6) |
Moreover,
(4.7) |
Substituting (4.6) in (4.7) gives
(4.8) |
Then from (4.8) it is obtained that
(4.9) |
Considering the third equation of system (4.5), from (4.9) we have
This can be written as
which is desired.
Conversely, assume that is timelike and the function in (4.3) is constant, i.e.,
We define
(4.10) |
Differentiating (4.10) with respect to and using (4.3) we have , i.e., is a constant vector. On the other hand . Thus, is a non-null -slant ruled surface.
If is considered as a spacelike ruled surface, then making the similar calculations, it is obtained that is a -slant ruled surface if and only if the function is constant.
At the following theorem we give a special case for which the first curvature is equal to 1 and obtain second curvature for to be a non-null -slant ruled surface.
Theorem 4.2. Let be a non-null ruled surface with first curvature . Then the central normal vector makes a constant angle with a fixed non-null direction , i.e., is a -slant ruled surface if and only if the second curvature is given as follows:
i) If is a timelike ruled surface then
(4.11) |
where
ii) If is a spacelike ruled surface then
(4.12) |
where
Proof. Let be a timelike ruled surface with first curvature and let be a -slant ruled surface. Then for a fixed constant timelike unit vector we have
(4.13) |
Differentiating (4.13) with respect to gives
(4.14) |
and from (4.14) we have
(4.15) |
If we put , we can write
(4.16) |
Since is unit, from (4.16) we have
(4.17) |
Then, the vector is given as follows
(4.18) |
Differentiating (4.14) with respect to , it follows
(4.19) |
Writing and (4.13) in (4.19) we have
(4.20) |
From (4.17) and (4.20) we obtain the following differential equation,
(4.21) |
By integration from (4.21) we get
(4.22) |
where is integration constant. By making parameter change , (4.22) can be written as
(4.23) |
which gives us .
Conversely, assume that holds and let us put
(4.24) |
where is constant. Then, . Let now consider the vector defined by
(4.25) |
By differentiating (4.25) and using Frenet formulae we have , i.e., the direction of is constant and . Then is a non-null -slant ruled surface.
If we assume that is spacelike then we have and making the similar calculations we obtain . Then we can write (4.11).
If the ruled surface is a spacelike ruled surface then following the same procedure, it is easily obtained that is a -slant ruled surface if and only if the second curvature is given by where , if is a timelike vector; and , if is a spacelike vector.
On the other hand, if the striction line is a geodesic on , then principal normal vector of and central normal vector of coincide. Then, we have the following corollary.
Corollary 4.1. Let the striction line be a geodesic on . Then is a non-null -slant ruled surface if and only if is a slant helix in .
If the non-null ruled surface is developable, then by Theorem 2.1, the Frenet frame of striction line coincides with the frame and we can give the following corollary.
Corollary 4.2. Let be a non-null developable surface in . Then is a -slant ruled surface if and only if striction line is a slant helix in .
In this section we introduce the definition of -slant ruled surfaces in .
Definition 5.1. Let be a non-null ruled surface in given by the parametrization
where is striction curve of and is arc length parameter of . Let the Frenet frame and non-zero invariants of be and , respectively. Then, is called a -slant ruled surface if the central tangent vector makes a constant angle with a fixed non-zero direction in the space, i.e.,
From (3.5) it is clear that a non-null ruled surface is -slant ruled surface if and only if it is a -slant ruled surface. So, all the theorems given in Section 3 also characterize the -slant ruled surfaces.
After these definitions and characterizations of non-null slant ruled surfaces we can give the followings:
Let and be two non-null ruled surfaces in with Frenet frames and , respectively. If and have common central normals i.e., at the corresponding points of their striction lines, then and are called Bertrand offsets [12]. Similarly, if at the corresponding points of their striction lines, then the surface is called a Mannheim offset of and the ruled surfaces and are called Mannheim offsets [20]. Considering these definitions we come to the following corollaries:
Corollary 5.1. Let be a non-null -slant ruled surface. Then the Bertrand offsets of form a family of non-null -slant ruled surfaces.
Corollary 5.2. Let and form a Mannheim offset. Then is a non-null -slant (or -slant) ruled surface if and only if is a non-null -slant ruled surface.
The notion of "slant" given for the curves is considered for the non-null ruled surfaces in the Minkowski 3-space and some new types of non-null ruled surfaces called "slant ruled surfaces" are defined and characterized. These characterizations are obtained according to the curvatures of the non-null surface which are defined on the striction line of the surface. Of course, new characterizations can be obtained for these surfaces and moreover, the subject can be studied in different spaces in which ruled surfaces are defined.
The author declares no conflict of interest.
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1. | Emel Karaca, On preserved properties for slant ruled surfaces under homothety in , 2024, 12, 2321-5666, 283, 10.26637/mjm1203/006 | |
2. | Emel Karaca, Non-null slant ruled surfaces and tangent bundle of pseudo-sphere, 2024, 9, 2473-6988, 22842, 10.3934/math.20241111 | |
3. | Onur Kaya, Constant Angle Ruled Surfaces Associated with Slant Ruled Surfaces, 2024, 19, 2731-8648, 989, 10.1007/s11464-023-0009-x | |
4. | Areej A. Almoneef, Rashad A. Abdel-Baky, Bertrand Offsets of Slant Ruled Surfaces in Euclidean 3-Space, 2024, 16, 2073-8994, 235, 10.3390/sym16020235 | |
5. | Areej A. Almoneef, Rashad A. Abdel-Baky, Slant spacelike ruled surfaces and their Bertrand offsets, 2024, 12, 2296-424X, 10.3389/fphy.2024.1484936 | |
6. | Emel Karaca, Slant ruled surfaces generated by the striction curves of the hyper-dual curves, 2024, 38, 0354-5180, 9463, 10.2298/FIL2427463K |