Evolution of Generalized Space Curve as a Function of Its Local Geometry

Abstract

Kinematics of moving generalized curves in a n-dimensional Euclidean space is formulated in terms of intrinsic geometries. The evolution equations of the orthonormal frame and higher curvatures are obtained. The integrability conditions for the evolutions are given. Finally, applications in R2 are given and plotted.

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Abdel-All, N. , Mohamed, S. and Al-Dossary, M. (2014) Evolution of Generalized Space Curve as a Function of Its Local Geometry. Applied Mathematics, 5, 2381-2392. doi: 10.4236/am.2014.515230.

1. Introduction

The flow of a curve is called inelastic if the arclength of this curve is preserved. Inelastic curve flows have an importance in many applications such as engineering, computer vision [1] [2] , computer animation [3] and even structural mechanics [4] . Physically, inelastic curve flows give rise to motion which no strain energy is induced. There exist such motions in many physical applications. G. S. Chirikjian and J. W. Burdick [5] studied applications of inelastic curve flows. M. Gage, R. S. Hamilton [6] and M. A. Grayson [7] investigated shrinking of closed plane curves to a circle via the heat equation. Also, D. Y. Kwon and F. C. Park [8] [9] derived the evolution equation for an inelastic plane and space curve. Latifi et al. [10] studied inextensible flows of curves in Minkowski 3-space.

The connection between integrable systems and differential geometry of curves and surfaces has been important topic of intense research [11] [12] . Goldstein and Petrich [13] showed that the celebrated mKdV equation naturally arises from inextensible motion of curves in Euclidean geometry. Nakayama, Segur and Wadati [14] set up a correspondence between the mKdV hierarchy and inextensible motions of plane curves in Euclidean geometry. Integrable systems satisfied by the curvatures of curves under inextensible motions in projective geometries are identified in [15] . Inextensible flows of curves in Galilean space are investigated in [16] .

In this paper, we shall present a general formulation of evolving generalized curves in. The outline of this paper is as follows: In Section 2, we give the local differential geometry of curves in. In Sections 3 and 4, we describe the motion of generalized curves in. In Section 5, the integrability conditions for the considered model are obtained. In Section 6, we specialized the motion of curves to motion of plane curves (curves in). Finally, Section 7 is devoted to conclusion.

2. Geometric Preliminaries

A generalized curve in a -dimensional Euclidean space can be regarded as a Riemannian submanifold of dimension 1 in [17] .

Definition 1 A differentiable manifold of dimension 1 immersed in is a topological hausdorff space with a differentiable structure with dimension one, where is an open interval in and is a diffeomorphism mapping:

and belongs to some index set.

Definition 2 A Generalized curve in is an image of a diffomorphism, where is an open interval of. The representation of in is given by

(1.1)

where is called the parameter of the curve.

The representation (1.1) is called the regular parametric representation of in, when

Also represents an immersion in if, the parameter, in this case, is called the arclength parameter and is denoted by, is called arclength parametrization.

Frenet Frame

A Frenet frame is a moving reference frame of orthonormal vectors which are used to describe the curve locally at each point. It is the main tool in differential geometric treatment of curves as it is far easier and more natural to describe the local properties (e.g. curvature and torsion) in terms of local reference system than using a global one like the Euclidean coordinates.

Give a curve in which is regular of order. The Frenet frame for the curve is the set of orthonormal vectors (Frenet vectors), and they are constructed from the derivatives of, which are linearly independent vectors,.

Using the Gram-Schmidt orthogonalization algorithm which convert linearly independent vectors

into the orthonormal one as follows:

where

By this way, one obtain an orthonormal -tuple of vectors at, called the Frenet -frame associated with the generalized curve at the point, with is of class if. The derivatives of the frenet -frame at satisfy the following Frenet formulas:

(1.2)

where These equations can be written in a matrix form:

(1.3)

where

(1.4)

and are higher curvature functions or Euclidean curvatures of the curve. The -th Euclidean curvature gives the speed of rotation of the osculation -plane around the osculating -plane.

3. Dynamics of Curves in

Consider a smooth curve in -dimension space. Assume that is the parameter along the curve in. Let denotes the position vector of a point on the curve at time. The metric on the curve is:

(1.5)

The arclength along the curve is given by:

(1.6)

we use as coordinates of a point on the curve. At, consider the orthonormal frame such that is the tangent vector and denote the normal vectors at any point on the curve.

Dynamics of the curve in (motion of a point on the curve) can be specified by the form:

(1.7)

where are the velocities along the frame. Consider a local motion that is the velocities depend only on the local values of the curvatures.

Lemma 1 The evolution equation for the metric is given by:

(1.8)

Proof 1 Take the derivative of (1.5) and derivative of (1.7), and since, commute, then we have:

Using (1.2), then we have

where,

(1.9)

Then

Hence the lemma holds.

Lemma 2 For a simple closed curve, the evolution of the length of the curve is given by:

Proof 2 From the definition of the length, we have

(1.10)

Substitute from (1.8) into (1.10), then the lemma holds.

4. Main Results

Definition 3 An inelastic curve is a curve whose length is preserved, i.e., it doesn't evolve in time.

(1.11)

The necessary and sufficient conditions for inelastic flow are then given by the following theorem:

Theorem 1 The flow of the curve is inelastic if and only if

Proof 3 Assume that the curve is inelastic.

From (1.6), the variation of the arclength is

(1.12)

Substitute from (1.8) into (1.12), then

Since the curve is inelastic, so, hence

Assume that Substitute from this equation into (1.8), so, then, this means that the arclength of the curve is preserved, hence the curve is inelastic.

Theorem 2 Consider an elastic curve. For the curve flow, then 1) The evolution for the frame, can be given in a matrix form:

(1.13)

where is the evolution matrix and it takes the form:

where the elements of the matrix are given explicitly by:

(1.14)

2) The evolution equations for the curvatures take the form:

(1.15)

Proof 4 Consider the elastic curve i.e.,. Take the derivative of (1.7), then we have:

(1.16)

Since, take the derivative of this equation, then we have

(1.17)

Since

(1.18)

Substitute from (1.9), (1.16) and (1.17) into (1.18), then we have

(1.19)

Take the derivative of (1.19), then we have:

(1.20)

Since, take the derivative of this equation, then we have

(1.21)

Since

(1.22)

Substitute from (1.20) and (1.21) into (1.22), then we have

(1.23)

Since, take the derivative of this equation, then we have

(1.24)

Take the derivative of (1.23), then we have

(1.25)

Since

(1.26)

Substitute from (1.24) and (1.25) into (1.26), and use the first equation of (1.23), then we have

(1.27)

Take the derivative of the second equation of (1.27), then we have

(1.28)

Since, take the derivative of this equation and use the first equation of (1.27), then we have

(1.29)

Since

(1.30)

Substitute from (1.28) and (1.29) into (1.30), then we have

(1.31)

By using the mathematical induction, we can extend the previous results to -dimensional space as follows:

where, and

(1.32)

We can rewrite (1.9) and the third equation of (1.23), (1.27) and (1.31) as follows:

(1.33)

Hence

(1.34)

So we can rewrite (1.32) in the following form:

(1.35)

By using the mathematical induction, we can extend the results in the first equation of (1.23), (1.27) and (1.31) as follows:

(1.36)

Hence the theorem holds.

Lemma 3 If the curve flow (1.7) is inelastic, then the evolution equations for curvatures (1.36) take the form:

(1.37)

Proof 5 If the curve flow (1.7) is inelastic, then

(1.38)

Then, substitute from (1.38) into (1.36), then the lemma holds.

5. Integrability Conditions

Theorem 3 The curve is inelastic curve if and only if the integrability condition (sometimes called the zero curvature condition) is given by:

(1.39)

where is the Lie bracket.

Proof 6 Consider the Frenet frame that satisfy (1.4) and (1.35). Since

(1.40)

Take the derivative of (1.40) and use (1.13), then we have

(1.41)

Differentiating (1.13) with respect to and use (1.40), then

(1.42)

From (1.41) and (1.42), then

(1.43)

First, If the curve is inelastic, so and and commute, then, hence

Second, Assume that the integrability condition (the zero curvature condition) is satisfied, then

(1.44)

From (1.4) and (1.35), we have

(1.45)

Differentiating (1.4) with respect to and (1.35) with respect to and use (1.36), then we have

(1.46)

Substitute from (1.45) and (1.46) into (1.44), then we have

(1.47)

Hence

Since for. Then. Hence, i.e., the arclength is preserved. Hence the curve is inelastic.

Theorem 4 In -dimensional Euclidean space, consider inelastic curve. If the matrices and are abelian, then the elements in the evolution matrix take the form:

Proof 7 Since the matrices and are abelian, so, then the integrability condition (1.39) takes the form:

(1.48)

Since the curve is inelastic, so, then

(1.49)

Substitute from (1.49) into (1.48), then for, we have

By using the mathematical induction, we can extend the previous results to -dimensional space, then we have

6. Applications

Here we give some applications for time evolution equations for plane curve. We are in a position to derive time evolution of geometrical quantities. For, and from (1.13), we have motion of the Frenet frame of the curve in the plane.

Lemma 4 In -dimensional Euclidean space, consider an elastic curve. The time evolution equation for the frame is given by

where,

Lemma 5 The time evolution equation for the curvature of the curve in is given explicitly by

(1.50)

This equation represents a quasilinear parabolic partial differential equation (PDE). This result coincide with [18] .

Example 1 If

Then (1.50) takes the form:

(1.51)

The solution of the PDE (1.51) is

where is constant. The curvature of the curve is plotted as a function of and (Figure 1(a)), and for different values of, the curvature is plotted (Figure 1(b)).

Example 2 If

Then (1.50) takes the form:

(1.52)

The solution of the PDE (1.52) is

where and are constants. The curvature of the curve is plotted as a function of and (Figure 2(a)), and for different values of, the curvature is plotted (Figure 2(b)).

Example 3 If

Then (1.50) takes the form:

(1.53)

The solution of the PDE (1.53) is

where and are constants. The curvature of the curve is plotted as a function of and (Figure 3(a)), and for different values of, the curvature is plotted (Figure 3(b)).

(a)(b)

Figure 1. The curvature of the curve for

(a)(b)

Figure 2. The curvature of the curve for

(a)(b)

Figure 3. The curvature of the curve for

7. Conclusion

In this paper, we have discussed the motion of curves in -dimension Euclidean space. We derived the evolution equations of the orthonormal frame and evolution equations for the higher curvatures. We get the integrability conditions for the evolutions. Moreover, we give some examples of motions of elastic curves in the plane.

Acknowledgements

The authors would like to thank the referee for helpful suggestions which improve the presentation of this work.

Conflicts of Interest

The authors declare no conflicts of interest.

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