Second Binormal Motions of Inextensible Curves in 4-dimensional Galilean Space

Galilean 3-space G₃ is simply defined as a Klein geometry of the product space ℝx𝔼² whose symmetry group is Galilean transformation group which has an important place in classical and modern physics. It is well known that the idea of world lines originates in physics and was pioneered by Einstein.

In the recent time, the study of the motion of inextensible curves has been arisen in a number of diverse engineering applications and has many applications in computer vision, snake-like, robots. The flow of a curve is said to be inextensible if the arc length is preserved.

Nowadays, many important and intensive studies are seen about inextensible curve flows in the different space. Physically, inextensible curve flows give rise to motions in which no strain energy is induced. The swinging motion of a cord of fixed length, for example, or of a piece of paper carried by the wind, can be described by inextensible curve and surface flows. Such motions arise quite naturally in a wide range of a physical applications. For example, both Chirikjian and Burdick [1] used novel and efficient kinematic modeling techniques for “hyper-redundant” robots that to determining the time varying backbone curve behavior and Mochiyama et al.

Inextensible flows of curves are studied surfaces in ℝ³ by Kwon in [2]. In addition, many researchers have studied on inextensible flows such as [3], [4] and [8]. In [7] and [19], the authors studied inextensible flows in Minkowski space-time E $14$ [_1^4 . In [10] is obtained a theoretical framework for controlling a manipulator with hyper degrees of freedom in which the shape control of hyper-redundant, or snake-like robots.

In [15], the generalization of Bertrand curves in Galilean 4-space is introduced and the characterization of the generalized Bertrand curves is obtained. In [16], the author constructed Frenet-Serret frame of a curve in the Galilean 4-space and obtained the mentioned curve’s Frenet-Serret equations. Inextensible curve and surface flows also arise in the context of many problems in computer vision [6], [12] and computer animation [9], and even structural mechanics [17]. Papers in [5], [11], [13], [14] and [18] are obtained some new characterizations of the inextensible curve flows curves. By drawing inspiration from them, in this paper, we consider the second binormal motions of inextensible curves in Galilean 4-space. We give the necessary and sufficient conditions to be inextensible flows and we find the evolution equations for the inextensible curve flows in 4-dimensional Galilean Space.

Geometric Preliminaries

The scalar product of two vectors $\vec{U} = (u_{1}, u_{2}, u_{3}, u_{4})$ \vec U = ({u_1},{u_2},{u_3},{u_4})\ and $\vec{V} = (v_{1}, v_{2}, v_{3}, v_{4})$ \vec V = ({v_1},{v_2},{v_3},{v_4})\ in G₄ is defined by (2.1) $< \vec{U}, \vec{V} > = {\begin{matrix} u_{1} v_{1}, if u_{1} \neq 0 or v_{1} \neq 0 \\ u_{2} v_{2} + u_{3} v_{3} + u_{4} v_{4}, if u_{1} = 0, & v_{1} = 0 \end{matrix} .$ [ < \overrightarrow U , \overrightarrow V > = \left\{ {\begin{array}{*{20}c} {u_{_1 } v_{_1 } ,\;\;\;\;\;\;\;\;\;\;\;\;{\rm{if}}u_{_1 } \ne 0\;{\rm{or}}v_{_1 } \ne 0} & {} & {} \\ {} & {} & {} \\ {u_{_2 } v_{_2 } + u_{_3 } v_{_3 } + u_{_4 } v_{_4 } ,\;\;\;{\rm{if}}u_{_1 } = 0,} & {v_{_1 } = 0} & {} \\\end{array}} \right..

The Galilean cross product in G₄ for the vectors $\vec{u} = (u_{1}, u_{2}, u_{3}, u_{4})$ \vec u = ({u_1},{u_2},{u_3},{u_4})\, $\vec{v} = (v_{1}, v_{2}, v_{3}, v_{4})$ \vec v = ({v_1},{v_2},{v_3},{v_4})\, and $\vec{w} = (w_{1}, w_{2}, w_{3}, w_{4})$ \vec w = ({w_1},{w_2},{w_3},{w_4})\ is defined by (2.2) $\begin{matrix} \vec{u} \land \vec{v} \land \vec{w} \end{matrix} = - | \begin{matrix} 0 & e_{2} & e_{3} & e_{4} \\ u_{1} & u_{2} & u_{3} & u_{4} \\ v_{1} & v_{2} & v_{3} & v_{4} \\ ω_{1} & ω_{2} & ω_{3} & ω_{4} \end{matrix} |,$ [\begin{array}{*{20}c} {\vec u \wedge \vec v \wedge \vec w} \\\end{array} = - \left| {\begin{array}{*{20}c} 0 & {e_2 } & {e_3 } & {e_4 } \\ {u_1 } & {u_2 } & {u_3 } & {u_4 } \\ {v_1 } & {v_2 } & {v_3 } & {v_4 } \\ {\omega _1 } & {\omega _2 } & {\omega _3 } & {\omega _4 } \\\end{array}} \right|, where e_i 1 ≤ i ≤ 4, are the standard basis vectors.

The norm of vector $\vec{U} = (u_{1}, u_{2}, u_{3}, u_{4})$ \vec U = ({u_1},{u_2},{u_3},{u_4})\ is defined by $∥ \vec{U} ∥ = \sqrt{| < \vec{U}, \vec{U} > |} [16] .$ [\parallel \vec U\parallel = \sqrt {| < \vec U,\vec U > |} {\rm{[16]}}{\rm{.}}

Let α : I ⊂ R → G₄, α(s) = (s,y(s),z(s),w(s)) be a curve parametrized by arclength s in G₄. . The first vector of the Frenet-Serret frame, that is, the tangent vector of α is defined by (2.3) $t = α^{'} (s) = (1, y^{'} (s), z^{'} (s), w^{'} (s)) .$ [{\bf{t}} = \alpha ^' (s) = (1,y^' (s),z^' (s),w^' (s)).

Since t is a unit vector, we can express (2.4) $< t, t > = 1 .$ [\; < {\bf{t}},{\bf{t}} > = 1.

Differentiating (2.4) with respect to s, we have (2.5) $< t^{'}, t > = 0 .$ [ < {\bf{t}}^' ,{\bf{t}} > = 0.

The vector function t^′ gives us the rotation measurement of the curve α. The real valued function (2.6) $κ (s) = ∥ t^{'} (s) ∥ = \sqrt{{(y^{''})}^{2} + {(z^{''})}^{2} + {(ω^{''})}^{2}}$ [\kappa (s) = \parallel {\bf{t}}^' (s)\parallel = \sqrt {(y^{''} )^2 + (z^{''} )^2 + (\omega ^{''} )^2 } is called the first curvature of the curve α. We assume that κ(s) ≠ 0 for all s ∈ l. Similar to space G₃, the principal vector is defined by $n (s) = \frac{t^{'} (s)}{κ (s)};$ [{\bf{n}}(s) = \frac{{{\bf{t}}^' (s)}}{{\kappa (s)}}; in other words (2.7) $n (s) = \frac{1}{κ (s)} (0, y^{''} (s), z^{''} (s), ω^{''} (s)) .$ [{\bf{n}}(s) = \frac{1}{{\kappa (s)}}\left( {0,\;y^{''} (s),\;z^{''} (s),\;\omega ^{''} (s)} \right).

By the aid of the differentiation of the principal normal vector given in (2.7), define the second curvature function that is defined by $τ (s) = ∥ n^{'} (s) ∥ .$ [\;\tau (s) = \parallel {\bf{n}}^' (s)\parallel .

This real valued function is called torsion of the curve α. The third vector field, namely, binormal vector field of the curve α, is defined by $b (s) = \frac{1}{τ (s)} (0, {(\frac{y^{''} (s)}{κ (s)})}^{'}, {(\frac{z^{''} (s)}{κ (s)})}^{'}, {(\frac{ω^{''} (s)}{κ (s)})}^{'}) .$ [\;{\bf{b}}(s) = \frac{1}{{\tau (s)}}\left( {0,\left( {\frac{{y^{''} (s)}}{{\kappa (s)}}} \right)^' ,\left( {\frac{{z^{''} (s)}}{{\kappa (s)}}} \right)^' ,\left( {\frac{{\omega ^{''} (s)}}{{\kappa (s)}}} \right)^' } \right).

Thus the vector b(s) is perpendicular to both t and n. The fourth unit vector is defined by $e (s) = μ t (s) \land n (s) \land b (s) .$ [\;{\bf{e}}(s) = \mu {\bf{t}}(s) \wedge {\bf{n}}(s) \wedge {\bf{b}}(s).

Here the coefficient μ is taken ±1 to make +1 determinant of the matrix [t,n,b,e]. The third curvature of the curve α by the Galilean inner product is defined by (2.8) $σ = 〈 b^{'}, e 〉 .$ [\sigma = \langle {\bf{b}}^' ,{\bf{e}}\rangle .

Here, as well known, the set {t,n,b,e,κ,τ,σ} is called the Frenet-Serret apparatus of the curve α. We know that the vectors {t,n,b,e} are mutually orthogonal vectors satisfying $\begin{array}{l} 〈 t, t 〉 = 〈 n, n 〉 = 〈 b, b 〉 = 〈 e, e 〉 = 1, \\ 〈 t, n 〉 = 〈 t, b 〉 = 〈 t, e 〉 = 〈 n, b 〉 = 〈 n, e 〉 = 〈 b, e 〉 = 0 . \end{array}$ [begin{array}{*{20}c} {\langle {\bf{t}},{\bf{t}}\rangle = \langle {\bf{n}},{\bf{n}}\rangle = \langle {\bf{b}},{\bf{b}}\rangle = \langle {\bf{e}},{\bf{e}}\rangle = 1,} \hfill \\ {\langle {\bf{t}},{\bf{n}}\rangle = \langle {\bf{t}},{\bf{b}}\rangle = \langle {\bf{t}},{\bf{e}}\rangle = \langle {\bf{n}},{\bf{b}}\rangle = \langle {\bf{n}},{\bf{e}}\rangle = \langle {\bf{b}},{\bf{e}}\rangle = 0.} \hfill \\\end{array}

For the curve α in G₄, we have the following Frenet-Serret equations [16]: $\begin{array}{l} t^{'} = κ (s) n (s), \\ n^{'} = τ (s) b (s), \\ b^{'} = - τ (s) n (s) + σ (s) e (s), \\ e^{'} = - σ (s) b (s) . \end{array}$ [\begin{array}{*{20}c} {\;{\bf{t}}^' = \kappa (s){\bf{n}}(s),} \hfill \\ {{\bf{n}}^' = \tau (s){\bf{b}}(s),} \hfill \\ {{\bf{b}}^' = - \tau (s){\bf{n}}(s) + \sigma (s){\bf{e}}(s),} \hfill \\ {{\bf{e}}^' = - \sigma (s){\bf{b}}(s).} \hfill \\\end{array}

Second Binormal motions of curves in the four-dimensional Galilean space G₄

Let ${\hat{γ}}_{0} : I \to G_{4}$ {\hat \gamma _0}:I \to {G_4}\, be a regular curve in four-dimensional Galilean space G₄.

Consider the family of curves $C_{t} : \hat{γ} (\hat{s}, t)$ {C_t}:\hat \gamma (\hat s,t)\, where $\hat{γ} (\hat{s}, t) : I \subset x \in [0, \infty) \to G_{4}$ \hat \gamma (\hat s,t):I \subset x \in [0,\infty ) \to {G_4}\, with intial curve ${\hat{γ}}_{0} = \hat{γ} (\hat{s}, 0)$ {\hat \gamma _0} = \hat \gamma (\hat s,0)\. Let $\hat{γ} (\hat{s}, t)$ \hat \gamma (\hat s,t)\ be the position vector of a point on the curve at the time t and at the arc length ŝ. The time parameter t is the parameter for the deformation C_t of the curve. The arc-length of the curve is defined by $\hat{s} (\hat{u}, t) = \int_{0}^{\hat{u}} \sqrt{g (\hat{σ}, t)} d \hat{σ} .$ [\hat s(\hat u,t) = \int_0^{\hat u} \sqrt {g(\widehat\sigma ,t)} {\rm{d}}\hat \sigma . where $\sqrt{g} = ∥ \hat{γ} (\hat{σ}, t) ∥ .$ [\sqrt g = \parallel \widehat\gamma (\widehat\sigma ,t)\parallel . . Then the element of the arc-length is $d \hat{s} = \sqrt{g (\hat{u}, t)} d \hat{u}$ [d\widehats = \sqrt {g(\widehatu,t)}{\rm{d}}\widehatu , and the operator $\frac{\partial}{\partial \hat{s}}$ [\frac{\partial }{{\partial \widehats}} satisfies the following $\frac{\partial}{\partial \hat{s}} = \frac{1}{\sqrt{\hat{g}}} \frac{1}{\partial \hat{u}}, \frac{\partial \hat{s}}{\partial \hat{u}} = \sqrt{g .}$ [\frac{\partial }{{\partial \widehats}} = \frac{1}{{\sqrt {\widehatg} }}\frac{1}{{\partial \widehatu}} , \frac{{\partial \widehats}}{{\partial \widehatu}} = \sqrt {g.}

Definition 3.1

The curve $\hat{γ} (\hat{s}, t)$ \hat \gamma (\hat s,t)\ and its flow $\frac{\partial \hat{γ} (\hat{s}, t)}{\partial t}$ [\frac{{\partial \hat \gamma (\widehats,t)}}{{\partial t}} in four-dimensional Galilean space G₄ are said to be inextensible if (3.1) $\dot{\hat{s}} = \frac{\partial}{\partial t} ∥ \hat{γ} (\dot{\hat{s}}, t) ∥ = 0, i . e . g_{t} = 0 [14] .$ [\mathop {\widehats}\limits^. = \frac{\partial }{{\partial t}}\parallel \widehat\gamma (\mathop {\widehats}\limits^. ,t)\parallel = 0, i.e. g_{t\;} = 0\;[14].

Hence, the arclength of curve $\hat{γ} (\hat{s}, t)$ \hat \gamma (\hat s,t)\ is preserved.

The second binormal motions of the curves can be expressed by the velocity vector field (3.2) ${\hat{γ}}_{t} = \frac{\partial \hat{γ}}{\partial t} = v e .$ [\widehat\gamma _t = \frac{{\partial \widehat\gamma }}{{\partial t}} = v {\bf{e}}. where {t,n,b,e} is the orthonormal Frenet Frame to the curve C_t and v is the velocity vector in the direction of second binormal vector e and it is a function of curvature $\hat{k} (\hat{s}, t)$ \hat k(\hat s,t)\, torsion $\hat{τ} (\hat{s}, t) .$ \hat \tau (\hat s,t).\.

Theorem 3.1

The time evolution equations of the curvature and torsion for the inextensible timelike curve Ĉ_t are given by: (3.3) $k_{t} = v σ τ$ [k_t = v\sigma \tau and (3.4) $τ_{t} = ψ_{s} - φ σ$ [\tau _t = \psi _s - \phi \sigma where $ϕ = \frac{[- 2 v_{s} σ - v σ]}{k}$ [\;\varphi = \frac{{\left[ { - 2v_s \sigma - v\sigma } \right]}}{k} and $φ = \frac{[v_{ss} - v σ^{2}]}{k}$ [ \phi = \frac{{\left[ {v_{ss} - v\sigma ^2 } \right]}}{k} .

Proof

Take the û derivative of (3.2), then (3.5) ${\hat{γ}}_{tu} = \frac{\partial v}{\partial s} \frac{\partial s}{\partial u} e + v \frac{\partial e}{\partial s} \frac{\partial s}{\partial u} = \sqrt{g} (v_{s} e - v σ b) .$ [\widehat\gamma _{tu} = \frac{{\partial v}}{{\partial s}}\frac{{\partial s}}{{\partial u}}{\bf{e}} + v\frac{{\partial {\bf{e}}}}{{\partial s}}\frac{{\partial s}}{{\partial u}} = \sqrt g (v_s {\bf{e}} - v\sigma {\bf{b}}).

Since ${\hat{γ}}_{u} = \frac{\partial \hat{γ}}{\partial s} \frac{\partial s}{\partial u} = \sqrt{g} t$ [\widehat\gamma _u = \frac{{\partial \widehat\gamma }}{{\partial s}}\frac{{\partial s}}{{\partial u}} = \sqrt g {\bf{t}} then (3.6) ${\hat{γ}}_{ut} = t_{t} \sqrt{g} + \frac{g_{t}}{2 \sqrt{g}} t .$ [\widehat\gamma _{ut} = {\bf{t}}_t \sqrt g + \frac{{g_t }}{{2\sqrt g }}{\bf{t}}{\bf{.}}

Since the derivatives with respect to u and t commute, then (3.7) ${\hat{γ}}_{ut} = {\hat{γ}}_{tu} .$ [\widehat\gamma _{ut} = \widehat\gamma _{tu} {\bf{.}}

Substituting from (3.7) and (3.5) into (3.6), then (3.8) $g_{t} = u, t_{t} = v_{s} e - v σ b .$ [{\bf{g}}_t = u,\,{\bf{t}}_t = v_s {\bf{e}} - v\sigma {\bf{b}}.

Take the u derivative of the second equation of (3.8), then (3.9) $t_{tu} = = \sqrt{g} (v σ τ n + (- 2 v_{s} σ - v σ) b + (v_{ss} - v σ^{2}) e) .$ [{\bf{t}}_{tu} = = \sqrt g (v\sigma \tau {\bf{n}} + ( - 2v_s \sigma - v\sigma ){\bf{b}} + (v_{ss} - v\sigma ^2 ){\bf{e}}).

Since (3.9) $t_{u} = \frac{\partial t}{\partial s} \frac{\partial s}{\partial u} = k n \sqrt{g} .$ [{\bf{t}}_u = \frac{{\partial {\bf{t}}}}{{\partial s}}\frac{{\partial s}}{{\partial u}} = k{\bf{n}}\sqrt g .

Taking the t derivative of (3.9), then we have (3.10) $t_{ut} = \sqrt{g} (k_{t} n + k n_{t}) .$ [{\bf{t}}_{ut} = \sqrt g (k_t {\bf{n}} + k{\bf{n}}_t ).

Since t_ut = t_tu, we obtain $k_{t} = v σ τ .$ [k_t = v\sigma \tau .

On the other hand, we write $k n_{t} = (- 2 v_{s} σ - v σ) b + (v_{ss} - v σ^{2}) e$ [k{\bf{n}}_t = ( - 2v_s \sigma - v\sigma ){\bf{b}} + (v_{ss} - v\sigma ^2 ){\bf{e}} or (3.11) $n_{t} = \frac{1}{k} [(- 2 v_{s} σ - v σ) b + (v_{ss} - v σ^{2}) e] .$ [{\bf{n}}_t = \frac{1}{k}\left[ {( - 2v_s \sigma - v\sigma ){\bf{b}} + (v_{ss} - v\sigma ^2 ){\bf{e}}} \right].

If we choose $ϕ = \frac{[- 2 v_{s} σ - v σ]}{k}$ [\varphi = \frac{{\left[ { - 2v_s \sigma - v\sigma } \right]}}{k} and $φ = \frac{[v_{ss} - v σ^{2}]}{k}$ [ \phi = \frac{{\left[ {v_{ss} - v\sigma ^2 } \right]}}{k} , then we rewrite the equation of (3.11) as following (3.12) $n_{t} = ψ b + φ e .$ [{\bf{n}}_t = \psi {\bf{b}} + \phi {\bf{e}}.

Taking the u derivative of (3.12), then we have (3.13) $n_{tu} = \sqrt{g} ((ψ_{s} - φ σ) b + (- τ ψ) n + (ψ σ + φ_{s}) e) .$ [{\bf{n}}_{tu} = \sqrt g \left( {\left( {\psi _s - \phi \sigma } \right){\bf{b}} + \left( { - \tau \psi } \right)n + \left( {\psi \sigma + \phi _s } \right){\bf{e}}} \right).

Since (3.14) $n_{u} = \frac{\partial n}{\partial s} \frac{\partial s}{\partial u} = \sqrt{g} τ b$ [{\bf{n}}_u = \frac{{\partial {\bf{n}}}}{{\partial s}}\frac{{\partial s}}{{\partial u}} = \sqrt g \tau {\bf{b}}

Taking the t derivative of (3.14), then we have (3.15) $n_{ut} = \sqrt{g} (τ_{t} b + τ b_{t}) .$ [{\bf{n}}_{ut} = \sqrt g \left( {\tau _t {\bf{b}} + \tau {\bf{b}}_t } \right).

From (3.13) and (3.15), then we have (3.16) $τ_{t} = ψ_{s} - φ σ .$ [\tau _t = \psi _s - \phi \sigma .

Thus (3.17) $τ b_{t} = (- τ ψ) n + (ψ σ + φ_{s}) e .$ [\tau {\bf{b}}_t = ( - \tau \psi ){\bf{n}} + (\psi \sigma + \phi _s ){\bf{e}}.

Since (3.18) $b_{t} = - ψ n + (\frac{ψ σ + φ_{s}}{τ}) e .$ [{\bf{b}}_t = - \psi {\bf{n}} + (\frac{{\psi \sigma + \phi _s }}{\tau }){\bf{e}}.

If we take $K = \frac{ψ σ + φ_{s}}{τ}$ [K = \frac{{\psi \sigma + \phi _s }}{\tau } , then the time evolution equation for the first binormal vector b to the curve Ĉ_t is given as follows: (3.19) $b_{t} = - ψ n + K e .$ [b_t = - \psi {\bf{n}} + K{\bf{e}}.

Taking the u derivative of (3.19), then we have (3.20) $b_{tu} = \sqrt{g} (- (ψ + K) τ b - ψ_{s} n + K_{s} e) .$ [{\bf{b}}_{tu} = \sqrt g ( - (\psi + K)\tau {\bf{b}} - \psi _s {\bf{n}} + K_s {\bf{e}}).

Since (3.21) $b_{u} = \frac{\partial b}{\partial s} \frac{\partial s}{\partial u} = \sqrt{g} (- τ n + σ e) .$ [{\bf{b}}_u = \frac{{\partial {\bf{b}}}}{{\partial s}}\frac{{\partial s}}{{\partial u}} = \sqrt g ( - \tau {\bf{n}} + \sigma {\bf{e}}).

Taking the t derivative of (3.21), then we have (3.22) $b_{ut} = \sqrt{g} (- τ_{t} n - τ n_{t} + σ_{t} e + σ e_{t}) .$ [{\bf{b}}_{ut} = \sqrt g ( - \tau _t {\bf{n}} - \tau {\bf{n}}_t + \sigma _t {\bf{e}} + \sigma {\bf{e}}_t ).

Since b_tu = b_ut, then by substituting from (3.20) and from (3.22) into this equation, then we have the time evolution equation for the torsion τ_t: (3.23) $τ_{t} = ψ_{s} and K_{s} = σ_{t} .$ [\tau _t = \psi _s {\rm{and}}K_s = \sigma _t .

Thus $- (ψ + K) τ b = - τ n_{t} + σ e_{t}$ [ - (\psi + K)\tau {\bf{b}} = - \tau {\bf{n}}_t + \sigma {\bf{e}}_t also $σ e_{t} = τ n_{t} - (ψ + K) τ b .$ [\sigma {\bf{e}}_t = \tau {\bf{n}}_t - (\psi + K)\tau {\bf{b}}. or $e_{t} = - \frac{τ}{σ} K b + \frac{τ}{σ} φ e$ [{\bf{e}}_t = - \frac{\tau }{\sigma }K{\bf{b}} + \frac{\tau }{\sigma }\phi {\bf{e}} where τ second curvature function, σ third curvature function, $K = \frac{ψ σ + φ_{s}}{τ}$ [K = \frac{{\psi \sigma + \phi _s }}{\tau } , $ψ = \frac{- 2 v_{s} σ - v σ}{K}$ [\psi = \frac{{ - 2v_s \sigma - v\sigma }}{K} and $φ = \frac{v_{ss} - v σ^{2}}{K}$ [\phi = \frac{{v_{ss} - v\sigma ^2 }}{K} . Hence, the theorem holds.

Theorem 3.2

The time evolution of the Serret Frenet frame for the curve can be given in matrix form: ${(\begin{matrix} t \\ n \\ b \\ e \end{matrix})}_{t} = (\begin{matrix} 0 & 0 & - v σ & v_{s} \\ 0 & 0 & ψ & φ \\ 0 & - ψ & 0 & K \\ 0 & 0 & - \frac{τ}{σ} K & \frac{τ}{σ} φ \end{matrix}) (\begin{matrix} t \\ n \\ b \\ e \end{matrix}) .$ [\left( {\begin{array}{*{20}c} t \\ n \\ b \\ e \\\end{array}} \right)_t = \left( {\begin{array}{*{20}c} 0 & 0 & { - v\sigma } & {v_s } \\ 0 & 0 & \psi & \phi \\ 0 & { - \psi } & 0 & K \\ 0 & 0 & { - \frac{\tau }{\sigma }K} & {\frac{\tau }{\sigma }\phi } \\\end{array}} \right)\left( {\begin{array}{*{20}c} t \\ n \\ b \\ e \\\end{array}} \right).\

eISSN:: 2444-8656
Język:: Angielski

Częstotliwość wydawania:: Volume Open
Dziedziny czasopisma:: Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics

Kanał RSS czasopisma

Second Binormal Motions of Inextensible Curves in 4-dimensional Galilean Space

Data publikacji: 31 mar 2020

Zakres stron: 249 - 254

Otrzymano: 15 mar 2019

Przyjęty: 15 maj 2019

DOI: https://doi.org/10.2478/amns.2020.1.00022

Słowa kluczoweinextensible curves, second binormal motions

© 2020 Fatma Bulut Korkmaz et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Słowa kluczowe
inextensible curves, second binormal motions