On Characterization of Inextensible Flows of Curves According to Type-2 Bishop Frame

In this paper, we study inextensible flows of curves according to type-2 Bishop frame in Euclidean 3-space. Necessary and sufficient conditions for an inextensible curve flow are expressed as a partial differential equation involving the curvature.


INTRODUCTION
The flow of a curve is called to be inextensible if the arc-length of a curve is preserved.Inextensible curve flows have growing importance in many applications such as engineering, computer vision, structural mechanics and computer animation [1][2][3][4][5].The terms "inextensible" and "extensible" mostly come up in physics.There are inextensible and extensible collisions in physics.In extensible collision, both the kinetic energy and momentum are conserved.In inextensible collision, the kinetic energy is not conserved in the collision; however, the momentum is conserved.One of the oldest topics in the calculus of variations is the study of the elastic rod which, according to Daniel Bernoulli's idealization, minimizes total squared curvature among curves of the same length and first order boundary data.The classical term extensible refers to a curve in the plane or 3   E which represents such a rod in equilibrium.
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 can be described by inextensible curve flows.Such motions arise quite naturally in a wide range of physical applications.For example, both Chirikjian and Burdick [6] and Mochiyama et al. [7] study the shape control of hyper-redundant, or snake-like, robots.Gage and Hamilton [8] and Grayson [9] investigated shrinking of closed plane curves to a circle via the heat equation.Kwon and Park [10] derived the evolution equation for an inextensible plane and space curve .Besides, Latifi, Razavi [11] studied inextensible flows of curves in Minkowskian space.In this paper, we study inextensible flows of curves according to type-2 Bishop frame in Euclidean 3-space 3 E .We hope that these results will be helpful to mathematicians who are specialized on this area.  .Denote by   ( ), ( ), ( ) T s N s B s the moving Frenet frame along the unit speed curve  .Then the Frenet formulas are given by 00 0 00

PRELIMINARIES
Here, , TN and B are the tangent, the principal normal and the binormal vector fields of the curves, respectively.
() s  and () s  are called, curvature and torsion of the curve ,  respectively.The Bishop frame or parallel transport frame is an alternative approach to defining a moving frame that is well defined even when the curve has vanishing second derivative.We can parallel transport an orthonormal frame along a curve simply by parallel transporting each component of the frame. is defined by [12,13] The relation matrix between Frenet-Serret and type-2 Bishop frames can be expressed Here, the type-2 Bishop curvatures are defined by 12 ( ) cos ( ), ( ) sin ( ). k s s k s s It can be also deduced as 1 The frame   ,.
The operator of Any flow of F can be given by 12 , where ,, f g h are tangential, principal normal, binormal speeds of the curve in 3 E , respectively.We put 0 ( , ) u s u t vdu   , which is called the arc-length variation of curve F .From this, the requirement that the curve not be subject to any elongation or compression can be expressed by condition 0 ( , ) 0 By using type-2 Bishop frame, we get by using features of inner product and after straightforward calculations from above equation, we get () Proof.Using definition of F , we have

TF fN gN hB t t s s
Using the type-2 Bishop frame and after straightforward calculations, we get Substituting ( 12) in (15) and If 1 N take instead of T , we have Now differentiate the 2-type Bishop frame by t , we obtain From the above and using () ) .

Bh fk gk N N ts
The following theorem states the conditions on the curvature and the torsion for the curve flow ( , ) F u t to be inextensible.
On the other hand, from type-2 Bishop frame we have Substituting ( 14) in ( 16) and after straightforward calculations, we get After straightforward calculations, we get

3 :
IE  be an arbitrary curve in3

be a unit speed regular curve in 3 E
. The type-2 Bishop frame of the () s (Necessary and Sufficient Conditions for an Inextensible Flow) Let