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

Fatma Bulut Korkmaz 1  and Mehmet Bektaandş 2
  • 1 Department of Mathematics, Faculty of Arts and Sciences, 13000, Bitlis
  • 2 Department of Mathematics, Faculty of Science, 23119
Fatma Bulut Korkmaz
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  • Department of Mathematics, Faculty of Arts and Sciences, Bitlis Eren University, 13000, Bitlis
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and Mehmet Bektaandş

Abstract

In our study, we give the associated evolution equations for curvature and torsion as a system of partial differential equations. In addition, we study second binormal motions of inextensible curves in 4-dimensional Galilean space.

1 Introduction

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 . 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.

2 Geometric Preliminaries

The scalar product of two vectors U=(u1,u2,u3,u4) and V=(v1,v2,v3,v4) in G4 is defined by

<U,V>={u1v1,ifu10orv10u2v2+u3v3+u4v4,ifu1=0,v1=0.

The Galilean cross product in G4 for the vectors u=(u1,u2,u3,u4), v=(v1,v2,v3,v4), and w=(w1,w2,w3,w4) is defined by

uvw=|0e2e3e4u1u2u3u4v1v2v3v4ω1ω2ω3ω4|,
where ei 1 ≤ i ≤ 4, are the standard basis vectors.

The norm of vector U=(u1,u2,u3,u4) is defined by

U=|<U,U>|[16].

Let α : IRG4, 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

t=α(s)=(1,y(s),z(s),w(s)).

Since t is a unit vector, we can express

<t,t>=1.

Differentiating (2.4) with respect to s, we have

<t,t>=0.

The vector function t gives us the rotation measurement of the curve α. The real valued function

κ(s)=t(s)=(y)2+(z)2+(ω)2
is called the first curvature of the curve α. We assume that κ(s) 0 for all sl. Similar to space G3, the principal vector is defined by
n(s)=t(s)κ(s);
in other words
n(s)=1κ(s)(0,y(s),z(s),ω(s)).

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).

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)=1τ(s)(0,(y(s)κ(s)),(z(s)κ(s)),(ω(s)κ(s))).

Thus the vector b(s) is perpendicular to both t and n. The fourth unit vector is defined by

e(s)=μt(s)n(s)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

σ=b,e.

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

t,t=n,n=b,b=e,e=1,t,n=t,b=t,e=n,b=n,e=b,e=0.

For the curve α in G4, we have the following Frenet-Serret equations [16]:

t=κ(s)n(s),n=τ(s)b(s),b=τ(s)n(s)+σ(s)e(s),e=σ(s)b(s).

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

Let γ^0:IG4, be a regular curve in four-dimensional Galilean space G4.

Consider the family of curves Ct:γ^(s^,t), where γ^(s^,t):Ix[0,)G4, with intial curve γ^0=γ^(s^,0). Let γ^(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 Ct of the curve. The arc-length of the curve is defined by

s^(u^,t)=0u^g(σ^,t)dσ^.
where g=γ^(σ^,t). . Then the element of the arc-length is ds^=g(u^,t)du^ , and the operator s^ satisfies the following
s^=1g^1u^,s^u^=g.

Definition 3.1

The curve γ^(s^,t) and its flow γ^(s^,t)t in four-dimensional Galilean space G4 are said to be inextensible if

s^.=tγ^(s^.,t)=0,i.e.gt=0[14].

Hence, the arclength of curve γ^(s^,t) is preserved.

The second binormal motions of the curves can be expressed by the velocity vector field

γ^t=γ^t=ve.
where {t,n,b,e} is the orthonormal Frenet Frame to the curve Ct and v is the velocity vector in the direction of second binormal vector e and it is a function of curvature k^(s^,t), torsion τ^(s^,t)..

Theorem 3.1

The time evolution equations of the curvature and torsion for the inextensible timelike curve Ĉt are given by:

kt=vστ
and
τt=ψsφσ
where ϕ=[2vsσvσ]k and φ=[vssvσ2]k .

Proof

Take the û derivative of (3.2), then

γ^tu=vssue+vessu=g(vsevσb).

Since

γ^u=γ^ssu=gt
then
γ^ut=ttg+gt2gt.

Since the derivatives with respect to u and t commute, then

γ^ut=γ^tu.

Substituting from (3.7) and (3.5) into (3.6), then

gt=u,tt=vsevσb.

Take the u derivative of the second equation of (3.8), then

ttu==g(vστn+(2vsσvσ)b+(vssvσ2)e).

Since

tu=tssu=kng.

Taking the t derivative of (3.9), then we have

tut=g(ktn+knt).

Since tut = ttu, we obtain

kt=vστ.

On the other hand, we write

knt=(2vsσvσ)b+(vssvσ2)e
or
nt=1k[(2vsσvσ)b+(vssvσ2)e].

If we choose ϕ=[2vsσvσ]k and φ=[vssvσ2]k , then we rewrite the equation of (3.11) as following

nt=ψb+φe.

Taking the u derivative of (3.12), then we have

ntu=g((ψsφσ)b+(τψ)n+(ψσ+φs)e).

Since

nu=nssu=gτb

Taking the t derivative of (3.14), then we have

nut=g(τtb+τbt).

From (3.13) and (3.15), then we have

τt=ψsφσ.

Thus

τbt=(τψ)n+(ψσ+φs)e.

Since

bt=ψn+(ψσ+φsτ)e.

If we take K=ψσ+φsτ , then the time evolution equation for the first binormal vector b to the curve Ĉt is given as follows:

bt=ψn+Ke.

Taking the u derivative of (3.19), then we have

btu=g((ψ+K)τbψsn+Kse).

Since

bu=bssu=g(τn+σe).

Taking the t derivative of (3.21), then we have

but=g(τtnτnt+σte+σet).

Since btu = but, then by substituting from (3.20) and from (3.22) into this equation, then we have the time evolution equation for the torsion τt:

τt=ψsandKs=σt.

Thus

(ψ+K)τb=τnt+σet
also
σet=τnt(ψ+K)τb.
or
et=τσKb+τσφe
where τ second curvature function, σ third curvature function, K=ψσ+φsτ , ψ=2vsσvσK and φ=vssvσ2K . Hence, the theorem holds.

Theorem 3.2

The time evolution of the Serret Frenet frame for the curve can be given in matrix form:

(tnbe)t=(00vσvs00ψφ0ψ0K00τσKτσφ)(tnbe).

References

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    G. Chirikjian, J. Burdick, A modal approach to hyper-redundant manipulator kinematics, IEEE Trans. Robot. Autom. 10, 343–354,1994.

  • [2]

    D.Y. Kwon, F.C. Park and D.P. Chi, Inextensible flows of curves and developable surfaces, Appl. Math. Lett. 2005, 18, 1156–1162.

  • [3]

    M. Ergut, E. Turhan, T. Körpinar, Characterization of inextensible flows of spacelike curves with Sabban frame in S 1 2 [_1^2 . Bol. Soc. Parana. Mat. Vol. 31 No. 2 (2013), 47–53.

  • [4]

    O., Gökmen, M. Tosun and S. Ozkaldı Karakus, A note on inextensible flows of curves in En. Int. Electron. J. Geom. Vol. 6 No. 2 (2013), 118–124.

  • [5]

    R. A. Hussien and S. G. Mohamed, Generated Surfaces via Inextensible Flows of Curves in R3, J. Appl. Math. Volume 2016, Article ID 6178961, 8 pages (2016), Article ID 6178961, 10.1155/2016/6178961.

  • [6]

    M. Kass, A. Witkin, D. Terzopoulos, Snakes: active contour models, in: Proc. 1st Int. Conference on Computer Vision, 259–268,1987.

  • [7]

    T. Körpinar, and E. Turhan A New Version of Inextensible Flows of Spacelike Curves with Timelike B2 in Minkowski Space-Time E 1 4 [_1^4 , Differ. Equ. Dyn. Syst., 2013, 21 (3), 281–290.

  • [8]

    T. Körpinar, and E. Turhan, Approximation for inextensible flows of curves in E3. Bol. Soc. Parana. Mat. Vol. (3) 32 No. 2 (2014), 45–54.

  • [9]

    M. Desbrun, M.P. Cani-Gascuel, Active implicit surface for animation, in: Proc. Graphics Interface Canadian Inf. Process. Soc., 143–150, 1998.

  • [10]

    H. Mochiyama, E. Shimemura, H. Kobayashi, Shape control of manipulators with hyper degrees of freedom, Int. J. Robot. Res., 18, 584–600, 1999.

  • [11]

    D. Latifi, A. Razavi A, Inextensible Flows of Curves in Minkowskian Space, Adv. Studies Theor. Phys. 2(16): 761–768, 2008.

  • [12]

    H.Q. Lu, J.S. Todhunter, T.W. Sze, Congruence conditions for nonplanar developable surfaces and their application to surface recognition, CVGIP, Image Underst. 56, 265–285, 1993.

  • [13]

    S. G. Mohamed, Inextensible Flows of Spacelike Curves in De-Sitter Space S 1 2 [_1^2 , Appl. Math. Inf. Sci. Lett. 6, No. 2, 75–83 (2018).

  • [14]

    S. G. Mohamed, Binormal motions of inextensible Curves in de-sitter space S 1 2 [_1^2 , Journal of Egyptian Mathematical Society 25 313–318, 2017.

  • [15]

    H. Öztekin, Special Bertrand Curves in 4D Galilean Space, Mathematical Problems in Engineering Volume 2014, Article ID 318458, 7 pages http://dx.doi.org/10.1155/2014/318458.

  • [16]

    S. Yilmaz, “Construction of the Frenet-Serret frame of a curve in 4D-Galilean space and some applications,” International Journal of Physical Sciences, vol. 5, no.8, pp. 1284–1289,2010.

  • [17]

    D.J. Unger, Developable surfaces in elastoplastic fracture mechanics, Int. J. Fract. 50, 33–38, 1991.

  • [18]

    M. Yeneroğlu, On new characterization of inextensible flows of space-like curves in de Sitter space, Open Mathemetics, Vol14(1), 2016.

  • [19]

    Z. Yüzbaşı and M. Bektaş, A Note on İnextensible Flows of Partially and Pseudo Null Curves In E 1 4 [E_1^4 . arXiv.org math arXiv:1303.2956v1

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  • [1]

    G. Chirikjian, J. Burdick, A modal approach to hyper-redundant manipulator kinematics, IEEE Trans. Robot. Autom. 10, 343–354,1994.

  • [2]

    D.Y. Kwon, F.C. Park and D.P. Chi, Inextensible flows of curves and developable surfaces, Appl. Math. Lett. 2005, 18, 1156–1162.

  • [3]

    M. Ergut, E. Turhan, T. Körpinar, Characterization of inextensible flows of spacelike curves with Sabban frame in S 1 2 [_1^2 . Bol. Soc. Parana. Mat. Vol. 31 No. 2 (2013), 47–53.

  • [4]

    O., Gökmen, M. Tosun and S. Ozkaldı Karakus, A note on inextensible flows of curves in En. Int. Electron. J. Geom. Vol. 6 No. 2 (2013), 118–124.

  • [5]

    R. A. Hussien and S. G. Mohamed, Generated Surfaces via Inextensible Flows of Curves in R3, J. Appl. Math. Volume 2016, Article ID 6178961, 8 pages (2016), Article ID 6178961, 10.1155/2016/6178961.

  • [6]

    M. Kass, A. Witkin, D. Terzopoulos, Snakes: active contour models, in: Proc. 1st Int. Conference on Computer Vision, 259–268,1987.

  • [7]

    T. Körpinar, and E. Turhan A New Version of Inextensible Flows of Spacelike Curves with Timelike B2 in Minkowski Space-Time E 1 4 [_1^4 , Differ. Equ. Dyn. Syst., 2013, 21 (3), 281–290.

  • [8]

    T. Körpinar, and E. Turhan, Approximation for inextensible flows of curves in E3. Bol. Soc. Parana. Mat. Vol. (3) 32 No. 2 (2014), 45–54.

  • [9]

    M. Desbrun, M.P. Cani-Gascuel, Active implicit surface for animation, in: Proc. Graphics Interface Canadian Inf. Process. Soc., 143–150, 1998.

  • [10]

    H. Mochiyama, E. Shimemura, H. Kobayashi, Shape control of manipulators with hyper degrees of freedom, Int. J. Robot. Res., 18, 584–600, 1999.

  • [11]

    D. Latifi, A. Razavi A, Inextensible Flows of Curves in Minkowskian Space, Adv. Studies Theor. Phys. 2(16): 761–768, 2008.

  • [12]

    H.Q. Lu, J.S. Todhunter, T.W. Sze, Congruence conditions for nonplanar developable surfaces and their application to surface recognition, CVGIP, Image Underst. 56, 265–285, 1993.

  • [13]

    S. G. Mohamed, Inextensible Flows of Spacelike Curves in De-Sitter Space S 1 2 [_1^2 , Appl. Math. Inf. Sci. Lett. 6, No. 2, 75–83 (2018).

  • [14]

    S. G. Mohamed, Binormal motions of inextensible Curves in de-sitter space S 1 2 [_1^2 , Journal of Egyptian Mathematical Society 25 313–318, 2017.

  • [15]

    H. Öztekin, Special Bertrand Curves in 4D Galilean Space, Mathematical Problems in Engineering Volume 2014, Article ID 318458, 7 pages http://dx.doi.org/10.1155/2014/318458.

  • [16]

    S. Yilmaz, “Construction of the Frenet-Serret frame of a curve in 4D-Galilean space and some applications,” International Journal of Physical Sciences, vol. 5, no.8, pp. 1284–1289,2010.

  • [17]

    D.J. Unger, Developable surfaces in elastoplastic fracture mechanics, Int. J. Fract. 50, 33–38, 1991.

  • [18]

    M. Yeneroğlu, On new characterization of inextensible flows of space-like curves in de Sitter space, Open Mathemetics, Vol14(1), 2016.

  • [19]

    Z. Yüzbaşı and M. Bektaş, A Note on İnextensible Flows of Partially and Pseudo Null Curves In E 1 4 [E_1^4 . arXiv.org math arXiv:1303.2956v1

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