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Research article

Sweeping surfaces generated by involutes of a spacelike curve with a timelike binormal in Minkowski 3-space

  • Received: 14 November 2024 Revised: 02 January 2025 Accepted: 07 January 2025 Published: 16 January 2025
  • MSC : 53A04, 53A05

  • In this paper, we studied the geometry of sweeping surfaces generated by the involutes of spacelike curves with a timelike binormal in Minkowski 3-space E31. First, we investigated the singularity concept, and the mean and Gaussian curvatures of these surfaces. Then, we provided the requirements for the surface to be developable (flat) and minimal. We also determined the sufficient and necessary conditions for the parameter curves of these surfaces to be geodesic and asymptotic. Moreover, we analyzed these surfaces when the parameter curves are lines of curvature on the surface. Finally, the examples of these surfaces were given and their corresponding figures were drawn.

    Citation: Özgür Boyacıoğlu Kalkan, Süleyman Şenyurt, Mustafa Bilici, Davut Canlı. Sweeping surfaces generated by involutes of a spacelike curve with a timelike binormal in Minkowski 3-space[J]. AIMS Mathematics, 2025, 10(1): 988-1007. doi: 10.3934/math.2025047

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  • In this paper, we studied the geometry of sweeping surfaces generated by the involutes of spacelike curves with a timelike binormal in Minkowski 3-space E31. First, we investigated the singularity concept, and the mean and Gaussian curvatures of these surfaces. Then, we provided the requirements for the surface to be developable (flat) and minimal. We also determined the sufficient and necessary conditions for the parameter curves of these surfaces to be geodesic and asymptotic. Moreover, we analyzed these surfaces when the parameter curves are lines of curvature on the surface. Finally, the examples of these surfaces were given and their corresponding figures were drawn.



    Sweeping surfaces play an important role in both theoretical geometric research and applied fields, particularly in geometric modeling, kinematics, robotics, and computer graphics. A sweeping surface is constructed by the movement of a given planar curve, known as the profile curve (generatrix), along a predetermined path, called the spine (trajectory) curve, where the movement of the plane curve always occurs in the direction of the normal to the plane [1]. Given a profile curve P(u), and a spine curve T(v), the parametric representation of a sweeping surface is given as S(u,v)=T(v)+M(v)P(u), where M(v) is the so-called transformation matrix. That is, the profile curve can be rotated and scaled depending on the parameter of the trajectory curve. Due to the nature of this parametric form of sweeping surfaces, several types of other well-known surfaces, such as ruled (a profile curve of straight lines) canal (or specifically pipe), tubular, swung (surface of revolutions), and string surfaces can be formed by carefully examining the transformation matrix M(v) [2,3].

    However, in most of the cases, the transformation matrix is considered to be an identity matrix for simplicity, and many researchers use a local orthonormal frame moving along the trajectory curve. For example, in [4], the possibility of the construction of sweeping surfaces was examined by means of the Frenet frame and its modified version in Euclidean space. Sweeping surfaces according to the Darboux frame and rotation minimizing Darboux frame were studied in [5,6], respectively. Further, by considering Bishop frames, the characterizations of sweeping surfaces can be found in [7], and the references therein. By utilizing the modified orthogonal frames, new sweeping surfaces were discussed in the very recent study [18].

    Recent studies show that the concept of sweeping surfaces has been extended to the non-Euclidean spaces, namely Lorentz-Minkowski spaces, where the theory of curves and surfaces in these spaces were well-studied in [8,9]. For example, timelike sweeping surfaces were studied in [10], whereas the devolopable spacelike sweeping surfaces were discussed in [11].

    Additionally, researchers also benefited from the theory of associated curves to construct sweeping surfaces. The involute-evolute curve pair is one of such associated curves where the theory of these curves can be found in [12,13] for Euclidean space, and in [14,15] for Minkowski space. By combining the concepts of associated curve and sweeping surfaces, the involutive sweeping surfaces according to the Frenet frame in Euclidean space were defined and characterized in [16].

    Apart from these, geodesic curves, asymptotic curves, and the line of curvature are important characteristic curves that lie on the surface and have been instrumental in surface analysis. The geodesic curve is defined as the shortest distance between two points on a surface. An asymptotic curve is a curve on a surface where its tangent vector at every point lies in the direction where the normal curvature is zero. Lastly, the line of curvature is a surface curve if all tangent vectors always points along with a principal direction. The studies on designing the surfaces on which a given specific curve lies as a characteristic curve can be found in [19,20,21,22] for Euclidean space and in [23,24,25] for Minkowski 3-space.

    Motivated by the given literature, in this study, the sweeping surfaces generated by the involutes of spacelike curves with a timelike binormal and spacelike Darboux vector in Minkowski 3-space are examined. The singularity, mean curvature and Gaussian curvature of the surfaces are discussed and the necessary relations for the new sweeping surfaces to be classified as developable and minimal are provided. The necessary and sufficient conditions for the parameter curves on these surfaces to be geodesic and asymptotic are also obtained. In addition, these surfaces are studied when the parameter curves are lines of curvature. Finally, some illustrative examples of these new types of sweeping surfaces are studied.

    Let R3 be a 3-dimensional real vector space. For any p=(p1,p2,p3) and q=(q1,q2,q3)R3, the Lorentzian inner product of p and q is defined by

    p,qL=p1q1+p2q2p3q3. (2.1)

    The Minkowski 3-space denoted by E31=(R3,,L) is defined as the real vector space R3 with the Lorentzian metric. The non-zero vector qE31 is spacelike, lightlike, or timelike if q,qL>0, q,qL=0, or q,qL<0, respectively. The norm of the vector qE31 is defined by q=|q,qL|. For any two vectors p,qE31, the vector product is defined by

    p×q=|e1e2e3p1p2p3q1q2q3|=(p3q2p2q3,p1q3p3q1,p1q2p2q1),

    where e1,e2,e3 is the canonical basis of E31 [9,12,17].

    Let φ(s), sI=[0,I], be a unit speed 3D spacelike curve with a timelike binormal in E31 with curvature κ(s) and torsion τ(s). Consider the Serret-Frenet frame {h1(s),h2(s),h3(s)} associated with the curve φ(s), and then the Serret-Frenet formulae is given by

    (h1(s)h2(s)h3(s))=(0κ(s)0κ(s)0τ(s)0τ(s)0)(h1(s)h2(s)h3(s))=ω(s)×(h1(s)h2(s)h3(s)),

    where ω(s)=τ(s)h1(s)κ(s)h3(s) defines the Darboux vector of the Serret-Frenet frame.

    The vectors h1(s)=φ(s),h2(s)=φ(s)/φ(s), and h3(s)=h1(s)×h2(s) are called the unit tangent vector, the principal normal vector, and the binormal vector, respectively. The Serret-Frenet vector fields satisfy the following relations [17]:

    h1,h1L=1,h2,h2L=1,h3,h3L=1,h1×h2=h3,h2×h3=h1,h3×h1=h2.

    Also, we have:

    a) If |τ|>|κ|, then ω is a spacelike vector and we can write

    κ=ωsinhϕτ=ωcoshϕ,ω,ωL=ω2=τ2κ2. (2.2)

    b) If |τ|<|κ|, then ω is a timelike vector and we can write

    κ=ωcoshϕτ=ωsinhϕ,ω,ωL=ω2=κ2τ2, (2.3)

    where ϕ=(ω,h3) [14].

    Let φ(s) and ˉφ(s), sI, be two curves such that ˉφ intersects the tangents of φ orthogonally. Then ˉφ is called an involute of φ. An involute of a curve φ(s) with arc length s is given by

    ˉφ(s)=φ(s)+λh1(s), (2.4)

    where λ=cs0 and c is a real constant [12,13].

    If φ(s) is a unit speed spacelike curve with a timelike binormal, then the involute curve ˉφ(s) must be a spacelike curve with a spacelike or timelike binormal. On the other hand, the relations between the Frenet frames of involute curve ˉφ and evolute curve φ are given as follows[14]:

    a) If ω is a spacelike vector:

    [h1h2h3]=[010sinhϕ0coshϕcoshϕ0sinhϕ][h1h2h3]. (2.5)

    b) If ω is a timelike vector:

    [h1h2h3]=[010coshϕ0sinhϕsinhϕ0coshϕ][h1h2h3]. (2.6)

    The parametric equation of the sweeping surface along the spine curve φ(s) can be given as follows:

    η(s,t)=φ(s)+H(s)x(t), (2.7)

    where x(t)=(0,x1(t),x2(t))T is called the planar profile (cross section); and T represents transposition, with another parameter tIR. The semiorthogonal matrix H(s)={h1(s),h2(s),h3(s)} represents a moving frame along φ(s) [4].

    Let η(s,t) be a sweeping surface in E31. Then the various definitions and fundamental concepts relevant to sweeping surfaces can be presented as follows:

    The normal vector is U(s,t)=ηs×ηtηs×ηt, where ηs=ηs and ηt=ηt.

    The Gaussian and mean curvatures of η(s,t) are defined by, respectively,

    K=εegf2EGF2,H=εEg2Ff+Ge2(EGF2), (2.8)

    where

    E=ηs,ηsL,F=ηs,ηtL,G=ηt,ηtL,e=ηss,UL,f=ηst,UL,g=ηtt,UL,

    and U(s,t),U(s,t)L=ε=±1 [8,9].

    A surface in Minkowski 3-space E31 is called a spacelike (resp. timelike) surface if the induced metric on the surface is a positive (resp. negative) definite Riemannian metric, which means the normal vector on the spacelike surface is timelike and the normal vector on the timelike surface is a spacelike vector. The surface is spacelike if EGF2>0 and it is timelike if EGF2<0 [8,9].

    Let ˉφ(s) be the involute of a regular 3D spacelike curve φ(s) with a timelike binormal and {h1(s),h2(s),h3(s)} is its Frenet frame in Minkowski 3-space E31. Also, consider the spacelike Darboux vector ω and the non-null planar profile curve. For the involute curve ˉφ(s) as a spine curve and a unit speed planar profile curve x(t)=(0,x1(t),x2(t))T, the sweeping surface can be given as

    ˉη(s,t)=ˉφ(s)+H(s)x(t)=ˉφ(s)+h2(s)x1(t)+h3x2(t), (3.1)

    where H(s)={h1(s),h2(s),h3(s)} is an orthogonal matrix representing a moving frame along the involute curve ˉφ(s). Then using Eqs (2.4) and (2.5), the equation of the sweeping surface ˉη(s,t) can be rearranged as follows:

    ˉη(s,t)=ˉφ(s)+H(s)x(t)=ˉφ(s)+h2(s)x1(t)+h3(s)x2(t)=φ(s)+λh1(s)+x1(t)(sinhϕh1(s)coshϕh3(s))+x2(t)(coshϕh1(s)+sinhϕh3(s)).

    If we take the partial derivatives of ˉφ(s) with respect to s and t, respectively, and using Eq (2.2), we obtain the partial differentiation with respect to s and t as follows:

    ˉηs=(x1ϕcoshθx2ϕsinhϕ)h1+ρh2+(x1ϕsinhϕ+x2ϕcoshϕ)h3,ˉηt=(x1sinhϕx2coshϕ)h1+(x1coshϕ+x2sinhϕ)h3, (3.2)

    where ρ(s,t)=λκx1ω.

    The second-order partial derivatives of ˉη(s,t) are found by

    ˉηss=((x1ϕcoshϕx2ϕsinhϕ)ρκ)h1+(x2ϕω+ρs)h2+((x1ϕsinhϕ+x2ϕcoshϕ)+ρτ)h3,ˉηst=(x1ϕcoshϕx2ϕsinhϕ)h1+ρth2+(x1ϕsinhϕ+x2ϕcoshϕ)h3,ˉηtt=(x1sinhϕx2coshϕ)h1+(x2sinhϕx1coshϕ)h3, (3.3)

    where ρs=κ+λκ and ρt=x1ω.

    By taking the vector product of ˉηs and ˉηt, we get

    ˉηs×ˉηt=ρ(x1coshϕx2sinhϕ)h1+mh2ρ(x1sinhϕx2coshϕ)h3, (3.4)

    where

    m(s,t)=ϕ(x2x2x1x1). (3.5)

    Remark 3.1 If ϕ=arctanh(κτ) is a constant angle, then the spacelike curve φ(s) with a timelike binormal is a helix and Eq (3.5) gives m(s,t)=0.

    Now, consider the cases when the planar profile curve is timelike and spacelike.

    Case 1. Let the planar profile curve x(t)=(0,x1(t),x2(t))T be a timelike curve, which means that x12x22=1.

    Hence, Eq (3.4) gives

    ˉηs×ˉηt=|ρ2(x12x22)+m2|=|m2ρ2|. (3.6)

    From Eqs (3.4) and (3.6), we can get the unit normal vector of ˉη(s,t) as follows:

    U(s,t)=ρ(x1coshϕx2sinhϕ)h1+mh2ρ(x1sinhϕx2coshϕ)h3|m2ρ2|. (3.7)

    By using the equations in (3.2), the components of the first fundamental form are obtained by

    E=(x21x22)ϕ2+ρ2,F=ϕ(x1x2x1x2),G=x12+x22=1. (3.8)

    Then by the aid of Eq (3.8), we obtain EGF2=ρ2m2. Hence, the surface is spacelike if ρ2m2>0 and the surface is timelike if ρ2m2<0.

    Thus, we can give the following theorem.

    Theorem 3.2. Let ˉφ(s) be the involute of a unit speed spacelike curve φ(s) with a timelike binormal and spacelike ω Darboux vector in E31. The sweeping surface ˉη(s,t) generated by the Frenet frame of ˉφ(s) has singularity on the point p=(s0,t0) if

    m(s0,t0)=±ρ(s0,t0). (3.9)

    Proof. Equation (3.6) shows that ˉη(s,t) has a singularity at p=(s0,t0) if it satisfies

    ˉηs×ˉηt=|m2(s0,t0)ρ2(s0,t0)|.

    Using this equation, we get

    m2(s0,t0)ρ2(s0,t0)=0,

    which implies that m(s0,t0)=±ρ(s0,t0).

    Theorem 3.3. Let ˉφ(s) be the involute of a unit speed spacelike curve φ(s) with a timelike binormal and spacelike ω Darboux vector in E31. The Gaussian and mean curvatures of the sweeping surface ˉη(s,t) generated by the Frenet frame of ˉφ(s) are, respectively,

    K=ε|m2ρ2|(ρ2m2)[ρ(ρ(x1x1x2x2)ϕ+ρ(x1x2x2x1)ϕ2ρ2x2ω+m(x2ϕω+ρs))(x1x2x1x2)(ρtmρϕ)2],H=ε2|m2ρ2|(ρ2m2)[ρ((x21x22)ϕ2+ρ2))(x1x2x1x2)+ρ(x1x1x2x2)ϕ+ρ(x1x2x2x1)ϕ2ρ2x2ω+m(x2ϕω+ρs)2ϕ(x2x1x1x2)(ρtmρϕ)]. (3.10)

    Proof. From Eq (3.3), we can compute the components of the second fundamental form as

    e=1|m2ρ2|(ρ(x1x1x2x2)ϕ+ρ(x1x2x2x1)ϕ2ρ2x2ω+m(x2ϕω+ρs)),f=1|m2ρ2|(mρtρϕ),g=ρ|m2ρ2|(x1x2x2x1), (3.11)

    where ρs=λκκ and ρt=x1ω. Thus, by substituting (3.8) and (3.11) into (2.8), the Gaussian curvature and the mean curvature of the sweeping surface ˉη(s,t) can be obtained as in (3.10).

    Theorem 3.4. Let ˉφ(s) be the involute of a unit speed spacelike curve φ(s) with a timelike binormal and spacelike ω Darboux vector in E31. The parameter curves of the sweeping surface ˉη(s,t) generated by the Frenet frame of ˉφ(s) are lines of curvature if and only if

    ϕ=constantorx1x2=constantϕ=mρtρ. (3.12)

    Proof. The parameter curves of the sweeping surface ˉη(s,t) are lines of curvature if F=f=0. Then, from Eqs (3.8) and (3.11), we get

    ϕ(x2x1x1x2)=0,andmρtρϕ=0. (3.13)

    Let the angle ϕ=arctanh(κτ) be constant. Then, the equations in (3.13) are satisfied. In this case, as κτ is a constant, φ(s) is a helix. If x1x2 is a constant, then the first equality in (3.13) is provided. If ϕ=mρtρ, then the second equality in (3.13) is provided. These results show that the conditions in (3.12) must be supplied in order to satisfy the two equations in (3.13). It is obvious that the converse is true.

    Theorem 3.5. Let ˉφ(s) be the involute of a unit speed spacelike curve φ(s) with a timelike binormal and ˉη(s,t) is a regular sweeping surface generated by the Frenet frame of ˉφ(s) in E31.

    i) The s parameter curve of ˉη(s,t) is an asymptotic curve if

    ρx1(x1ϕx2ϕ2)+ρx2(x1ϕ2x2ϕρω)+m(x2ϕω+ρs)=0. (3.14)

    ii) The t parameter curve of ˉη(s,t) is an asymptotic curve if

    ρ=0,orx2x1x1x2=0. (3.15)

    Proof. i) The s parameter curve of ˉη(s,t) is an asymptotic curve if ηss,UL=0. Then, by using (3.3) and (3.7), we obtain (3.14).

    ii) The t parameter curve of ˉη(s,t) is an asymptotic curve if ηtt,UL=0. Then, by using (3.3) and (3.7), equation ρ(x1x2x1x2)=0 gives (3.15).

    Theorem 3.6. Let ˉφ(s) be the involute of a unit speed spacelike curve φ(s) with a timelike binormal and ˉη(s,t) is a regular sweeping surface generated by the Frenet frame of ˉφ(s) in E31.

    i) The s parameter curve of ˉη(s,t) is a geodesic curve if

    m((x2ϕcoshϕx1ϕsinhϕ)+ρτ)+ρ(x2ϕω+ρs)(x1sinhϕx2coshϕ)=0,ρx1(x1ϕ2x2ϕ)+ρx2(x1ϕx2ϕ2)ρ2x1ω=0,ρ(x2ϕω+ρs)(x1coshϕx2sinhϕ)m((x1ϕcoshϕx2ϕsinhϕ)ρκ)=0. (3.16)

    ii) The t parameter curve of ˉη(s,t) is a geodesic curve if

    m=0orx1coshϕx2sinhϕ=0,ρ=0orx2x2x1x1=0,x2coshϕx1sinhϕ=0. (3.17)

    Proof. i) The s parameter curve of ˉη(s,t) is a geodesic curve if U×ˉηss=0. Then, by using (3.3) and (3.7),

    {m((x2ϕcoshϕx1ϕsinhϕ)+ρτ)+ρ(x2ϕω+ρs)(x1sinhϕx2coshϕ)}h1+{ρx1(x1ϕ2x2ϕ)+ρx2(x1ϕx2ϕ2)ρ2x1ω}h2+{ρ(x2ϕω+ρs)(x1coshϕx2sinhϕ)m((x1ϕcoshϕx2ϕsinhϕ)ρκ)}h3=0,

    satisfying the equations in (3.16).

    ii) The t parameter curve of ˉη(s,t) is a geodesic curve if U×ˉηtt=0. Then, by using (3.3) and (3.7),

    U×ˉηtt=(m(x1coshϕx2sinhϕ))h1+(ρ(x2x2x1x1))h2+(m(x2coshϕx1sinhϕ))h3=0,

    satisfying the equations in (3.17).

    Now, we assume that the parameter curves of the sweeping surface ˉη(s,t) are its lines of curvature. For simplicity, we suppose that ρ=λκx1ω>0. In this case, ˉηs×ˉηt=|ρ2|=ρ0, which means that ˉη(s,t) is a regular surface. Let us begin by examining the singular points of the sweeping surface ˉη(s,t).

    Theorem 3.7. Let ˉφ(s) be the involute of a unit speed spacelike curve φ(s) with a timelike binormal and the parameter curves of the sweeping surface ˉη(s,t) generated by the Frenet frame of ˉφ(s) are its lines of curvature in E31. ˉη(s,t) has singularity on the point p=(s0,t0) if

    x1(s0,t0)=(cs0)κ(s0)τ2(s0)κ2(s0).

    Proof. ˉη(s,t) has singularity on the point p=(s0,t0) if ˉηs×ˉηt(s0,t0)=0. Then, from Remark 3.1 and (3.9), the equation can be expressed as follows:

    ˉηs×ˉηt(s0,t0)=ρ(s0,t0)=0(cs0)κ(s0)x1(t0)τ(s0)κ(s0)=0x1(t0)=(cs0)κ(s0)τ(s0)κ(s0).

    Theorem 3.8. Let ˉφ(s) be the involute of a unit speed spacelike curve φ(s) with a timelike binormal and the parameter curves of the sweeping surface ˉη(s,t) generated by the Frenet frame of ˉφ(s) are its lines of curvature in E31. Then, the Gaussian and mean curvatures of ˉη(s,t) are, respectively,

    K=εx2ω(x2x1x1x2)ρ,H=ερ(x1x2x1x2)x2ω2ρ. (3.18)

    Proof. By utilizing Eqs (3.8), (3.11), and (3.12), we can determine the fundamental coefficients of the surface of ˉη(s,t) as

    E=ρ2,F=0,G=1,e=ρx2ω,f=0,g=x1x2x1x2. (3.19)

    By substituting these equations into Eq (2.8), the equations can be acquired.

    Theorem 3.9. Let ˉφ(s) be the involute of a unit speed spacelike curve φ(s) with a timelike binormal and the parameter curves of the sweeping surface ˉη(s,t) generated by the Frenet frame of ˉφ(s) are its lines of curvature in E31. Then, the principal curvatures of ˉη(s,t) are

    k1=x2ωρ,k2=x1x2x2x1. (3.20)

    Proof. Let the parameter curves of the sweeping surface ˉη(s,t) be lines of curvature. Then, the principal curvatures of ˉη(s,t) can be obtained as follows:

    k1=eE=x2ωρ,k2=gG=x1x2x2x1. (3.21)

    Theorem 3.10. Let ˉφ(s) be the involute of a unit speed spacelike curve φ(s) with a timelike binormal and the parameter curves of the sweeping surface ˉη(s,t) generated by the Frenet frame of ˉφ(s) are its lines of curvature in E31. Then, the relation between the Gaussian curvature K and the mean curvature H of ˉη(s,t) is given by

    H=12(εg+Kg). (3.22)

    Proof. By utilizing Eqs (3.18) and (3.19), Eq (3.22) is obtained.

    Theorem 3.11. Let ˉφ(s) be the involute of a unit speed spacelike curve φ(s) with a timelike binormal and the parameter curves of the sweeping surface ˉη(s,t) generated by the Frenet frame of ˉφ(s) are its lines of curvature in E31. Then, ˉη(s,t) is a flat surface if

    x2=constant,ork2=0. (3.23)

    Proof. The surface ˉη(s,t) is a flat surface when the Gaussian curvature vanishes. Hence, Eq (3.18) gives

    x2ω(x2x1x1x2)=0.

    Since ω0, by using Eq (3.20), we obtain the equations in (3.23).

    Theorem 3.12. Let ˉφ(s) be the involute of a unit speed spacelike curve φ(s) with a timelike binormal and the parameter curves of the sweeping surface ˉη(s,t) generated by the Frenet frame of ˉφ(s) are its lines of curvature in E31. Then ˉη(s,t) is a minimal surface if

    x2=ρgω. (3.24)

    Proof. The surface ˉη(s,t) is a flat surface when the mean curvature vanishes. Hence, Eq (3.18) gives

    ρ(x1x2x1x2)x2ω=0.

    By using Eq (3.19), we obtain Eq (3.24).

    In the next part of this study, we examine the sweeping surfaces for a spacelike profile curve.

    Case 2. Let the planar profile curve x(t)=(0,x1(t),x2(t))T be a spacelike curve which means that x21x22=1.

    By using Eq (3.4), we get

    ˉηs×ˉηt=|ρ2(x12x22)+m2|=ρ2+m2. (3.25)

    From Eqs (3.4) and (3.25), we can get the unit normal vector of ˉη(s,t) as

    U(s,t)=ρ(x1coshϕx2sinhϕ)h1+mh2ρ(x1sinhϕx2coshϕ)h3m2+ρ2. (3.26)

    By using Eq (3.2), the components of the fundamental forms are obtained by

    E=(x21x22)ϕ2+ρ2,F=ϕ(x1x2x1x2),G=x12+x22=1. (3.27)

    Then by the aid of Eq (3.27), we obtain EGF2=(ρ2+m2)<0. Hence, the surface is timelike.

    Hence, we can give the following theorem.

    Theorem 3.13. Let ˉφ(s) be the involute of a unit speed spacelike curve φ(s) with a timelike binormal and spacelike ω Darboux vector in E31. The sweeping surface ˉη(s,t) generated by the Frenet frame of ˉφ(s) has singularity on the point p=(s0,t0) if

    m(s0,t0)=ρ(s0,t0)=0. (3.28)

    Proof. Equation (3.25) gives that ˉη(s,t) has a singularity at p=(s0,t0) if it satisfies

    ˉηs×ˉηt=ρ2(s0,t0)+m2(s0,t0)=0.

    Using this equation, we get

    ρ2(s0,t0)+m2(s0,t0)=0m(s0,t0)=ρ(s0,t0)=0.

    Theorem 3.14. Let ˉφ(s) be the involute of a unit speed spacelike curve φ(s) with a timelike binormal and spacelike ω Darboux vector in E31. The Gaussian and mean curvatures of the sweeping surface ˉη(s,t) generated by the Frenet frame of ˉφ(s) are, respectively,

    K=1(m2+ρ2)2[ρ(ρ(x1x1x2x2)ϕρ(x1x2x2x1)ϕ2+ρ2x2ωm(x2ϕω+ρs))(x1x2x1x2)+(ρtm+ρϕ)2],H=12(m2+ρ2)3/322[ρ((x21x22)ϕ2+ρ2)(x1x2x1x2)+ρ(x1x1x2x2)ϕ+ρ(x1x2x2x1)ϕ2ρ2x2ω+m(x2ϕω+ρs)+2ϕ(x2x1x1x2)(ρtm+ρϕ)]. (3.29)

    Proof. From Eqs (3.3) and (3.26), we can compute the components of the second fundamental form as

    e=1m2+ρ2(ρ(x1x1x2x2)ϕ+ρ(x1x2x2x1)ϕ2ρ2x2ω+m(x2ϕω+ρs)),f=1m2+ρ2(mρt+ρϕ),g=ρm2+ρ2(x1x2x2x1). (3.30)

    Thus, by substituting (3.27) and (3.30) into (2.8), the Gaussian curvature and the mean curvature of sweeping surface ˉη(s,t) can be obtained as in (3.29).

    Theorem 3.15. Let ˉφ(s) be the involute of a unit speed spacelike curve φ(s) with a timelike binormal and spacelike ω Darboux vector in E31. The parameter curves of the sweeping surface ˉη(s,t) generated by the Frenet frame of ˉφ(s) are lines of curvature if and only if

    ϕ=constantorx1x2=constantϕ=mρtρ. (3.31)

    Proof. The parameter curves of the sweeping surface ˉη(s,t) are lines of curvature if F=f=0. Then, using the same computations as in the proof of Theorem 3.4, from Eqs (3.27) and (3.30), we obtain the equations in (3.31).

    Theorem 3.16. Let ˉφ(s) be the involute of a unit speed spacelike curve φ(s) with a timelike binormal and ˉη(s,t) is a regular sweeping surface generated by the Frenet frame of ˉφ(s) in E31.

    i) The s parameter curve of ˉη(s,t) is an asymptotic curve if

    ρx1(x1ϕx2ϕ2)+ρx2(x1ϕ2x2ϕρω)+m(x2ϕω+ρs)=0. (3.32)

    ii) The t parameter curve of ˉη(s,t) is an asymptotic curve if

    ρ=0,orx1x2x1x2=0. (3.33)

    Proof. Using the same computations as in the proof of Theorem 3.5, by using (3.3) and (3.26), the equations in (3.32) and (3.33) are obtained.

    Theorem 3.17. Let ˉφ(s) be the involute of a unit speed spacelike curve φ(s) with a timelike binormal and ˉη(s,t) is a regular sweeping surface generated by the Frenet frame of ˉφ(s) in E31.\\ i) The s parameter curve of ˉη(s,t) is a geodesic curve if

    m((x2ϕcoshϕx1ϕsinhϕ)+ρτ)+ρ(x2ϕω+ρs)(x1sinhϕx2coshϕ)=0,ρx1(x1ϕ2x2ϕ)+ρx2(x1ϕx2ϕ2)ρ2x1ω=0,+ρ(x2ϕω+ρs)(x1coshϕx2sinhϕ)m((x1ϕcoshϕx2ϕsinhϕ)ρκ)=0. (3.34)

    ii) The t parameter curve of ˉη(s,t) is a geodesic curve if

    m=0orx1coshϕx2sinhϕ=0,ρ=0orx2x2x1x1=0,x2coshϕx1sinhϕ=0. (3.35)

    Proof. Using the same computations as the proof of Theorem 3.6, by using (3.3) and (3.26), the equations in (3.34) and (3.35) are obtained.

    Now, assume that the parameter curves of the sweeping surface ˉη(s,t) are its lines of curvature. For simplicity, we take ρ=λκx1ω>0. In this case, ˉηs×ˉηt=ρ2=ρ0, which means that ˉη(s,t) is a regular surface.

    Theorem 3.18. Let ˉφ(s) be the involute of a unit speed spacelike curve φ(s) with a timelike binormal and the parameter curves of the sweeping surface ˉη(s,t) generated by the Frenet frame of ˉφ(s) are its lines of curvature in E31. ˉη(s,t) has singularity on the point p=(s0,t0) if

    x1(s0,t0)=(cs0)κ(s0)τ2(s0)κ2(s0). (3.36)

    Proof. ˉη(s,t) has singularity on the point p=(s0,t0) if ˉηs×ˉηt(s0,t0)=0. Then, from Remark 3.1 and Eq (3.28), ρ(s0,t0)=0 satisfies (3.36).

    Theorem 3.19. Let ˉφ(s) be the involute of a unit speed spacelike curve φ(s) with a timelike binormal and spacelike ω Darboux vector in E31. The Gaussian and mean curvatures of the sweeping surface ˉη(s,t) generated by the Frenet frame of ˉφ(s) are, respectively,

    K=x2ω(x1x2x1x2)ρ,H=ρ(x1x2x1x2)x2ω2ρ. (3.37)

    Proof. By utilizing Eqs (3.27), (3.30), and (3.31), we can determine the fundamental coefficients of the surface of ˉη(s,t) as

    E=ρ2,F=0,G=1,e=ρx2ω,f=0,g=x1x2x1x2. (3.38)

    By substituting these equations into Eq (2.8), the equations can be acquired.

    Theorem 3.20. Let ˉφ(s) be the involute of a unit speed spacelike curve φ(s) with a timelike binormal and the parameter curves of the sweeping surface ˉη(s,t) generated by the Frenet frame of ˉφ(s) are its lines of curvature in E31. Then, the principal curvatures of ˉη(s,t) are

    k1=x2ωρ,k2=x1x2x1x2. (3.39)

    Proof. Let the parameter curves of the sweeping surface ˉη(s,t) be lines of curvature. Then, the principal curvatures of ˉη(s,t) are

    k1=eE=x2ωρ,k2=gG=x1x2x1x2. (3.40)

    Theorem 3.21. Let ˉφ(s) be the involute of a unit speed spacelike curve φ(s) with a timelike binormal and the parameter curves of the sweeping surface ˉη(s,t) generated by the Frenet frame of ˉφ(s) are its lines of curvature in E31. Then, the relation between the Gaussian curvature K and the mean curvature H of ˉη(s,t) is given by

    H=12(gKg). (3.41)

    Proof. By utilizing Eqs (3.37) and (3.38), Eq (3.41) is obtained.

    Theorem 3.22. Let ˉφ(s) be the involute of a unit speed spacelike curve φ(s) with a timelike binormal and the parameter curves of the sweeping surface ˉη(s,t) generated by the Frenet frame of ˉφ(s) are its lines of curvature in E31. Then ˉη(s,t) is a flat surface if

    x2=constant,ork2=0. (3.42)

    Proof. The surface ˉη(s,t) is a flat surface when the Gaussian curvature vanishes. Hence, Eq (3.37) gives

    x2ω(x2x1x1x2)=0.

    Since ω0, by using Eq (3.39), the equations in (3.42) are obtained.

    Theorem 3.23. Let ˉφ(s) be the involute of a unit speed spacelike curve φ(s) with a timelike binormal and the parameter curves of the sweeping surface ˉη(s,t) generated by the Frenet frame of ˉφ(s) are its lines of curvature in E31. Then, ˉη(s,t) is a minimal surface if

    x2=ρgω. (3.43)

    Proof. The surface ˉη(s,t) is a flat surface when the mean curvature vanishes. Hence, Eq (3.37) gives

    ρ(x1x2x1x2)+x2ω=0.

    By using Eq (3.38), we obtain Eq (3.43).

    Example 3.24. Let us consider a spacelike curve with a timelike binormal parametrized as

    φ(s)=(53s,49cosh3s,49sinh3s). (3.44)

    Then, the Frenet vectors of φ(s) are given by

    h1(s)=(53,43sinh3s,43cosh3s),h2(s)=(0,cosh3s,sinh3s),h3(s)=(43,53sinh3s,53cosh3s),κ(s)=4,τ(s)=5,ω(s)=τ(s)h1(s)κ(s)h3(s)=(3,0,0). (3.45)

    Hence, using (2.2), we obtain sinhϕ(s)=43 and coshϕ(s)=53. By putting c=0 in (2.4), we get the involute curve of φ(s) as

    ˉφ(s)=(0,49cosh3s43ssinh3s,49sinh3s43scosh3s), (3.46)

    with the Frenet vectors based on Eqs (2.7) and (3.1), the sweeping surfaces η1(s,t), and the ˉη1(s,t) generated by moving a timelike profile curve along φ(s) and ˉφ(s) expressed by the following equations:

    η1(s,t)=(53s,49cosh3s,49sinh3s)+(0,sinht,cosht)(5343sinh3s43cosh3s0cosh3ssinh3s4353sinh3s53cosh3s)=(53s+43cosht,49cosh3s+sinhtcosh3s+53coshtsinh3s,49sinh3s+sinhtsinh3s+53coshtcosh3s), (3.47)

    and

    ˉη1(s,t)=(0,49cosh3s43ssinh3s,49sinh3s43scosh3s)+(0,sinht,cosht)(0cosh3ssinh3s0sinh3scosh3s100)=(cosht,49cosh3s43ssinh3ssinhtsinh3s,49sinh3s43scosh3ssinhtcosh3s), (3.48)

    respectively. The curve φ(s), its involute ˉφ(s) as the trajectory curve, and a planar timelike profile curve are shown in Figure 1, whereas the surface η1(s,t) is shown in Figure 2, and the surface ˉη1(s,t) is shown in Figure 3. Finally, both surfaces η1(s,t) and ˉη1(s,t) are shown in Figure 4 where 1 and - 3 \leqslant t \leqslant 3 .

    Figure 1.  Timelike profile curve (dark green), the curve \varphi (s) (red), and its involute curve \bar \varphi (s) as the spine (trajectory) curve (blue).
    Figure 2.  Sweeping surface {\eta _1}(s, t) generated by \varphi (s) (red) with the timelike profile curve (dark green).
    Figure 3.  Sweeping surface {\bar \eta _1}(s, t) generated by \bar \varphi (s) (blue) with the timelike profile curve (dark green).
    Figure 4.  Sweeping surface {\eta _1}(s, t) generated by \varphi (s) (on the left) and the surface {\bar \eta _1}(s, t) generated by \bar \varphi (s) (on the right) with the timelike profile curve.

    The sweeping surfaces {\eta _2}(s, t) and {\bar \eta _2}(s, t) generated by moving a spacelike profile curve along \varphi (s) and \bar \varphi (s) are expressed by the following equations:

    \begin{equation} \begin{aligned} {\eta _2}(s, t) & = \left( {\frac{5}{3}s, \frac{4}{9}\cosh 3s, \frac{4}{9}\sinh 3s} \right) \hfill \\ & \qquad + (0, \cosh t, \sinh t)\left( {\begin{array}{*{20}{c}} {\frac{5}{3}}&{\frac{4}{3}\sinh 3s}&{\frac{4}{3}\cosh 3s} \\ 0&{\cosh 3s}&{\sinh 3s} \\ {\frac{4}{3}}&{\frac{5}{3}\sinh 3s}&{\frac{5}{3}\cosh 3s} \end{array}} \right) \\ & = \left( \begin{gathered} \frac{5}{3}s + \frac{4}{3}\sinh t, \frac{4}{9}\cosh 3s + \cosh t\cosh 3s + \frac{5}{3}\sinh t\sinh 3s, \hfill \\ \frac{4}{9}\sinh 3s + \cosh t\sinh 3s + \frac{5}{3}\sinh t\cosh 3s \hfill \\ \end{gathered} \right), \end{aligned} \end{equation} (3.49)

    and

    \begin{equation} \begin{aligned} {{\bar \eta }_2}(s, t) & = \left( {0, \frac{4}{9}\cosh 3s - \frac{4}{3}s\sinh 3s, \frac{4}{9}\sinh 3s - \frac{4}{3}s\cosh 3s} \right) \hfill \\ & \qquad + (0, \cosh t, \sinh t)\left( {\begin{array}{*{20}{c}} 0&{\mp \cosh 3s}&{\mp \sinh 3s} \\ 0&{ \mp \sinh 3s}&{\mp \cosh 3s} \\ { - 1}&0&0 \end{array}} \right) \\ & = \left( \begin{gathered} - \sinh t, \frac{4}{9}\cosh 3s - \frac{4}{3}s\sinh 3s \mp \cosh t\sinh 3s, \hfill \\ \frac{4}{9}\sinh 3s - \frac{4}{3}s\cosh 3s \mp \cosh t\cosh 3s \end{gathered} \right). \end{aligned} \end{equation} (3.50)

    The curve \varphi (s) , its involute \bar \varphi (s) as the trajectory curve, and a planar spacelike profile curve are shown in Figure 5, whereas the surface { \eta _2}(s, t) is shown in Figure 6, and the surface {\bar \eta _2}(s, t) is given in Figure 7. Lastly, both of the two surfaces {\eta _2}(s, t) and {\bar \eta _2}(s, t) are shown in Figure 8 where - 1 \leqslant s \leqslant 1 and - 3 \leqslant t \leqslant 3 .

    Figure 5.  Spacelike profile curve (turquoise), the curve \varphi (s) (red), and its involute \bar \varphi (s) as the spine (trajectory) curve (blue).
    Figure 6.  Sweeping surface {\eta _2}(s, t) generated by \varphi (s) (red) with the spacelike profile curve (turquoise).
    Figure 7.  Sweeping surface {\bar \eta _2}(s, t) generated by \bar \varphi (s) (blue) with the spacelike profile curve (turquoise).
    Figure 8.  Sweeping surface {\eta _2}(s, t) generated by \varphi (s) and the surface {\bar \eta _2}(s, t) generated by \bar \varphi (s) with the spacelike profile curve.

    This study analyzes the construction of sweeping surfaces generated by moving a non-null planar profile curve along the involute curve \bar \varphi (s) of the main curve \varphi (s) considered in Euclidean 3-space and now defined in Minkowski 3-space, taking into account the causal character of the curves. First, the sweeping surfaces generated by the involutes of spacelike curves with a timelike binormal and spacelike Darboux vector are defined in Minkowski 3-space E_1^3. Then, the singularity, and the mean and Gaussian curvatures of these surfaces are calculated and the conditions for these surfaces to be flat and minimal are examined. Also, the conditions for the parameter curves on the surface to be asymptotic and geodesic are investigated. Finally, the cases where the parameter curves are lines of curvature on the surface are obtained. Additionally, the examples of these surfaces are presented and their graphics are illustrated. Therefore, new surfaces are contributed to the literature of surface theory in Minkowski 3-space.

    Özgür Boyacıoğlu Kalkan: Methodology, conceptualization, writing-original draft preparation, supervision, writing, formal analysis, resources; Süleyman Şenyurt: Supervision, formal analysis, validation; Mustafa Bilici: Methodology, investigation, conceptualization, reviewing, editing; Davut Canlı: Investigation, formal analysis, software, validation, visualization. All authors have read and approved the final version of the manuscript for publication.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors declare no conflict of interest in this paper.



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