Abstract
We study the inextensible flows of curves in 3-dimensional Euclidean space . The main purpose of this paper is constructing and plotting the surfaces that are generated from the motion of inextensible curves in . Also, we study some geometric properties of those surfaces. We give some examples about the inextensible flows of curves in and we determine the curves from their intrinsic equations (curvature and torsion). Finally, we determine and plot the surfaces that are generated by the motion of those curves by using Mathematica 7.
1. Introduction
The evolution of curves and surfaces has important applications in many fields such as computer vision [1, 2], computer animation [3], and image processing [4]. The motion of curves and surfaces in leads to nonlinear evolution equations, which are often integrable. The connection between integrable systems and the differential geometry of curves has been studied extensively. Some integrable systems arise from invariant curve flows in certain geometries such as affine and centroaffine geometries [5–7] and similarity and projective geometries [8, 9]. Motion of curves in Minkowski space is studied in [10–12]. Hashimoto [13] showed that the integrable nonlinear Schrodinger equation (NLS)is equivalent to the system for the curvatures and of curves in :via the so-called Hashimoto transformation . System (2) is equivalent to the vortex filament equation where is the binormal vector of .
In [14], Schief and Rogers studied the binormal motion of curves of constant curvature or torsion.
This line of research has been extended to motions of curves in three-dimensional space forms.
Recently, Abdel-All et al. [15–18] constructed new geometrical models of motion of plane curves. Also, they constructed a Hashimoto surface from its fundamental coefficients via numerical integration of Gauss-Weingarten equations and fundamental theorem of surfaces. Also, they studied kinematics of moving generalized curves in -dimensional Euclidean space in terms of intrinsic geometries. Mohamed [19] studied the motions of inextensible curves in spherical space .
In this paper, we will present the flows of curves in . The outline of this paper is as follows.
In Section 2, we study the geometry of curves in . In Sections 3 and 4, we study the motion of curves in and we get the time evolution of Serret-Frenet frame and the evolution of curvatures. In Section 5, we study the geometric properties of the surfaces that are generated by the motion of the family of curves . In Section 6, we give some examples of motions of inextensible curves in , and we plot the surfaces that are generated by the motion of those curves. For these surfaces, we study the Gaussian and Mean curvatures. Finally, Section 7 is devoted to conclusion.
2. Geometric Preliminaries
Consider a smooth curve in a -dimensional space. Assume that is the parameter along the curve in . Let denote the position vector of a point on the curve. The metric on the curve iswhere is the parameter of the curve. The arc length along the curve is given bywe use as coordinates of a point on the curve.
Consider the orthonormal frame , such that is the tangent vector and denote the normal vectors at any point on the curve.
Lemma 1. The Frenet frame for the curve in satisfies the following:where
Lemma 2. Consider the curve with an arbitrary parameter . Then the Serret-Frenet frame satisfieswhere and are given as in (6).
3. Curve Evolution
An evolving curve can be considered as a family of curves parametrized by time. This means that each curve in the family is a mapping that assigns for each space parameter and each time parameter ; there is a point . An evolution equation which is a differential equation that describes the evolution of in time can be specified by the formwhere are the velocities along the frame . Consider a local motion; that is, the velocities depend only on the local values of the curvatures .
4. Main Results
From [18], we considered the curve . For the curve flow where are the velocities in the direction of , we had the following.
Lemma 3. The evolution equation for the metric is given bywhere
Theorem 4. The evolution for the frame can be given in a matrix formwhere is the evolution matrix and it takes the form where the elements of the matrix are given explicitly by
Theorem 5. The time evolution equations for the curvatures take the formFor , we can study the motion of curves in ; we choose , and ; then we have the following.
Theorem 6. The time evolution of the Serret-Frenet frame can be written in matrix form as follows:where
Theorem 7. The time evolution of the curvature and torsion of the curve can be given by
Definition 8. An inextensible curve is a curve whose length is preserved; that is, it does not evolve in time:
The necessary and sufficient conditions for inextensible flows are then given by the following theorem.
Theorem 9. The flow of the curve is inextensible if and only if
Lemma 10. If the curve is inextensible (), then the evolution equations for the curvature and torsion (19) are
Then the PDE system (21) can be written explicitly in the following form:
5. Examples of Inextensible Flows of Curves in
Example 1. Ifthe PDE system (22) takes the formOne solution of this system iswhere , , and are constants.
The curvature of the family of curves as a function of the coordinates and is plotted in Figure 1.
Substitute (23) and (25) into systems (6) and (16) and solve them numerically. Then we can get the family of curves , so we can determine the surface that is generated by this family of curves (Figure 2).
Example 2. Ifand assuming that , then PDE system (22) takes the form One solution of this system iswhere , , , and are constants.
The curvature of the family of curves as a function of the coordinates and is plotted in Figure 3.
Substitute (26) and (28) into systems (6) and (16) and solve them numerically. Then we can get the family of curves , so we can determine the surface that is generated by this family of curves (Figure 4).
Example 3. Ifthen the PDE system (22) takes the form One solution of this system iswhere and are constants.
The curvature of the family of curves as a function of the coordinates and is plotted in Figure 5.
Substitute (29) and (31) into systems (6) and (16) and solve them numerically. Then we can get the family of curves , so we can determine the surface that is generated by this family of curves (Figure 6).
5.1. Examples of Binormal Motion of Inextensible Curves
Consider the binormal motion of inextensible curves in , so . Then the evolution equation (9) takes the form
Lemma 11. Consider the binormal motion of inextensible curves in ; then the time evolution of the Serret-Frenet frame can be written in matrix form as follows:whereThe PDE system (22) takes the form
Example 4. The famous example of binormal motion is the motion of the vortex filament in , where the binormal velocity equals the curvature of the curve , and the evolution equation is If , then the PDE system (35) takes the formThe solution of the PDE (37) iswhere , and are constants.
The curvature of the family of curves as a function of the coordinates and is plotted in Figures 7 and 8.
Substitute (38) into (6), (33) for and solve them numerically. Then we can get the family of curves , so we can determine the surface that is generated by this family of curves. For see (Figure 9), and for see (Figure 10).
Example 5. If , then (35) takes the formThe solution of the PDE system (39) iswhere , , , , and are constants.
The curvature of the family of curves as a function of the coordinates and is plotted in Figure 11.
Substitute (40) into (6), (33) and solve them numerically. Then we can get the family of curves , so we can determine the surface that is generated by this family of curves (Figure 12).
6. Geometric Properties of the Generated Surfaces
Let be the surface that is generated by the motion of the family of curves . In this section, we study some geometric properties of these surfaces.
Lemma 12. The first fundamental form of the surface in is given by where are the first fundamental quantities and they are given by
Lemma 13. The unit normal vector to the surface in at a point on the surface is given by
Lemma 14. The second fundamental form of the surface in is given by where are the second fundamental quantities and they are given by where , , and are given from (18).
Lemma 15. For the surface in , the Gaussian and the Mean curvatures and , respectively, are given by
Lemma 16. The Gaussian curvature and the Mean curvature for the surface in (Figure 2) are given by
Lemma 17. The Gaussian curvature and the Mean curvature for the surface in (Figure 4) are given by Hence the surface in Figure 4 is developable surface.
Lemma 18. The Gaussian curvature and the Mean curvature for the surface in (Figure 6) are given by
Lemma 19. The Gaussian curvature and the Mean curvature for the surfaces in Figures 9 and 10 are given by
Lemma 20. The Gaussian curvature and the Mean curvature for the surface in (Figure 12) are given by Hence the surface in (Figure 12) is developable surface.
7. Conclusion
In this paper we studied the inextensible flows of curves in . We constructed and plotted the surfaces that are generated from the motion of inextensible curves in . Also, we studied some geometric properties of those surfaces.
Competing Interests
The authors declare that there are no competing interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank Professor Dr. Nassar Hassan Abdel-All, Emeritus Professor, Department of Mathematics, Faculty of Science, Assiut University, for helpful discussions and valuable suggestions about the topic of this paper. The authors also would like to thank the referees for their helpful comments and suggestions.