Figure 1.
Helical Curve over Clothoid.
Citation: Kenneth R. Paap, Oliver M. Sawi, Chirag Dalibar, Jack Darrow, Hunter A. Johnson. Beyond Panglossian Optimism: Larger N2 Amplitudes Probably Signal a Bilingual Disadvantage in Conflict Monitoring[J]. AIMS Neuroscience, 2015, 2(1): 1-6. doi: 10.3934/Neuroscience.2015.1.1
| [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 $ \mathbb{R}^{3} $. 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] | Kemal Eren, Soley Ersoy, Mohammad Nazrul Islam Khan . Simultaneous characterizations of alternative partner-ruled surfaces. AIMS Mathematics, 2025, 10(4): 8891-8906. doi: 10.3934/math.2025407 |
| [6] | Kemal Eren, Soley Ersoy, Mohammad Nazrul Islam Khan . Novel theorems on constant angle ruled surfaces with Sasai's interpretation. AIMS Mathematics, 2025, 10(4): 8364-8381. doi: 10.3934/math.2025385 |
| [7] | 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 $ \mathrm{E}_1^3 $. AIMS Mathematics, 2024, 9(5): 13028-13042. doi: 10.3934/math.2024635 |
| [8] | Ayman Elsharkawy, Ahmer Ali, Muhammad Hanif, Fatimah Alghamdi . Exploring quaternionic Bertrand curves: involutes and evolutes in $ \mathbb{E}^{4} $. AIMS Mathematics, 2025, 10(3): 4598-4619. doi: 10.3934/math.2025213 |
| [9] | 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 |
| [10] | Maryam T. Aldossary, Rashad A. Abdel-Baky . Sweeping surface due to rotation minimizing Darboux frame in Euclidean 3-space $ \mathbb{E}^{3} $. AIMS Mathematics, 2023, 8(1): 447-462. doi: 10.3934/math.2023021 |
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 $ < , > = dx_1^2 + dx_2^2 + dx_3^2 $ where $ x = \left({{x_1}, {x_2}, {x_3}} \right) \in {E^3} $. The norm of a vector $ x \in {E^3} $ is $ \left\| x \right\| = \sqrt {\left| { < x, x > } \right|} $. For any $ \alpha $ curve, if $ \left\| {\alpha '(s)} \right\| = 1 $, then $ \alpha $ 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 $ \alpha $ be a space curve with respect to the arc-length $ s $ in Euclidean 3-space $ {E^3} $. $ t $, $ n $ and $ b $ are tangent, principal normal and binormal unit vectors at each point $ \alpha (s) $ of a curve $ \alpha $, respectively. Then there exists an orthogonal frame $ \left\{ {t, n, b} \right\} $ which satisfies the Frenet-Serret equation
|
$ t′=κn,n′=−κt+τb,b′=−τn $
|
(2.1) |
where $ \kappa $ is the curvature, $ \tau $ is the torsion.
The fundamental theorem of regular curves states that if $ \kappa > 0 $ and $ \tau $ are differentiable functions then there exists a unit speed curve whose curvature and torsion are $ \kappa $ and $ \tau $, 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 $ \alpha $ be an analytic curve of which curvature has discrete zero points in Euclidean 3-space. Under the assumption $ \kappa \left(s \right) $ of $ \alpha $ is not identically zero, the elements of modified orthogonal frame are given by
| $T = \frac{{d\alpha }}{{ds}}, \quad N = \frac{{dT}}{{ds}}, \quad B = T \wedge N$ |
where $ s $ is the arc-length parameter and $ T \wedge N $ is the vector product of $ T $ and $ N $.
The relations between the Frenet frame $ \left\{ {t, n, b} \right\} $ and modified orthogonal frame $ \left\{ {T, N, B} \right\} $ at non-zero points of $ \kappa $ are
| $T = t, \quad N = \kappa n, \quad B = \kappa b.$ |
In the course of time, this orthogonal frame is called the modified orthogonal frame [22]. The modified orthogonal frame $ \left\{ {T, N, B} \right\} $ satisfies
| $\left\langle {T, T} \right\rangle = 1, {\rm{ }}\left\langle {N, N} \right\rangle = \left\langle {B, B} \right\rangle = {\kappa ^2}, {\rm{ }}\left\langle {T, N} \right\rangle = \left\langle {T, B} \right\rangle = \left\langle {N, B} \right\rangle = 0.$ |
such that $ \left\langle, \right\rangle $ is the Euclidean inner product. From these equations, the differentiation formula for the modified orthogonal frame $ \left\{ {T, N, B} \right\} $ is given by
|
$ T′(s)=N(s),N′(s)=−κ2T(s)+κsκN(s)+τB(s),B′(s)=−τN(s)+κsκB(s), $
|
(2.2) |
where $ {\kappa _s} $ denotes the differentiation with respect to $ s $ and $ \tau = \frac{{\det \left({\alpha ', \alpha '', \alpha '''} \right)}}{{{\kappa ^2}}} $ is the torsion of $ \alpha $. Here, the essential quantities $ {\kappa ^2} $ and $ \tau $ are analytic in [21,22].
Example 2.1. Let us consider a curve given by the parametric equation
| $\alpha \left( s \right) = \left( {\frac{1}{{\sqrt 2 }}\int\limits_0^s {\cos \left( {\frac{{\pi {t^2}}}{2}} \right)dt} , \frac{1}{{\sqrt 2 }}\int\limits_0^s {\sin \left( {\frac{{\pi {t^2}}}{2}} \right)dt} , \frac{s}{{\sqrt 2 }}} \right)$ |
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 $ \int\limits_0^s {\cos \left({\frac{{\pi {t^2}}}{2}} \right)dt} $ and $ \int\limits_0^s {\sin \left({\frac{{\pi {t^2}}}{2}} \right)dt} $ are called Fresnel integrals. The elements of the Frenet trihedron of the curve $ \alpha $ are obtained as
| $t\left( s \right) = \left( {\frac{1}{{\sqrt 2 }}\cos \left( {\frac{{\pi {s^2}}}{2}} \right), \frac{1}{{\sqrt 2 }}\sin \left( {\frac{{\pi {s^2}}}{2}} \right), \frac{1}{{\sqrt 2 }}} \right), $ |
| $n\left( s \right) = \left( { - \frac{s}{{\left| s \right|}}\sin \left( {\frac{{\pi {s^2}}}{2}} \right), \frac{s}{{\left| s \right|}}\cos \left( {\frac{{\pi {s^2}}}{2}} \right), 0} \right), $ |
| $b\left( s \right) = \left( { - \frac{s}{{\sqrt 2 \left| s \right|}}\cos \left( {\frac{{\pi {s^2}}}{2}} \right), - \frac{s}{{\sqrt 2 \left| s \right|}}\sin \left( {\frac{{\pi {s^2}}}{2}} \right), \frac{s}{{\sqrt 2 \left| s \right|}}} \right)$ |
and the curvature is $ \kappa \left(s \right) = \frac{{\pi \left| s \right|}}{{\sqrt 2 }} $. Besides the curvature is not differentiable, the principal normal and binormal vectors are discontinuous at $ s = 0 $ since $ {n_ + } \ne {n_ - } $ and $ {b_ + } \ne {b_ - } $ for $ {{n}_{+}} = \lim\limits_{s\to {{0}^{+}}}\lim\limits_\, n\left(s \right) $, $ {{n}_{-}} = \lim\limits_{s\to {{0}^{-}}}\lim\limits_\, n\left(s \right) $ and $ {{b}_{+}} = \lim\limits_{s\to {{0}^{+}}}\, b\left(s \right) $, $ {{b}_{-}} = \lim\limits_{s\to {{0}^{-}}}\lim\limits_\, b\left(s \right) $.
Whenever the curvature is considered as a signed quantity $ \kappa \left(s \right) = \mp \frac{{\pi s}}{{\sqrt 2 }} $, the curve forms a symmetrical double spiral, see Figure 1.
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\left( s \right) = \left( \frac{1}{\sqrt{2}}\cos \left( \frac{\pi {{s}^{2}}}{2} \right), \frac{1}{\sqrt{2}}\sin \left( \frac{\pi {{s}^{2}}}{2} \right), \frac{1}{\sqrt{2}} \right), $ |
| $ N\left( s \right) = \left( -\frac{\pi s}{\sqrt{2}}\sin \left( \frac{\pi {{s}^{2}}}{2} \right), \frac{\pi s}{\sqrt{2}}\cos \left( \frac{\pi {{s}^{2}}}{2} \right), 0 \right), $ |
| $ B\left( s \right) = \left( -\frac{\pi s}{2}\cos \left( \frac{\pi {{s}^{2}}}{2} \right), -\frac{\pi s}{2}\sin \left( \frac{\pi {{s}^{2}}}{2} \right), \frac{\pi s}{2} \right), $ |
under the assumption that the curvature $ \kappa \left(s \right) $ of $ \alpha $ is not zero. Here the essential quantities are obtained as $ {\kappa ^2} = \frac{{{\pi ^2}{s^2}}}{2} $ and $ \tau \left(s \right) = \frac{\pi s}{\sqrt{2}} $.
A curve $ \alpha $ in Euclidean 3-space is a vector-valued function $ \alpha \left({s, t} \right) \in {E^3} $ 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=αs∧αss, $
|
(3.1) |
where the subscripts indicate the partial differential. Let $ \alpha $ 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 $ \left\{ {T, N, B} \right\} $ can be written in matrix form as follows:
|
$ [TNB]t=[0ηβ−ηκκtγ−β−γκκt][TNB] $
|
(3.2) |
where $ \alpha $, $ \beta $ and $ \gamma $ are smooth functions. By considering the curvature $ \kappa $ of the curve $ \alpha $ is not identically zero and using the equations $ {T_{st}} = {T_{ts}}, {\rm{ }}{N_{st}} = {N_{ts}}, {\rm{ }}{B_{st}} = {B_{ts}} $, we obtain
|
$ ηs=τβ−κsκη+κκt,βs=γ−τη−κsκβ,γs=τt−β. $
|
(3.3) |
We suppose that the velocity according to the curve $ \alpha $ is given by
|
$ αt=dαdt=aT+bN+cB. $
|
(3.4) |
From equation $ {\alpha _{st}} = {\alpha _{ts}} $, we find the following equations
|
$ 0=as−bκ2,η=a+bs+κsκb−τc,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+κsκb−τc)+κsκ(τb+cs+κsκc). $
|
(3.6) |
For a solution of smoke ring equation, the velocity vector is given by
|
$ αt=αs∧αss=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=κssκ−τ2. $
|
(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\left( {s, v} \right) = \alpha (s) + vl\left( s \right), $ |
where $ \alpha (s) $ is called the base curve and $ l(s) $ is the director curve. If the curves $ \alpha (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 $ \alpha $ 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
| $ \varphi \left( {s, v} \right) = \left( {\frac{1}{{\sqrt 2 }}\left( {\int\limits_0^s {\cos \left( {\frac{{\pi {t^2}}}{2}} \right)dt} + v\cos \left( {\frac{{\pi {s^2}}}{2}} \right)} \right), \frac{1}{{\sqrt 2 }}\left( {\int\limits_0^s {\sin \left( {\frac{{\pi {t^2}}}{2}} \right)dt + v\sin \left( {\frac{{\pi {s^2}}}{2}} \right)} } \right), \frac{{s + v}}{{\sqrt 2 }}} \right), $ |
see Figure 2.
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=Xs∧Xv‖Xs∧Xv‖=−Bκ. $
|
(4.4) |
The first fundamental form of the tangent ruled surface in Euclidean space is given by
| $I = \left\langle {dX, dX} \right\rangle = \left\langle {{X_s}ds + {X_v}dv, {X_s}ds + {X_v}dv} \right\rangle = Ed{s^2} + 2Fdsdv + Gd{v^2}$ |
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
| $II = \left\langle {dX, dU} \right\rangle = - \left\langle {dX, dU} \right\rangle = \left\langle {{X_s}ds + {X_v}dv, {U_s}ds + {U_v}dv} \right\rangle = ed{s^2} + 2fdsdv + gd{t^2}$ |
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\left({s, v, t} \right) $ 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 = \frac{{eg - {f^2}}}{{EG - {F^2}}} = 0, $ |
| $H = \frac{1}{2}\frac{{Eg - 2Ff + Ge}}{{EG - {F^2}}} = \frac{{ - \tau }}{{2v\kappa }}$ |
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 $ \tau = 0 $.
The ruled surface generated by the motion of the normal vector $ N $ of the curve $ \alpha $ 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
| $ {\varphi _1}\left( {s, v} \right) = \left( {\frac{1}{{\sqrt 2 }}\int\limits_0^s {\cos \left( {\frac{{\pi {t^2}}}{2}} \right)dt} - v\frac{s}{{\left| s \right|}}\sin \left( {\frac{{\pi {s^2}}}{2}} \right), \frac{1}{{\sqrt 2 }}\int\limits_0^s {\sin \left( {\frac{{\pi {t^2}}}{2}} \right)dt + v} \frac{s}{{\left| s \right|}}\cos \left( {\frac{{\pi {s^2}}}{2}} \right), \frac{s}{{\sqrt 2 }}} \right) $ |
and
| $ {\varphi _2}\left( {s, v} \right) = \left( {\frac{1}{{\sqrt 2 }}\left( {\int\limits_0^s {\cos \left( {\frac{{\pi {t^2}}}{2}} \right)dt} - v\pi s\sin \left( {\frac{{\pi {s^2}}}{2}} \right)} \right), \frac{1}{{\sqrt 2 }}\left( {\int\limits_0^s {\sin \left( {\frac{{\pi {t^2}}}{2}} \right)dt + v} \pi s\cos \left( {\frac{{\pi {s^2}}}{2}} \right)} \right), \frac{s}{{\sqrt 2 }}} \right), $ |
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.
The derivatives of the normal ruled surface with respect to $ s $ and $ v $ are
|
$ Xs(s,v,t)=(1−vκ2)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=Xs∧Xv‖Xs∧Xv‖=−vτT+(1−vκ2)B√(1−vκ2)2κ2+(vτ)2. $
|
(4.12) |
The first fundamental form of the normal ruled surface in Euclidean space is given by
| $I = \left\langle {dX, dX} \right\rangle = \left\langle {{X_s}ds + {X_v}dv, {X_s}ds + {X_v}dv} \right\rangle = Ed{s^2} + 2Fdsdv + Gd{v^2}$ |
where the coefficients of the first fundamental form are
|
$ E=⟨Xs,Xs⟩=(1−vκ2)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+(1−vκ2+vκssκ−vτ2)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
| $II = \left\langle {dX, dU} \right\rangle = - \left\langle {dX, dU} \right\rangle = \left\langle {{X_s}ds + {X_v}dv, {U_s}ds + {U_v}dv} \right\rangle = ed{s^2} + 2fdsdv + gd{t^2}$ |
where the coefficients of the second fundamental form are
|
$ e=⟨Xss,U⟩=vκ(3vτκs+(1−vκ2)(κτs+2τκs))√(κ(1−vκ2))2+(vτ)2,f=⟨Xsv,U⟩=κ2τ(1+v−vκ2)√(κ(1−vκ2))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\left({s, v, t} \right) $ are
|
$ K=−τ2κ2(1+v−vκ2)2((κ(1−vκ2))2+(vτ)2)((1−vκ2)2+(vτκ)2), $
|
(4.16) |
|
$ H=vκ(vτκs+(1−vκ2)κτs)2((κ(1−vκ2))2+(vτ)2)12((1−vκ2)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 = \frac{{eg - {f^2}}}{{EG - {F^2}}} = \frac{{ - {\tau ^2}{\kappa ^2}{{\left( {1 + v - v{\kappa ^2}} \right)}^2}}}{{\left( {{{\left( {\kappa \left( {1 - v{\kappa ^2}} \right)} \right)}^2} + {{\left( {v\tau } \right)}^2}} \right)\left( {{{\left( {1 - v{\kappa ^2}} \right)}^2} + {{\left( {v\tau \kappa } \right)}^2}} \right)}}, $ |
| $H = \frac{1}{2}\frac{{Eg - 2Ff + Ge}}{{EG - {F^2}}} = \frac{{v\kappa \left( {v\tau {\kappa _s} + \left( {1 - v{\kappa ^2}} \right)\kappa {\tau _s}} \right)}}{{2{{\left( {{{\left( {\kappa \left( {1 - v{\kappa ^2}} \right)} \right)}^2} + {{\left( {v\tau } \right)}^2}} \right)}^{\frac{1}{2}}}\left( {{{\left( {1 - v{\kappa ^2}} \right)}^2} + {{\left( {v\tau \kappa } \right)}^2}} \right)}}.$ |
From the Eqs. (4.16) and (4.17), the following result is obvious.
Corollary 4.4. The normal ruled surface is developable iff $ \tau = 0 $ and minimal iff $ v\tau {\kappa _s} + \left({1 - v{\kappa ^2}} \right)\kappa {\tau _s} = 0. $
The ruled surface generated by the motion of the binormal vector $ B $ of the curve $ \alpha $ 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
| $ {\varphi _3}\left( {s, v} \right) = \left( {\frac{1}{{\sqrt 2 }}\left( {\int\limits_0^s {\cos \left( {\frac{{\pi {t^2}}}{2}} \right)dt} - v\frac{s}{{\left| s \right|}}\cos \left( {\frac{{\pi {s^2}}}{2}} \right)} \right), \frac{1}{{\sqrt 2 }}\left( {\int\limits_0^s {\sin \left( {\frac{{\pi {t^2}}}{2}} \right)dt - v} \frac{s}{{\left| s \right|}}\cos \left( {\frac{{\pi {s^2}}}{2}} \right)} \right), \frac{1}{{\sqrt 2 }}\left( {s + v\frac{s}{{\left| s \right|}}} \right)} \right) $ |
and
| $ {\varphi _4}\left( {s, v} \right) = \left( {\frac{1}{{\sqrt 2 }}\int\limits_0^s {\cos \left( {\frac{{\pi {t^2}}}{2}} \right)dt} - v\frac{{\pi s}}{2}\cos \left( {\frac{{\pi {s^2}}}{2}} \right), \frac{1}{{\sqrt 2 }}\int\limits_0^s {\sin \left( {\frac{{\pi {t^2}}}{2}} \right)dt - v} \frac{{\pi s}}{2}\sin \left( {\frac{{\pi {s^2}}}{2}} \right), \frac{s}{{\sqrt 2 }}\left( {1 + \frac{{v\pi }}{{\sqrt 2 }}} \right)} \right), $ |
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.
The tangent vectors for the binormal ruled surface are
|
$ Xs(s,v,t)=T−vτN+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=Xs∧Xv‖Xs∧Xv‖=−vτT−N√κ2+(vτ)2. $
|
(4.20) |
The first fundamental form of the normal ruled surface in Euclidean space is given by
| $I = \left\langle {dX, dX} \right\rangle = \left\langle {{X_s}ds + {X_v}dv, {X_s}ds + {X_v}dv} \right\rangle = Ed{s^2} + 2Fdsdv + Gd{v^2}$ |
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+(1−vτs−2vτκsκ)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
| $II = \left\langle {dX, dU} \right\rangle = - \left\langle {dX, dU} \right\rangle = \left\langle {{X_s}ds + {X_v}dv, {U_s}ds + {U_v}dv} \right\rangle = ed{s^2} + 2fdsdv + gd{t^2}$ |
where the coefficients of the second fundamental form are
|
$ e=⟨Xss,U⟩=κ2(−v2τ2−1+vτs+2vτκsκ)√κ2+(vτ)2,f=⟨Xsv,U⟩=κ2τ√κ2+(vτ)2,g=⟨Xvv,U⟩=0. $
|
(4.23) |
Corollary 4.5. For the binormal ruled surface $ X = X\left({s, v, t} \right) $ the Gaussian and mean the curvatures are
|
$ K=−τ2(κ2+(vτ)2)(1+(vτκ)2), $
|
(4.24) |
|
$ H=κ2(vτs−(vτ)2−1)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 = \frac{{eg - {f^2}}}{{EG - {F^2}}} = - \frac{{{\tau ^2}}}{{\left( {{\kappa ^2} + {{\left( {v\tau } \right)}^2}} \right)\left( {1 + {{\left( {v\tau \kappa } \right)}^2}} \right)}}, $ |
| $H = \frac{1}{2}\frac{{Eg - 2Ff + Ge}}{{EG - {F^2}}} = \frac{{{\kappa ^2}\left( {v{\tau _s} - {{\left( {v\tau } \right)}^2} - 1} \right)}}{{2{{\left( {{\kappa ^2} + {{\left( {v\tau } \right)}^2}} \right)}^{\frac{1}{2}}}\left( {1 + {{\left( {v\tau \kappa } \right)}^2}} \right)}}.$ |
Corollary 4.6. From the Eqs. (4.16) and (4.17), the normal ruled surface is developable iff $ \tau = 0 $ and minimal iff $ v{\tau _s} - {\left({v\tau } \right)^2} - 1 = 0 $.
The authors declare no conflict of interest.
| [1] |
Paap KR, Sawi OM, Dalibar C, et al. (2014) The Brain Mechanisms Underlying the Cognitive Benefits of Bilingualism may be Extraordinarily Difficult to Discover. AIMS Neuroscience 1:245-256. doi: 10.3934/Neuroscience.2014.3.245
|
| [2] | Fernandez M, Acosta J, Douglass K, et al. (2014) Speaking Two Languages Enhances an Auditory but not a Visual Neural Marker of Cognitive Inhibition. AIMS Neuroscience 1:145-157. |
| [3] |
Paap KR, Greenberg ZI (2013) There is no coherent evidence for a bilingual advantage in executive processing. Cogn Psychol 66: 232-258. doi: 10.1016/j.cogpsych.2012.12.002
|
| [4] |
Paap KR, Johnson HA, Sawi O (2014) Are bilingual advantages dependent upon specific tasks or specific bilingual experiences? J Cogn Psychol 26: 615-639. doi: 10.1080/20445911.2014.944914
|
| [5] | Paap KR (2014) The role of componential analysis, categorical hypothesizing, replicability and confirmation bias in testing for bilingual advantages in executive functioning. J CognPsychol 26:242-255. |
| [6] |
Paap KR, Liu Y (2014) Conflict resolution in sentence processing is the same for bilinguals and monolinguals: The role of confirmation bias in testing for bilingual advantages. J Neurolinguistics 27: 50-74. doi: 10.1016/j.jneuroling.2013.09.002
|
| [7] | Paap KR, Sawi O (2014) Bilingual advantages in executive functioning: problems in convergent validity, discriminant validity, and the identification of the theoretical constructs. Frontiers in Psychology 5: 962. |
| [8] |
Rouder JN, Speckman PL, Sun D, et al. (2009) Bayesian t tests for accepting and rejecting the null hypothesis. Psychonomic Bull Rev 16: 225-237. doi: 10.3758/PBR.16.2.225
|
| [9] | Valian V (2014) Bilingualism and cognition. Biling Lang Cogn 18: 3-24. |
| [10] | Paap KR (2014) Do many hones dull the bilingual whetstone? Biling Lang Cogn 18: 41-42. |
| [11] |
Fernandez M, Tartar JL, Padron D, et al. (2013) Neurophysiological marker of inhibition distinguishes language groups on a non-linguistic executive function test. Brain Cognition 83:330-336. doi: 10.1016/j.bandc.2013.09.010
|
| [12] | Falkenstein M, Hoormann J, Johnsbein J (1999) ERP components in Go/NoGo tasks and their relation to inhibition. Acta Psychol 101. |
| [13] |
Lamm C, Zelazo PD, Lewis MD (2006) Neural correlates of cognitive control in childhood and adolescence: disentangling the contributions of age and executive function. Neuropsychologia 44:2139-2148. doi: 10.1016/j.neuropsychologia.2005.10.013
|
| [14] | Espinet SD, Anderson JE, Zelazo PD (2012) N2 amplitude as a neural marker of executive function in young children: an ERP study of children who switch versus perseverate on the Dimensional Change Card Sort. Dev Cogn Neurosci 2 Suppl 1: S49-58. |
| [15] |
Kousaie S, Phillips NA (2012) Conflict monitoring and resolution: Are two languages better than one? Brain Res 1446: 71-90. doi: 10.1016/j.brainres.2012.01.052
|
| [16] |
Kornblum S, Hasbroucq T, Osman A (1990) Dimensional Overlap: Cognitive Basis for Stimulus-Response Compatibility - A Model and Taxonomy. Psychol Rev 97: 253-270. doi: 10.1037/0033-295X.97.2.253
|
| [17] |
Nigg JT (2000) On inhibition/disinhibition in developmental psychopathology: Views from cognitive and personality psychology and a working inhibition taxonomy. Psychol Bulletin 126:220-246. doi: 10.1037/0033-2909.126.2.220
|
| [18] |
Bunge SA, Dudukovic NM, Thomason ME, et al. (2002) Immature frontal lobe contributions to cogntive control in children: evidence from fMRI. Neuron 33: 301-311. doi: 10.1016/S0896-6273(01)00583-9
|
| [19] |
Luk G, Anderson JA, Craik FI, et al. (2010) Distinct neural correlates for two types of inhibition in bilinguals: response inhibition versus interference suppression. Brain Cogn 74: 347-357. doi: 10.1016/j.bandc.2010.09.004
|
| [20] | Friedman NP, Miyake A (2004) The Relations Among Inhibition and Interference Control Functions: A Latent-Variable Analysis. J ExpPsychol Gen133: 101-135. |
| [21] |
Duckworth L, Kern ML (2011) A meta-analysis of the convergent validity of self-control measures. J Res Personal 45: 259-268. doi: 10.1016/j.jrp.2011.02.004
|
| [22] |
Votruba KL, Langenecker SA (2013) Factor structure, construct validity, and age- and education-based normative data for the Parametric Go/No-Go Test. J Clin Exp Neuropsychol 35:132-146. doi: 10.1080/13803395.2012.758239
|
| 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 $\mathbb{E}^{3}$, 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 $ \mathbb{R}^{2, 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 | |
| 16. | Kemal Eren, Soley Ersoy, Mohammad Nazrul Islam Khan, Novel theorems on constant angle ruled surfaces with Sasai's interpretation, 2025, 10, 2473-6988, 8364, 10.3934/math.2025385 |
Kenneth R. Paap, Oliver M. Sawi, Chirag Dalibar, Jack Darrow, Hunter A. Johnson. Beyond Panglossian Optimism: Larger N2 Amplitudes Probably Signal a Bilingual Disadvantage in Conflict Monitoring[J]. AIMS Neuroscience, 2015, 2(1): 1-6. doi: 10.3934/Neuroscience.2015.1.1
DownLoad:
