Research article

Non-null slant ruled surfaces

  • Received: 20 December 2018 Accepted: 02 April 2019 Published: 19 April 2019
  • MSC : 53A25, 53C50, 14J26

  • In this study, we define some new types of non-null ruled surfaces called slant ruled surfaces in the Minkowski 3-space E13. We introduce some characterizations for a non-null ruled surface to be a slant ruled surface in E13. Moreover, we obtain some corollaries which give the relationships between a non-null slant ruled surface and its striction line.

    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 this study, we define some new types of non-null ruled surfaces called slant ruled surfaces in the Minkowski 3-space E13. We introduce some characterizations for a non-null ruled surface to be a slant ruled surface in E13. Moreover, we obtain some corollaries which give the relationships between a non-null slant ruled surface and its striction line.


    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 κ and τ of the curve is constant, i.e., κ/τ 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 E3 [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 E3 has been studied by Ali [4]. Then the corresponding characterizations for the position vector of a timelike slant helix in Minkowski 3-space E13 have been given by Ali and Turgut [3]. Moreover, Ali and Lopez have also given some new characterizations of slant helices in E13 [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 E3 [10,11,23]. They have also studied this subject for null scrolls and defined slant null scrolls in E13 [22].

    In this paper, we define non-null slant ruled surfaces by considering the Frenet vectors of timelike and spacelike ruled surfaces in E13. We give the conditions for a non-null ruled surface to be a slant ruled surface.

    Let E13 be a Minkowski 3-space with natural Lorentz metric ,=dx12+dx22+dx32, where (x1,x2,x3) is a rectangular coordinate system of E13. Since this metric is not positive definite, for an arbitrary vector v=(v1,v2,v3) in E13 we have (ⅰ) v,v>0 and v=0, (ⅱ) v,v<0 (ⅲ) v,v=0 and v0. Then we have three types of vectors: spacelike, timelike or null (lightlike) if (ⅰ), (ⅱ) or (ⅲ) holds, respectively [16]. Similarly, an arbitrary curve α=α(s) can locally be spacelike, timelike or null (lightlike), if all of its velocity vectors α(s) satisfy (ⅰ), (ⅱ) or (ⅲ), respectively. For the vectors x=(x1,x2,x3) and y=(y1,y2,y3) in E13, the vector product of x and y is defined by

    x×y=(x2y3x3y2,x1y3x3y1,x2y1x1y2).

    Analogue to the curves, a surface can be timelike or spacelike in E13. The Lorentzian character of a surface in E13 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 I be an open interval in the real line IR. Let k=k(u) be a curve in E13 defined on I and q=q(u) be a unit direction vector of an oriented line in E13. Then we have following parametrization for a ruled surface N,

    r(u,v)=k(u)+vq(u). (2.1)

    The straight lines of surface are called rulings and the curve k=k(u) is called base curve or generating curve. In particular, if the direction of q is constant, then ruled surface is said to be cylindrical, and non-cylindrical otherwise.

    The function defined by

    δ=det(dk,q,dq)dq,dq

    is called distribution parameter (or drall) of ruled surface. Then, N is called developable surface if and only if δ=0 [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 det(dk,q,dq)0, then the tangent planes of N are distinct at all points of same ruling which is called nontorsal [19,21].

    Let consider the unit normal vector m of N defined by m=ru×rvru×rv. So, at the points of a nontorsal ruling u=u1 we have

    a=limvm(u1,v)=dq×qdq.

    which is called central tangent. The point at which the vectors a and m are orthogonal is called the striction point (or central point) C and the set of striction points of all rulings is called striction curve which has parametric representation

    c(u)=k(u)dq,dkdq,dqq(u). (2.2)

    It is clear that base curve is same with striction curve if and only if dq,dk=0.

    Since the vectors a and q are orthogonal, we can define an orthonormal frame on surface. For this purpose, let write h=a×q. The unit vector h is called central normal and the orthonormal frame {C;q,h,a} at central point C is called Frenet frame of N.

    According to the Lorentzian casual characters of ruling and central normal, the Lorentzian character of surface N is classified as follows:

    ⅰ) If the central normal vector h is spacelike and q is timelike, then the ruled surface N is said to be of type N.

    ⅱ) If the central normal vector h and the ruling q are both spacelike, then the ruled surface N is said to be of type N+.

    ⅲ) If the central normal vector h is timelike, then the ruled surface N is said to be of type N× [19,21].

    The ruled surfaces of type N and N+ are clearly timelike and ruled surface of type N× is spacelike. By using these classifications and taking striction curve as base curve, parametrization of ruled surface N can be given as follows,

    r(s,v)=c(s)+vq(s), (2.3)

    where q,q=ε(=±1),h,h=±1 and s is arc length of striction curve.

    For the derivatives of vectors of Frenet frame {C;q,h,a} of ruled surface N with respect to arc length s of striction curve we have followings:

    ⅰ) If the ruled surface N is a timelike ruled surface then we have

    [dq/dsdh/dsda/ds]=[0k10εk10k20εk20][qha], (2.4)

    and q×h=εa,h×a=εq,a×q=h [19].

    ⅱ)If the ruled surface N is spacelike ruled surface then we have

    [dq/dsdh/dsda/ds]=[0k10k10k20k20][qha], (2.5)

    and q×h=a,h×a=q,a×q=h [21].

    In the equations (2.4) and (2.5), k1=ds1ds, k2=ds3ds and s1, s3 are arc lengths of spherical curves generated by unit vectors q and a, respectively.

    Theorem 2.1. ([17]) Let the striction curve c=c(s) of ruled surface N be a unit speed curve and has the same Lorentzian casual character with the ruling and let also c(s) be the base curve of the surface. Then N 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 q-slant ruled surfaces in E13. First, we give the following definition.

    Definition 3.1. Let N be a non-null ruled surface in E13 given by the parametrization

    r(s,v)=c(s)+vq(s),q(s)=1, (3.1)

    where c(s) is striction curve of N and s is arc length parameter of c(s). Let the Frenet frame and non-zero invariants of N be {q,h,a} and k1,k2, respectively. Then, N is called a q-slant ruled surface if the ruling q(s) makes a constant angle with a fixed non-null unit direction u in the space, i.e.,

    q,u=cq=constant. (3.2)

    Then we give following characterizations for q-slant ruled surfaces in E13. Whenever we talk about N we will mean that the surface has properties as assumed in Definition 3.1.

    Theorem 3.1. The ruled surface N is a q-slant ruled surface if and only if the function k1/k2 is constant and given by

    k1/k2={εca/cq,Nistimelikeca/cq,Nisspacelike (3.3)

    where cq=q,u,ca=a,u are non-zero constants.

    Proof. Let N be a q-slant ruled surface in E13. Then denoting by u the unit vector of fixed direction, N satisfies

    q,u=cq=constant. (3.4)

    Differentiating (3.4) with respect to s gives h,u=0. Therefore, u lies on the plane spanned by the vectors q and a, i.e.,

    u=cqq+caa, (3.5)

    where cq and ca are real constants. By differentiating (3.5) with respect to s it follows

    0={(cqk1+εcak2)h;Nistimelike,(cqk1+cak2)h;Nisspacelike, (3.6)

    and then we have that the function

    k1/k2={εca/cq,Nistimelikeca/cq,Nisspacelike

    is constant.

    Conversely, given a non-null ruled surface N, let the equation (3.3) is satisfied. We define

    u=cqq+caa, (3.7)

    where q,u=cq,a,u=ca are non-zero constants. Differentiating (3.7) and using (3.3) it follows u=0, i.e., u is a constant vector. On the other hand q,u=cq=constant. Then N is a q-slant ruled surface in E13.

    Theorem 3.2. Non-null ruled surface N is a q-slant ruled surface if and only if det(q,q,q)=0.

    Proof. From the Frenet formulae in (2.4) and (2.5) we have

    det(q,q,q)={εk13k22(k1k2);Nistimelikek13k22(k1k2);Nisspacelike (3.8)

    Let now N be a q-slant ruled surface in E13. By Theorem 3.1 we have k1/k2 is constant. Then from (3.8) it follows that det(q,q,q)=0.

    Conversely, if det(q,q,q)=0, since the curvatures are non-zero, from (3.8) it is obtained that k1/k2 is constant and Theorem 3.1 gives that N is a q-slant ruled surface in E13.

    Theorem 3.3. Non-null ruled surface N is a q-slant ruled surface if and only if det(a,a,a)=0.

    Proof. From the Frenet formulae in (2.4) and (2.5) it follows

    det(a,a,a)={k25(k1k2);Nistimelike,k25(k1k2);Nisspacelike. (3.9)

    Let now N be a q-slant ruled surface in E13. By Theorem 3.1, we have k1/k2 is constant. Then from (3.9) it follows that det(a,a,a)=0.

    Conversely, if det(a,a,a)=0, since the curvature k2 is non-zero, from (3.9) it is obtained that k1/k2 is constant and Theorem 3.1 gives that N is a q-slant ruled surface in E13.

    Theorem 3.4. Non-null ruled surface N is a q-slant ruled surface if and only if

    q+mq=3k1h, (3.10)

    holds where

    m={k1k1+ε(k12k22);Nistimelike,(k1k1+k12+k22);Nisspacelike.

    Proof. Assume that N is a timelike q-slant ruled surface. From (2.4) we get followings

    q=εk12q+k1h+k1k2a, (3.11)
    q=(3εk1k1)q+(k1+εk1k22)h+(2k1k2+k1k2)a(εk12)q. (3.12)

    Since N is a q-slant ruled surface, k1/k2 is constant and by differentiation we have

    k1k2=k2k1, (3.13)

    and from (2.4)

    h=1k1q. (3.14)

    Substituting (3.13) and (3.14) in (3.12) gives

    q=(k1k1+ε(k22k12))q+3k1(εk1q+k2a). (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

    h=(k1k12)q+(1k1)q, (3.16)

    and so,

    h=(k1k12)q2(k1k12)q+(1k1)q. (3.17)

    Substituting (3.10) in (3.17) it follows

    h=2(k1k12)q[(k1k12)+mk1]q+3(k1k1)h. (3.18)

    Now, writing (3.11) in (3.18) and using (2.4) we have

    h=[(k1k12)+mk1]q(εk)q2(k1k1)2h+(k2k1k1)a. (3.19)

    On the other hand, from (2.4) it is obtained

    h=(εk1)q(εk1)q+(εk22)h+k2a. (3.20)

    Substituting (3.20) in (3.19) we have

    k2k2=k1k1. (3.21)

    Integrating (3.21), we get that k1/k2 is constant and by Theorem 3.1, N is a q-slant ruled surface.

    If N is a spacelike ruled surface, then by the similar way it is obtained that N is a q-slant ruled surface if and only if (3.10) holds for m=(k1k1+k12+k22).

    Theorem 3.5. Let N be a developable non-null ruled surface in E13. Then N is a q-slant ruled surface if and only if the striction line c(s) is a general helix in E13.

    Proof. Since N is a developable non-null ruled surface in E13, from Theorem 2.1 we have c(s)=t(s)=q(s) where t(s) is the unit tangent of c(s). Then from Definition 3.1, it is clear that N is a q-slant ruled surface if and only if the striction line c(s) is a general helix in E13.

    In this section, we introduce the definition and characterizations of non-null h-slant ruled surfaces in E13. First, we give the following definition.

    Definition 4.1. Let N be a non-null ruled surface in E13 given by the parametrization

    r(s,v)=c(s)+vq(s),q(s)=1, (4.1)

    where c(s) is striction curve of N and s is arc length parameter of c(s). Let the Frenet frame and non-zero invariants of N be {q,h,a} and k1,k2, respectively. Then, N is called a h-slant ruled surface if the central normal vector h makes a constant angle with a fixed non-zero unit direction u in the space, i.e.,

    h,u=ch=constant. (4.2)

    Then, under the assumptions given in Definition 4.1, we can give the following theorems characterizing non-null h-slant ruled surfaces.

    Theorem 4.1. N is a non-null h-slant ruled surface if and only if the function

    f={k12(ε(k22k12))32(k2k1);Nistimelike,k12(k12+k22)32(k2k1);Nisspacelike. (4.3)

    is constant.

    Proof. Assume that N is a non-null h-slant ruled surface and let N be timelike. Let u be a fixed non-zero constant vector such that h,u=ch=constant. Then for the vector u we have

    u=b1(s)q(s)+chh(s)+b2(s)a(s), (4.4)

    where b1=b1(s) and b2=b2(s) are smooth functions of arc length parameter s. Since u is constant, differentiation of (4.4) gives

    {b1εchk1=0,b1k1+εb2k2=0,b2+chk2=0. (4.5)

    From the second equation of system (4.5) we have

    b1=εb2k2k1. (4.6)

    Moreover,

    u,u=εb12+ch2εb22=constant. (4.7)

    Substituting (4.6) in (4.7) gives

    εb22((k2k1)21)=n2=constant. (4.8)

    Then from (4.8) it is obtained that

    b2=±nε[(k2k1)21]. (4.9)

    Considering the third equation of system (4.5), from (4.9) we have

    dds[±nε[(k2k1)21]]=chk2.

    This can be written as

    k12(ε(k22k12))32(k2k1)=chn=constant,

    which is desired.

    Conversely, assume that N is timelike and the function in (4.3) is constant, i.e.,

    k12(ε(k22k12))32(k2k1)=constant=d.

    We define

    u=k2ε(k22k12)qdhεk1ε(k22k12)a,ε(k22k12)>0. (4.10)

    Differentiating (4.10) with respect to s and using (4.3) we have u=0, i.e., u is a constant vector. On the other hand h,u=constant. Thus, N is a non-null h-slant ruled surface.

    If N is considered as a spacelike ruled surface, then making the similar calculations, it is obtained that N is a h-slant ruled surface if and only if the function k12(k12+k22)32(k2k1) is constant.

    At the following theorem we give a special case for which the first curvature k1 is equal to 1 and obtain second curvature for N to be a non-null h-slant ruled surface.

    Theorem 4.2. Let N be a non-null ruled surface with first curvature k11. Then the central normal vector h makes a constant angle θ with a fixed non-null direction u, i.e., N is a h-slant ruled surface if and only if the second curvature is given as follows:

    i) If N is a timelike ruled surface then

    k2(s)=±ss2+εμ2(θ),(s2+εμ2(θ)>0), (4.11)

    where

    μ(θ)={cothθ;ifuisatimelikevector,tanhθ;ifuisaspacelikevector.

    ii) If N is a spacelike ruled surface then

    k2(s)=±sη2(θ)s2,(η2(θ)s2>0), (4.12)

    where

    η(θ)={tanhθ;ifuisatimelikevector,cothθ;ifuisaspacelikevector.

    Proof. Let N be a timelike ruled surface with first curvature k11 and let N be a h-slant ruled surface. Then for a fixed constant timelike unit vector u we have

    h,u=sinhθ=constant. (4.13)

    Differentiating (4.13) with respect to s gives

    εq+k2a,u=0, (4.14)

    and from (4.14) we have

    q,u=εk2a,u. (4.15)

    If we put a,u=εx, we can write

    u=(εk2x)q+(sinhθ)hxa. (4.16)

    Since u is unit, from (4.16) we have

    x=±coshθε(1k22),(ε(1k22)>0). (4.17)

    Then, the vector u is given as follows

    u=±εk2coshθε(1k22)q+(sinhθ)hcoshθε(1k22)a. (4.18)

    Differentiating (4.14) with respect to s, it follows

    ε(1k22)h+k2a,u=0. (4.19)

    Writing a,u=εx and (4.13) in (4.19) we have

    x=(1k22)sinhθk2. (4.20)

    From (4.17) and (4.20) we obtain the following differential equation,

    ±cothθεk2(ε(1k22))3/21=0. (4.21)

    By integration from (4.21) we get

    ±cothθεk2ε(1k22)s+c=0, (4.22)

    where c is integration constant. By making parameter change ss+c, (4.22) can be written as

    ±cothθεk2ε(1k22)=s (4.23)

    which gives us k2(s)=±ss2+εcoth2θ.

    Conversely, assume that k2(s)=±ss2+εcoth2θ holds and let us put

    x=±coshθε(1k22)=±coshθεεs2s2+εcoth2θ=±sinhθs2+εcoth2θ, (4.24)

    where θ is constant. Then, k2x=ssinhθ. Let now consider the vector u defined by

    u=sinhθ(εsq+h(s2+εcoth2θ)a) (4.25)

    By differentiating (4.25) and using Frenet formulae we have u=0, i.e., the direction of u is constant and h,u=sinhθ=constant. Then N is a non-null h-slant ruled surface.

    If we assume that u is spacelike then we have h,u=coshθ=constant and making the similar calculations we obtain k2(s)=±ss2+εcoth2θ. Then we can write (4.11).

    If the ruled surface N is a spacelike ruled surface then following the same procedure, it is easily obtained that N is a h-slant ruled surface if and only if the second curvature is given by k2(s)=±sη2(θ)s2 where η(θ)=tanhθ, if u is a timelike vector; and η(θ)=cothθ, if u is a spacelike vector.

    On the other hand, if the striction line c(s) is a geodesic on N, then principal normal vector n of c(s) and central normal vector h of N coincide. Then, we have the following corollary.

    Corollary 4.1. Let the striction line c(s) be a geodesic on N. Then N is a non-null h-slant ruled surface if and only if c(s) is a slant helix in E13.

    If the non-null ruled surface N is developable, then by Theorem 2.1, the Frenet frame {t,n,b} of striction line c(s) coincides with the frame {q,h,a} and we can give the following corollary.

    Corollary 4.2. Let N be a non-null developable surface in E13. Then N is a h-slant ruled surface if and only if striction line is a slant helix in E13.

    In this section we introduce the definition of a-slant ruled surfaces in E13.

    Definition 5.1. Let N be a non-null ruled surface in E13 given by the parametrization

    r(s,v)=c(s)+vq(s),q(s)=1,

    where c(s) is striction curve of N and s is arc length parameter of c(s). Let the Frenet frame and non-zero invariants of N be {q,h,a} and k1,k2, respectively. Then, N is called a a-slant ruled surface if the central tangent vector a makes a constant angle with a fixed non-zero direction u in the space, i.e.,

    a,u=ca=constant.

    From (3.5) it is clear that a non-null ruled surface N is a-slant ruled surface if and only if it is a q-slant ruled surface. So, all the theorems given in Section 3 also characterize the a-slant ruled surfaces.

    After these definitions and characterizations of non-null slant ruled surfaces we can give the followings:

    Let N1 and N2 be two non-null ruled surfaces in E13 with Frenet frames {q1,h1,a1} and {q2,h2,a2}, respectively. If N1 and N2 have common central normals i.e., h1=h2 at the corresponding points of their striction lines, then N1 and N2 are called Bertrand offsets [12]. Similarly, if a1=h2 at the corresponding points of their striction lines, then the surface N2 is called a Mannheim offset of N1 and the ruled surfaces N1 and N2 are called Mannheim offsets [20]. Considering these definitions we come to the following corollaries:

    Corollary 5.1. Let N1 be a non-null h-slant ruled surface. Then the Bertrand offsets of N1 form a family of non-null h-slant ruled surfaces.

    Corollary 5.2. Let N1 and N2 form a Mannheim offset. Then N1 is a non-null q-slant (or a-slant) ruled surface if and only if N2 is a non-null h-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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