Research article Special Issues

Evolution of space curves and the special ruled surfaces with modified orthogonal frame

  • In this study, we investigate the evolution of space curves and the special ruled surfaces with modified orthogonal frame. Firstly, we recall the relations between the Serret-Frenet frame and the modified orthogonal frame. Then we investigate the evolution of space curves relative to these relations. Moreover, we determine the first and second fundamental forms, the Gaussian and mean curvatures of the tangent, normal, and binormal ruled surfaces according to the modified frame. In these regards, we obtain the characterizations of the minimal and developable ruled surfaces based on modified orthogonal frame.

    Citation: Kemal Eren, Hidayet Huda Kosal. Evolution of space curves and the special ruled surfaces with modified orthogonal frame[J]. AIMS Mathematics, 2020, 5(3): 2027-2039. doi: 10.3934/math.2020134

    Related Papers:

    [1] Atakan Tuğkan Yakut, Alperen Kızılay . On the curve evolution with a new modified orthogonal Saban frame. AIMS Mathematics, 2024, 9(10): 29573-29586. doi: 10.3934/math.20241432
    [2] Gülnur Şaffak Atalay . A new approach to special curved surface families according to modified orthogonal frame. AIMS Mathematics, 2024, 9(8): 20662-20676. doi: 10.3934/math.20241004
    [3] Samah Gaber, Asmahan Essa Alajyan, Adel H. Sorour . Construction and analysis of the quasi-ruled surfaces based on the quasi-focal curves in R3. AIMS Mathematics, 2025, 10(3): 5583-5611. doi: 10.3934/math.2025258
    [4] Yanlin Li, H. S. Abdel-Aziz, H. M. Serry, F. M. El-Adawy, M. Khalifa Saad . Geometric visualization of evolved ruled surfaces via alternative frame in Lorentz-Minkowski 3-space. AIMS Mathematics, 2024, 9(9): 25619-25635. doi: 10.3934/math.20241251
    [5] Emad Solouma, Ibrahim Al-Dayel, Meraj Ali Khan, Youssef A. A. Lazer . Characterization of imbricate-ruled surfaces via rotation-minimizing Darboux frame in Minkowski 3-space E31. AIMS Mathematics, 2024, 9(5): 13028-13042. doi: 10.3934/math.2024635
    [6] Ayman Elsharkawy, Ahmer Ali, Muhammad Hanif, Fatimah Alghamdi . Exploring quaternionic Bertrand curves: involutes and evolutes in E4. AIMS Mathematics, 2025, 10(3): 4598-4619. doi: 10.3934/math.2025213
    [7] Ayman Elsharkawy, Clemente Cesarano, Abdelrhman Tawfiq, Abdul Aziz Ismail . The non-linear Schrödinger equation associated with the soliton surfaces in Minkowski 3-space. AIMS Mathematics, 2022, 7(10): 17879-17893. doi: 10.3934/math.2022985
    [8] Maryam T. Aldossary, Rashad A. Abdel-Baky . Sweeping surface due to rotation minimizing Darboux frame in Euclidean 3-space E3. AIMS Mathematics, 2023, 8(1): 447-462. doi: 10.3934/math.2023021
    [9] Yanlin Li, Kemal Eren, Kebire Hilal Ayvacı, Soley Ersoy . Simultaneous characterizations of partner ruled surfaces using Flc frame. AIMS Mathematics, 2022, 7(11): 20213-20229. doi: 10.3934/math.20221106
    [10] Emad Solouma, Mohamed Abdelkawy . Family of ruled surfaces generated by equiform Bishop spherical image in Minkowski 3-space. AIMS Mathematics, 2023, 8(2): 4372-4389. doi: 10.3934/math.2023218
  • In this study, we investigate the evolution of space curves and the special ruled surfaces with modified orthogonal frame. Firstly, we recall the relations between the Serret-Frenet frame and the modified orthogonal frame. Then we investigate the evolution of space curves relative to these relations. Moreover, we determine the first and second fundamental forms, the Gaussian and mean curvatures of the tangent, normal, and binormal ruled surfaces according to the modified frame. In these regards, we obtain the characterizations of the minimal and developable ruled surfaces based on modified orthogonal frame.


    The surface theory is an indispensable research topic for scientists in the fields of differential geometry, physics, engineering, and designation. One type of well-known surfaces is the ruled surfaces which are obtained by the continuous movement of a line along the curve [1,2]. Various approaches have been put forward on ruled surfaces considering the characterizations, geometric and algebraic properties of ruled the surfaces. Ruled surfaces have applications in a number of domains including geometric modeling [5] or computer aided geometric designing [3,4]. Further, the technical advantages of ruled surfaces in the realization of free-form architecture and complex shapes can be seen in [6]. Such as surfaces, the theory of curves is also an important research topic regarding various disciplines. A lot of investigation has been done on the motion, evolution, and integrability of curves. Moreover, the geometric characterizations for these integrable curves and the evolution of inelastic plane curves have been studied widely [7,8,9,10,11]. Hasimoto [12] handled the motion of the vortex filament equation has been studied by and the time evolution of the curves has been obtained. It also has been proved that the vortex filament equation (smoke ring) is equal to the nonlinear Schrödinger equation. The motion of the curves is a predecessor part of soliton theory. The geometric applications of Bäcklund and Darboux transformations between curves have been discussed in soliton theory [12,13,14]. Many researchers have examined the surfaces obtained from these curves. For instance, the evolution of translation surfaces has been studied by [15]. Additionally, by considering the inextensible flows of curves, the developable surfaces and tangent developable surfaces have been considered in [16,17,18,19] and the evolution of special ruled surfaces according to the Serret-Frenet frame has been clarified in [20]. On the other hand, the Serret-Frenet frames are inadequate for studying analytic space curves of which curvatures have discrete zero points since the principal normal and binormal vectors may be discontinuous at zero points of the curvature. For the solution of this problem, Sasai [21] has introduced an orthogonal frame and obtained a formula, which corresponds to the Frenet-Serret equation. Recently, in Minkowski 3-space, the modified orthogonal frame with non-zero curvature and torsion of a space curve has been described by Bukcu and Karacan [22]. Then, using this the modified orthogonal frame, spherical curves, Mannheim curves and some special curves have been reconsidered [23,24,25]. In the light of recent events given above, the aim of this study is to study the evolution of analytic space curve according to the modified orthogonal frame and the geometric properties of special ruled surfaces generated by the motion of these curves.

    In Euclidean 3-space, Euclidean inner product is given by <,>=dx21+dx22+dx23 where x=(x1,x2,x3)E3. The norm of a vector xE3 is x=|<x,x>|. For any α curve, if α(s)=1, then α curve is unit speed curve in Euclidean space. The most well-known and used Frenet frame on a curve plays an important role in differential geometry. Let α be a space curve with respect to the arc-length s in Euclidean 3-space E3. t, n and b are tangent, principal normal and binormal unit vectors at each point α(s) of a curve α, respectively. Then there exists an orthogonal frame {t,n,b} which satisfies the Frenet-Serret equation

    t=κn,n=κt+τb,b=τn (2.1)

    where κ is the curvature, τ is the torsion.

    The fundamental theorem of regular curves states that if κ>0 and τ are differentiable functions then there exists a unit speed curve whose curvature and torsion are κ and τ, respectively [1]. However, the principal normal and binormal vectors are discontinuous at zero points of the curvature in general and the curvature is not always differentiable even if the curve is analytic. In that case, the formulation of the Frenet frame of a space curve generally established causes ambiguity for an analytical space curve at a point where the curvature vanishes, see Example 2.1.

    This problem was considered by Hord [26] and Sasai [21,27] for analytic space curves of which the curvatures have discrete zero points. With a simple but convenient approach, an orthogonal frame was introduced by Sasai [21]. Although this modified orthogonal frame seems like a Frenet frame with scaled normal and binormal vectors, it allows to use a new formula corresponding to the Frenet-Serret equation for the aforementioned case and is also useful for investigating analytic curves with singularities.

    Let α be an analytic curve of which curvature has discrete zero points in Euclidean 3-space. Under the assumption κ(s) of α is not identically zero, the elements of modified orthogonal frame are given by

    T=dαds,N=dTds,B=TN

    where s is the arc-length parameter and TN is the vector product of T and N.

    The relations between the Frenet frame {t,n,b} and modified orthogonal frame {T,N,B} at non-zero points of κ are

    T=t,N=κn,B=κb.

    In the course of time, this orthogonal frame is called the modified orthogonal frame [22]. The modified orthogonal frame {T,N,B} satisfies

    T,T=1,N,N=B,B=κ2,T,N=T,B=N,B=0.

    such that , is the Euclidean inner product. From these equations, the differentiation formula for the modified orthogonal frame {T,N,B} is given by

    T(s)=N(s),N(s)=κ2T(s)+κsκN(s)+τB(s),B(s)=τN(s)+κsκB(s), (2.2)

    where κs denotes the differentiation with respect to s and τ=det(α,α,α)κ2 is the torsion of α. Here, the essential quantities κ2 and τ are analytic in [21,22].

    Example 2.1. Let us consider a curve given by the parametric equation

    α(s)=(12s0cos(πt22)dt,12s0sin(πt22)dt,s2)

    which is a helical curve over clothoid (Cornu spiral or Euler spiral) [2] and has various applications in real life such as the highway, railway route design or roller coasters, etc. Here the components s0cos(πt22)dt and s0sin(πt22)dt are called Fresnel integrals. The elements of the Frenet trihedron of the curve α are obtained as

    t(s)=(12cos(πs22),12sin(πs22),12),
    n(s)=(s|s|sin(πs22),s|s|cos(πs22),0),
    b(s)=(s2|s|cos(πs22),s2|s|sin(πs22),s2|s|)

    and the curvature is κ(s)=π|s|2. Besides the curvature is not differentiable, the principal normal and binormal vectors are discontinuous at s=0 since n+n and b+b for n+=lims0+limn(s), n=lims0limn(s) and b+=lims0+b(s), b=lims0limb(s).

    Whenever the curvature is considered as a signed quantity κ(s)=πs2, the curve forms a symmetrical double spiral, see Figure 1.

    Figure 1.  Helical Curve over Clothoid.

    To prevent the occurrence of two reverse oriented principal normal vectors and binormal vectors, it is useful to refer to the modified orthogonal frame with unique elements

    T(s)=(12cos(πs22),12sin(πs22),12),
    N(s)=(πs2sin(πs22),πs2cos(πs22),0),
    B(s)=(πs2cos(πs22),πs2sin(πs22),πs2),

    under the assumption that the curvature κ(s) of α is not zero. Here the essential quantities are obtained as κ2=π2s22 and τ(s)=πs2.

    A curve α in Euclidean 3-space is a vector-valued function α(s,t)E3 where s is the arc-length parameter and t is the time parameter, then the equation of the vortex filament (smoke ring equation) is given by

    αt=αsαss, (3.1)

    where the subscripts indicate the partial differential. Let α be an analytic curve with curvature having discrete zero points. To at non-zero points of the curve, the time evolution of the modified orthogonal frame {T,N,B} can be written in matrix form as follows:

    [TNB]t=[0ηβηκκtγβγκκt][TNB] (3.2)

    where α, β and γ are smooth functions. By considering the curvature κ of the curve α is not identically zero and using the equations Tst=Tts,Nst=Nts,Bst=Bts, we obtain

    ηs=τβκsκη+κκt,βs=γτηκsκβ,γs=τtβ. (3.3)

    We suppose that the velocity according to the curve α is given by

    αt=dαdt=aT+bN+cB. (3.4)

    From equation αst=αts, we find the following equations

    0=asbκ2,η=a+bs+κsκbτc,β=τb+cs+κsκc (3.5)

    where a, b and c are the coefficients of the tangent, normal and binormal vectors of the velocity, respectively. Substituting the Eq. (3.5) into the second Equation of (3.3), we get

    γ=(τb+cs+κsκc)s+τ(a+bs+κsκbτc)+κsκ(τb+cs+κsκc). (3.6)

    For a solution of smoke ring equation, the velocity vector is given by

    αt=αsαss=B. (3.7)

    Thus, from the Eqs. (3.4) and (3.7), we get

    a=0,b=0,c=1. (3.8)

    Substituting the Eq. (3.8) into the Eqs. (3.5) and (3.6), we get

    η=τ,β=κsκ,γ=κssκτ2. (3.9)

    Thus, according to the modified orthogonal frame, the Eq. (3.9) represents the time evolution of the curve and the motion of the curve

    In this section, we study the tangent, normal and binormal ruled surfaces using the modified orthogonal frame along an analytic space curve. The parametric equation of the ruled surface is given by

    X(s,v)=α(s)+vl(s),

    where α(s) is called the base curve and l(s) is the director curve. If the curves α(s) and l(s) move with time t, then the equation of the ruled surface is as follows

    X(s,v,t)=α(s,t)+vl(s,t). (4.1)

    The ruled surface generated by the motion of the tangent vector T of a curve α is called the tangent ruled surface and the equation of this surface is represented by

    X(s,v,t)=α(s,t)+vT(s,t). (4.2)

    Example 4.1. Let us consider the helical curve over clothoid given in Example 2.1. Then the parametric equation of tangent ruled surface is

    φ(s,v)=(12(s0cos(πt22)dt+vcos(πs22)),12(s0sin(πt22)dt+vsin(πs22)),s+v2),

    see Figure 2.

    Figure 2.  Tangent Ruled Surface.

    The partial differentiations of the equations of the tangent ruled surface are

    Xs(s,v,t)=T+vN,Xv(s,v,t)=T, (4.3)

    By using the Eq. (4.3), we get the unit normal field of this surface as

    U=XsXvXsXv=Bκ. (4.4)

    The first fundamental form of the tangent ruled surface in Euclidean space is given by

    I=dX,dX=Xsds+Xvdv,Xsds+Xvdv=Eds2+2Fdsdv+Gdv2

    where the coefficients of the first fundamental form are

    E=Xs,Xs=1+vκ2,F=Xs,Xv=1,G=Xv,Xv=1. (4.5)

    From Eq. (4.3), the second derivatives are found and given as

    Xss(s,v,t)=vκ2T+(1+vκsκ)N+vτB,Xsv(s,v,t)=N,Xvv(s,v,t)=0. (4.6)

    The second fundamental form of the normal surface is given by

    II=dX,dU=dX,dU=Xsds+Xvdv,Usds+Uvdv=eds2+2fdsdv+gdt2

    where the coefficients of the second fundamental form are

    e=Xss,U=vτκ,f=Xsv,U=0,g=Xvv,U=0. (4.7)

    Corollary 4.1. The Gaussian and mean the curvatures of the tangent ruled surface X=X(s,v,t) are

    K=0, (4.8)
    H=τ2vκ, (4.9)

    respectively.

    Proof. From the Eqs. (4.5) and (4.7), we easily obtain the Gaussian curvature and the mean curvature, respectively, as follows

    K=egf2EGF2=0,
    H=12Eg2Ff+GeEGF2=τ2vκ

    From the Eqs. (4.8) and (4.9) the followings are obvious.

    Corollary 4.2.

    ⅰ. The tangent ruled surface is developable.

    ⅱ. The tangent ruled surface is minimal surface if τ=0.

    The ruled surface generated by the motion of the normal vector N of the curve α is called the normal ruled surface and the equation of this surface is

    X(s,v,t)=α(s,t)+vN(s,t). (4.10)

    Example 4.2. If we take the curve given in Example 2.1, then the parametric equations of normal ruled surfaces generated by normal vectors of Frenet frame and modified orthogonal frame are

    φ1(s,v)=(12s0cos(πt22)dtvs|s|sin(πs22),12s0sin(πt22)dt+vs|s|cos(πs22),s2)

    and

    φ2(s,v)=(12(s0cos(πt22)dtvπssin(πs22)),12(s0sin(πt22)dt+vπscos(πs22)),s2),

    respectively. The first normal ruled surface is generated by the normal vector of Frenet frame and it is discontinuous at s=0 and the second one is generated by the normal vector of the modified orthogonal frame, see Figure 3.

    Figure 3.  Normal Ruled Surfaces.

    The derivatives of the normal ruled surface with respect to s and v are

    Xs(s,v,t)=(1vκ2)T+vκsκN+vτB,Xv(s,v,t)=N, (4.11)

    respectively. Using Eq. (4.11), we get the unit normal field of this surface is found as

    U=XsXvXsXv=vτT+(1vκ2)B(1vκ2)2κ2+(vτ)2. (4.12)

    The first fundamental form of the normal ruled surface in Euclidean space is given by

    I=dX,dX=Xsds+Xvdv,Xsds+Xvdv=Eds2+2Fdsdv+Gdv2

    where the coefficients of the first fundamental form are

    E=Xs,Xs=(1vκ2)2+v2κs2+v2τ2κ2,F=Xs,Xv=vκsκ,G=Xv,Xv=κ2. (4.13)

    From Eq. (4.11), the second derivative is found as

    Xss(s,v,t)=3vκκsT+(1vκ2+vκssκvτ2)N+(vτs+2vτκsκ)B,Xsv(s,v,t)=κ2T+κsκN+τB,Xvv(s,v,t)=0. (4.14)

    The second fundamental form of the normal surface is given by

    II=dX,dU=dX,dU=Xsds+Xvdv,Usds+Uvdv=eds2+2fdsdv+gdt2

    where the coefficients of the second fundamental form are

    e=Xss,U=vκ(3vτκs+(1vκ2)(κτs+2τκs))(κ(1vκ2))2+(vτ)2,f=Xsv,U=κ2τ(1+vvκ2)(κ(1vκ2))2+(vτ)2,g=Xvv,U=0. (4.15)

    Corollary 4.3. The Gaussian and mean the curvatures of a normal ruled surface X=X(s,v,t) are

    K=τ2κ2(1+vvκ2)2((κ(1vκ2))2+(vτ)2)((1vκ2)2+(vτκ)2), (4.16)
    H=vκ(vτκs+(1vκ2)κτs)2((κ(1vκ2))2+(vτ)2)12((1vκ2)2+(vτκ)2), (4.17)

    respectively.

    Proof. From the Eqs. (4.13) and (4.15), we easily obtain the Gaussian curvature and the mean curvature respectively as follows

    K=egf2EGF2=τ2κ2(1+vvκ2)2((κ(1vκ2))2+(vτ)2)((1vκ2)2+(vτκ)2),
    H=12Eg2Ff+GeEGF2=vκ(vτκs+(1vκ2)κτs)2((κ(1vκ2))2+(vτ)2)12((1vκ2)2+(vτκ)2).

    From the Eqs. (4.16) and (4.17), the following result is obvious.

    Corollary 4.4. The normal ruled surface is developable iff τ=0 and minimal iff vτκs+(1vκ2)κτs=0.

    The ruled surface generated by the motion of the binormal vector B of the curve α is called the binormal ruled surface and the equation of this surface is

    X(s,v,t)=α(s,t)+vB(s,t). (4.18)

    Example 4.3. The parametric equations of binormal ruled surfaces generated by binormal vectors of Frenet frame and modified orthogonal frame the curve given in Example 2.1 are

    φ3(s,v)=(12(s0cos(πt22)dtvs|s|cos(πs22)),12(s0sin(πt22)dtvs|s|cos(πs22)),12(s+vs|s|))

    and

    φ4(s,v)=(12s0cos(πt22)dtvπs2cos(πs22),12s0sin(πt22)dtvπs2sin(πs22),s2(1+vπ2)),

    respectively. The first surface (generated by binormal vector of Frenet frame) is discontinuous at s=0 and the second one is generated by binormal vector of modified orthogonal frame, see Figure 4.

    Figure 4.  Binormal Ruled Surfaces.

    The tangent vectors for the binormal ruled surface are

    Xs(s,v,t)=TvτN+vκsκB,Xv(s,v,t)=B, (4.19)

    where the subscripts s and v represent partial derivatives of the binormal ruled surface. Using Eq. (4.19), we get the unit normal field of this surface is found as

    U=XsXvXsXv=vτTNκ2+(vτ)2. (4.20)

    The first fundamental form of the normal ruled surface in Euclidean space is given by

    I=dX,dX=Xsds+Xvdv,Xsds+Xvdv=Eds2+2Fdsdv+Gdv2

    where the coefficients of the first fundamental form are

    E=Xs,Xs=1+v2τ2κ2+v2κs2,F=Xs,Xv=vκsκ,G=Xv,Xv=κ2. (4.21)

    From the Eq. (4.19), the second derivative is found and given as

    Xss(s,v,t)=(vτκ2)T+(1vτs2vτκsκ)N+(vτ2+vκssκ)B,Xsv(s,v,t)=τN+κsκB,Xvv(s,v,t)=0. (4.22)

    The second fundamental form of the normal surface is given by

    II=dX,dU=dX,dU=Xsds+Xvdv,Usds+Uvdv=eds2+2fdsdv+gdt2

    where the coefficients of the second fundamental form are

    e=Xss,U=κ2(v2τ21+vτs+2vτκsκ)κ2+(vτ)2,f=Xsv,U=κ2τκ2+(vτ)2,g=Xvv,U=0. (4.23)

    Corollary 4.5. For the binormal ruled surface X=X(s,v,t) the Gaussian and mean the curvatures are

    K=τ2(κ2+(vτ)2)(1+(vτκ)2), (4.24)
    H=κ2(vτs(vτ)21)2(κ2+(vτ)2)12(1+(vτκ)2), (4.25)

    respectively.

    Proof. From the Eqs. (4.21) and (4.23), we easily obtain the Gaussian curvature and the mean curvature respectively as follows

    K=egf2EGF2=τ2(κ2+(vτ)2)(1+(vτκ)2),
    H=12Eg2Ff+GeEGF2=κ2(vτs(vτ)21)2(κ2+(vτ)2)12(1+(vτκ)2).

    Corollary 4.6. From the Eqs. (4.16) and (4.17), the normal ruled surface is developable iff τ=0 and minimal iff vτs(vτ)21=0.

    The authors declare no conflict of interest.



    [1] M. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, 1976.
    [2] A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, CRC, New York, 1998.
    [3] K. Sprott, B. Ravani, Kinematic generation of ruled surfaces, Adv. Comput. Math., 17 (2002), 115-133. doi: 10.1023/A:1015211729988
    [4] B. van Sosin, M. Barton, G. Elber, Accessibility for line-cutting in freeform surfaces, Comput. Aided Design, 114 (2019), 202-214. doi: 10.1016/j.cad.2019.05.014
    [5] H. Y. Chen, H. Pottmann, Approximation by ruled surfaces, J. Comput. Appl. Math., 102 (1999), 143-156. doi: 10.1016/S0377-0427(98)00212-X
    [6] S. Flöry, H. Pottmann, Ruled Surfaces for Rationalization and Design in Architecture, In: LIFE in:formation. On Responsive Information and Variations in Architecture, 2010, 103-109.
    [7] K. Nakayama, M. Wadati, Motion of curves in the plane, J. Phys. Soc. JPN, 62 (1993), 473-479. doi: 10.1143/JPSJ.62.473
    [8] K. Nakayama, H. Segur, M. Wadati, Integrability and the motion of curves, Phys. Rev. Lett., 69 (1992), 2603-2606. doi: 10.1103/PhysRevLett.69.2603
    [9] N. H. Abdel-All, R. A. Hussien, T. Youssef, Evolution of curves via the velocities of the moving frame, J. Math. Comput. Sci., 2 (2012), 1170-1185.
    [10] A. Doliwa, P. Santini, An elementary geometric characterization of the integrable motions of a curve, Phys. Lett. A, 185 (1994), 373-384. doi: 10.1016/0375-9601(94)90170-8
    [11] D. Y. Kwon, F. C. Park, Evolution of inelastic plane curves, Appl. Math. Lett., 12 (1999), 115-119.
    [12] H. Hasimoto, A soliton on a vortex filament, J. Fluid Mech., 52 (1972), 477-485.
    [13] G. L. Lamb, Solitons on moving space curves, J. Math. Phys., 18 (1977), 1654-1661. doi: 10.1063/1.523453
    [14] C. Rogers, W. K. Schief, Bäcklund and Darboux Transformations, In: Geometry and Modern Applications in Soliton Theory, Cambridge University Press, 2002.
    [15] H. N. Abd-Ellah, Evolution of translation surfaces in Euclidean 3-space, Appl. Math. Inform. Sci., 9 (2015), 661-668.
    [16] R. A. Hussien, S. G. Mohamed, Generated surfaces via inextensible flows of curves in R3, J. Appl. Math., 1 (2016), 6178961.
    [17] D. Y. Kwon, F. C. Park, Inextensible flows of curves and developable surfaces, Appl. Math. Lett., 18 (2005), 1156-1162. doi: 10.1016/j.aml.2005.02.004
    [18] O. G. Yıldız, S. Ersoy, M. A. Masal, Note on inextensible flows of curves on oriented surface, Cubo, 16 (2014), 11-19. doi: 10.4067/S0719-06462014000300002
    [19] K. Nakayama, M. Wadati, The motion of surfaces, J. Phys. Soc. JPN, 62 (1993), 1895-1901. doi: 10.1143/JPSJ.62.1895
    [20] R. A. Hussien, T. Youssef, Evolution of special ruled surfaces via the evolution of their directrices in Euclidean 3-Space E3, Appl. Math. Inform. Sci., 10 (2016), 1949-1956.
    [21] T. Sasai, The fundamental theorem of analytic space curves And apparent singularities of Fuchsian differential equations, Tohoku Math. J., 36 (1984), 17-24. doi: 10.2748/tmj/1178228899
    [22] B. Bükcü, M. K. Karacan, On the modified orthogonal frame with curvature and torsion in 3- Space, Math. Sci. Appl. E-Notes, 4 (2016), 184-188.
    [23] M. S. Lone, E. S Hasan, M. K. Karacan, et al., Mannheim curves with modified orthogonal frame in Euclidean 3-Space, Turk. J. Math., 43 (2019), 648-663. doi: 10.3906/mat-1807-177
    [24] M. S. Lone, E. S Hasan, M. K. Karacan, et al., On some curves with modified orthogonal frame in Euclidean 3-Space, Iran. J. Sci. Technol. Trans. A. Sci., 43 (2019), 1905-1916. doi: 10.1007/s40995-018-0661-2
    [25] B. Bükcü, M. K. Karacan, Spherical curves with modified orthogonal frame, J. New Res. Sci., 10 (2016), 60-68.
    [26] R. A. Hord, Torsion at an inflection point of a space curve, Am. Math. Mon., 79 (1972), 371-374. doi: 10.1080/00029890.1972.11993052
    [27] T. Sasai, Geometry of analytic space curves with singularities and regular singularities of differential equations, Funkcial. Ekvac., 30 (1987), 283-303.
  • This article has been cited by:

    1. Alev KELLECİ, Kemal EREN, On Evolution of Some Associated Type Ruled Surfaces, 2020, 2147-6268, 10.36753/mathenot.750639
    2. Kahraman Esen ÖZEN, Murat TOSUN, A New Moving Frame for Trajectories with Non-Vanishing Angular Momentum, 2021, 2636-8692, 10.33187/jmsm.869698
    3. Gu¨l UG˘ UR KAYMANLI, Cumali EKICI, Evolutions of the Ruled Surfaces along a Spacelike Space Curve, 2022, 1016-2526, 221, 10.52280/pujm.2022.540401
    4. Sezai Kiziltug, Ali Cakmak, Tulay Erisir, Gokhan Mumcu, On tubular surfaces with modified orthogonal frame in Galilean space G3, 2022, 26, 0354-9836, 571, 10.2298/TSCI22S2571K
    5. Sümeyye GÜR MAZLUM, Süleyman ŞENYURT, Mehmet BEKTAŞ, Salkowski Curves and Their Modified Orthogonal Frames in E3, 2022, 2149-1402, 12, 10.53570/jnt.1140546
    6. Nural Yüksel, Burçin Saltık, On inextensible ruled surfaces generated via a curve derived from a curve with constant torsion, 2023, 8, 2473-6988, 11312, 10.3934/math.2023573
    7. Abdussamet ÇALIŞKAN, Characterizations of Unit Darboux Ruled Surface with Quaternions, 2023, 2149-1402, 43, 10.53570/jnt.1194990
    8. Wei Zhang, Pengcheng Li, Donghe Pei, Circular evolutes and involutes of spacelike framed curves and their duality relations in Minkowski 3-space, 2024, 9, 2473-6988, 5688, 10.3934/math.2024276
    9. Zhen Zhang, Fangchen Yin, Hui Huang, Guoqin Huang, Changcai Cui, An industrial robot-based sawing method for natural stone sculpture, 2024, 11, 2288-5048, 75, 10.1093/jcde/qwae094
    10. Emel KARACA, An Examination for the Intersection of Two Ruled Surfaces, 2023, 6, 2645-8845, 70, 10.33401/fujma.1235668
    11. Samah Gaber, Norah Alfadhli, Elsayed I. Mahmoud, A Study on Normal Motion of the Torus of Revolution in ℝ3 , 2024, 9, 2444-8656, 10.2478/amns.2023.2.00079
    12. Yanlin Li, Kemal Eren, Soley Ersoy, Ana Savić, Modified Sweeping Surfaces in Euclidean 3-Space, 2024, 13, 2075-1680, 800, 10.3390/axioms13110800
    13. Burçin Saltık Baek, Esra Damar, Nurdan Oğraş, Nural Yüksel, Ruled Surfaces of Adjoint Curve with the Modified Orthogonal Frame, 2024, 2149-1402, 69, 10.53570/jnt.1583283
    14. Samah Gaber, Abeer Al Elaiw, Evolution of null Cartan and pseudo null curves via the Bishop frame in Minkowski space R2,1, 2025, 10, 2473-6988, 3691, 10.3934/math.2025171
    15. Emad Solouma, Ibrahim Al-Dayel, Mohamed A. Abdelkawy, Ruled Surfaces and Their Geometric Invariants via the Orthogonal Modified Frame in Minkowski 3-Space, 2025, 13, 2227-7390, 940, 10.3390/math13060940
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(4902) PDF downloads(530) Cited by(15)

Figures and Tables

Figures(4)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog