Journal of Applied Mathematics
Volume 2015 (2015), Article ID 218956, 11 pages
http://dx.doi.org/10.1155/2015/218956

Research Article

Motion of Bishop Frenet Offsets of Ruled Surfaces in

H. N. Abd-Ellah

Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt

Received 7 August 2015; Revised 11 October 2015; Accepted 20 October 2015

Academic Editor: Hui-Shen Shen

Copyright © 2015 H. N. Abd-Ellah. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The main goal of this paper is to study the motion of two associated ruled surfaces in Euclidean 3-space . In particular, the motion of Bishop Frenet offsets of ruled surfaces is investigated. Additionally, the characteristic properties for such ruled surfaces are given. Finally, an application is presented and plotted using computer aided geometric design.

1. Introduction

Motion is to add the time element to our curves and surfaces. Motion theory has received a great deal of attention from mathematical physics, biology, dynamical systems, image processing, and computer vision. The problem is interesting since we may set two different subjects on the same theoretical basis. One is a geometrical interpretation of integrable systems. Connections between the differential geometry of curve motions and the integrable systems have been discussed. The analysis is extended to more general types of motion and other integrable systems [1–3]. The other is surface dynamics, the dynamics of shapes in physical and biological systems, as in crystal growth.

A variety of dynamics of shapes in physics, chemistry, and biology are modeled in terms of motion of surfaces and interfaces, and some dynamics of shapes are reduced to motion of curves. These models are specified by velocity fields or acceleration fields which are local or nonlocal functionals of the intrinsic quantities of curves. In physics, it is very interesting to describe motions of patterns such as interfaces, wave fronts, and defects [4]. Applications include kinematics of interfaces in crystal growth [5, 6], deformation of vortex filaments in inviscid fluid, and viscous fingering in a Hele-Shaw cell [7, 8]. The subject of how space curves or surfaces evolve in time is of great interest and has been investigated by many authors [9–22].

Classical differential geometry of the curves may be surrounded by the topics of general helices, involute-evolute curve couples, spherical curves, and Bertrand curves. Such special curves are investigated and used in some real world problems like mechanical design or robotics by the well-known Frenet-Serret equations because we think of curves as the path of a moving particle in the Euclidean space [23]. Thereafter researchers aimed to determine another moving frame for a regular curve. In 1975, Bishop pioneered “Bishop frame” by means of parallel vector fields. This special frame is also called a “parallel” or “alternative” frame of the curves [24].

A practical application of Bishop frame is that it is used in the area of biology and computer graphics. For example, it may be possible to compute information about the shape of sequences of DNA using a curve defined by Bishop frame. The Bishop frame may also provide a new way to control virtual cameras in computer animations [25]. Nowadays a good deal of research has been done on Bishop frames in Euclidean space [26, 27], in Minkowski space [28, 29], and in dual space [30]. Recently, the authors in [31] introduced a new version of the Bishop frame and called it a “type 2 Bishop frame” and this special frame is extended to study many surfaces [32, 33].

Studies related to offset profiles date back to the nineteenth century. Offsets curves play an important role in areas of CAD/CAM, robotics, cam design, and many industrial applications. In particular they are used in mathematical modeling of cutting paths milling machines. The classic work in this area is that of Bertrand [34], who studied curve pairs which have common principal normals. Such curves are referred to as Bertrand curves and can be considered as offsets of one another. The theory of the Mannheim curves has been extended in the three-dimensional Euclidean space by [35, 36].

Recently, there have been a number of studies of offsets ruled surfaces [37, 38], studied Bertrand and Mannheim offsets of ruled surfaces. Pottmann et al. [39] presented classical and circular offsets of rational ruled surfaces. More recently, Soliman et al. [40] studied geometric properties and invariants of Mannheim offsets of timelike ruled surface with timelike ruling.

The aim of this paper is to use the new version of type 2 Bishop frame which is studied in [23, 31, 41] and Frenet frame to construct offsets base curves of two ruled surfaces. Thus, the kinematics of such surfaces in terms of their intrinsic geometric formulas are established. An application of these surfaces and their motions is considered and plotted.

2. Geometry of Motion Curves and Surfaces in

2.1. Motion of Curves

Let be an arbitrary curve in . Recall that the curve is said to be of unit speed if , where is the standard scalar (inner) product of . Denote by the moving Frenet frame along the unit speed curve . Then the Frenet formulas are given by [42]

Here, , , and are the tangent, the principal normal, and the binormal vector fields of the curve , respectively. and are called curvature and torsion of the curve , respectively.

In the rest of the paper, we suppose everywhere that and .

Let be a unit speed regular curve in . The type 2 Bishop formulas of are defined by [23, 31, 41]

Here, , , and are the tangent, the principal normal, and the binormal vector fields of the curve , respectively.

The Bishop frame or parallel transport frame is an alternative to the Frenet frame. Thus, the matrix relation between type 2 Bishop and Frenet-Serret frames can be expressed as

Here, the type 2 Bishop curvatures are defined by

It can be also deduced as

The frame is properly oriented, and and are polar coordinates for the curve . We will call the set type 2 Bishop invariants of the curve .

Using Frenet formulas (1) many geometries [1–3, 9, 15–19] studied connections between integrable evolution and the motion of curves in a 3-dimensional Euclidean space. They considered that denote a point on a space curve at the time . The conventional geometrical model is specified by the velocity fields

where , , and are the unit tangent, normal, and binormal vectors along the curve and , , and are the tangential, normal, and binormal velocities, respectively. Velocity fields are functionals of the intrinsic quantities of curves, for example, curvature, , torsion , and their derivatives.

The time evolution equations for Frenet frame , , and are given by

where

Using type 2 Bishop frame, Kiziltuğ [41] considered the flow of the curve as the following:

and in view of type 2 Bishop formulas (2), Kiziltuğ [41] obtained the time evolution equations for such frame as follows:

where

2.2. Motion of Surfaces

Here, and in the sequel, we assume that the indices run over the ranges . The Einstein summation convention will be used; that is, repeated indices, with one upper index and one lower index, denoted summation over its range.

Let our surface, moving in 3-dimensional Euclidean space , be given at time by the position vector

where are the Cartesian coordinates in some fixed in time Cartesian frame and are convective curvilinear coordinates. Then, the two tangent vectors and the unit normal vector to the surface are given by

respectively. Thus, the metric and the coefficients of the second fundamental form are given by

where is the Euclidean inner product.

Thus, the Gaussian curvature and the mean curvature are given by

respectively, where is the associated contravariant metric tensor field of the covariant metric tensor field ; that is, .

As one moves along the surface (at a fixed time), the tangent and normal vectors change according to the Gauss-Weingarten equations,

where are called the Christoffel symbols of the 2nd kind, which are given as

From the compatibility conditions of (16), we get the Gauss-Codazzi equations,

where is the Riemann tensor and is the covariant derivative,

Nakayama et al. [11, 14, 21, 22] introduced the dynamics of the surface, where the velocity of the surface is expressed by

where and are the tangential and the normal velocities, respectively.

Using (13), (16), and (20) we can obtain the time evolution equations for the local frame,

Thus, using (13), (16), one can see that the time evolution equations for and are given by

respectively.

2.3. Bishop Frenet Offsets of Ruled Surfaces

In view of relation (3) and inspired by the concepts of Bertrand and Mannheim offsets of ruled surfaces [35–38], we can reformulate the following definitions of Bishop Frenet offsets for ruled surfaces.

A pair of curves and are said to be Bishop Frenet curves if there exists a one-to-one correspondence between their points such that both curves have a common binormal vector at their corresponding points . Such curves will be referred to as “Bishop Frenet offsets.”

Thus, we can write the relation between the curves and as

where is distance between corresponding points on the curves and .

If we take the derivative of the above equation and adopt the relation , we can see that . On the other hand, from the distance function between two points, we have

Thus, we can say that is a nonzero positive constant.

The ruled surface is said to be Bishop Frenet offset of the ruled surface if there exists a one-to-one correspondence between their rulings such that the binormal vector of the base curve of is the binormal vector of the base curve of . In this case, is called a pair of Bishop Frenet ruled surfaces.

Thus, we can write the parametric representation of the ruled surfaces and as follows:

where and are the base curves of and , respectively.

3. Motion of Frenet Ruled Surface

In this section, the fundamental quantities , and their evolution of ruled surfaces (26) and (28) are obtained, respectively. Thus the Gaussian, mean curvatures, and their evolution of such surfaces are given. For this purpose, let a ruled surface generated by the binormal vector of the Frenet frame, moving in 3-dimensional Euclidean space , be given at time by the parametrization [11]:

where , , and .

3.1. Curvatures of

From (26) and using Serret-Frenet formulas (1) it is easily checked that the coefficients of the first fundamental form of are given by

where .

Using (13), the unit normal vector field to the surface is given by

This leads to the coefficients of the second fundamental form of given by

Thus, using (15) one can see that the Gaussian and mean curvature functions of are given, respectively, by the following.

Lemma 1. Consider

From (17) one can see that the Christoffel symbols of are given by the following.

Lemma 2. Consider

and other components equal zero.

3.2. Curvatures’ Evolution of

Actually, here and in the sequel is a remarkable fact that when we calculate and of , we have to compute the velocities of . Thus, using (6), (7), and (20) with the assumption that velocities of the curve are , , and , one can see that the tangential velocities and the normal velocity of are given by the following.

Corollary 3. Consider

where and .
Using (19), (33), and (34) one can obtain the following:

where and .

Based on the above results, we have the following.

Corollary 4. The evolution equations for the metric tensor of are given by

In view of (19), (31), and (34), one can obtain

Using (23) and combining (31), ((34) and (35)) with (37), we have the following.

Corollary 5. The evolution equations for of are given by

Taking (15) into account and using the above results one could have the evolution of the Gaussian and mean curvatures of as follows:

respectively.

4. Motion of Bishop Frenet Offset of Ruled Surface

In this section, the fundamental quantities and of Bishop Frenet offset are obtained. The velocities of are obtained. Thus the formulas of the evolution of the 1st and 2nd fundamental quantities of are derived. For this purpose, let a Bishop Frenet offset of ruled surface generated by binormal vector of the type 2 Bishop frame, moving in 3-dimensional Euclidean space , be given at time by the parametrization [1]:

where , , and .

4.1. Curvatures of

From (27) and using Bishop formulas (2) it is easily checked that the coefficients of the first fundamental form of Bishop Frenet offset are given by

where and , are the type 2 Bishop curvatures of the curve given from (4).

Using (13), the unit normal vector field of the Bishop Frenet offset is given by

This leads to the coefficients of the second fundamental form of Bishop Frenet offset given by

According to (15) one can get the Gaussian and mean curvatures of Bishop Frenet offset , respectively, as follows.

Lemma 6. Consider

From (17), one can obtain the Christoffel symbols of Bishop Frenet offset as follows.

Lemma 7. Consider

and other components equal zero.

4.2. Curvatures’ Evolution of

By a similar manner to Section 3.1, to calculate and , we have to compute the velocities of . Thus, using (9), (10), and (20) with the assumption that velocities of the curve are , , and , one can get the tangential velocities and the normal velocity of as the following.

Corollary 8. Consider

where , , and .
Having (19) and (33) in mind and taking into account the above corollary and after straightforward computations we get the following:

Considering the above obtained results, we can formulate the following.

Corollary 9. The evolution equations for the metric tensor of are given by

Using (19), (43), and (46), one can get the following:

From the foregoing results, using (23), and after straightforward computations we conclude the following.

Corollary 10. The evolution equations for of are given by

Regarding the above obtained results, the evolution of the Gaussian and mean curvatures of can be obtained using the same technique as in (39).

5. Application

In this section, we give an example of how the evolving offsets base curves and generators of a pair of Bishop Frenet ruled surface will look like after a period of time and the effect of the evolution on curves’ frames. This is given through some illustrated figures. For this purpose we consider a helix curve as a base curve of the ruled surface :

Thus, its binormal vector is given by

Using relation (24), the parametrization of the base curve of the ruled surface takes the following form:

Thus and using the parametrizations of surfaces (26), (27), (28), and (40) we can show Figures 1, 2, and 3 as follows.

Figure 1: The original pair , , , , , .

Figure 2: The evolution pair , , , , , .

Figure 3: The evolution pair , , , , , .

Figure 1 shows the original pair of Bishop Frenet ruled surface at and it is noticed that the vector is in direction of the vector .

Figure 2 shows the evolution of the pair of Bishop Frenet ruled surface. In this case, the function affects the pair and these surfaces are plotted for different values of the time . The evolution of the pair expands and collapses under these values of the time and it is noticed that the motion of the pair conserves the motion of the moving frames associated with the offsets base curves , respectively; that is, the vector is in direction of the vector .

Figure 3 shows the evolution of the pair of Bishop Frenet ruled surface. In this case, the function affects the pair and these surfaces are plotted for different values of the time . Also, the same result as in Figure 2 is obtained; that is, the vector is in direction of the vector .

6. Conclusion

We conclude that, by changing the applied time-varying functions on the pair , it is found that the moving frames associated with the offsets base curves describe a parallel transport frame motion. This study plays an important role in the construction and motion of new offsets of ruled surfaces by giving a new frame. We can apply this idea on many different surfaces using different methods. The field is developing rapidly, and there are a lot of problems to be solved and more work is needed to establish different results of new surfaces in different spaces. We hope that this idea will be helpful to mathematicians who are specialized in this area.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author wishes to express profound thanks and appreciation to Professor Dr. Nassar H. Abdel All, Emeritus Professor, Department of Mathematics, Assiut University, Egypt, for his strong support, continuous encouragement, revising this paper carefully, and his helpful suggestions. The author would like also to thank Mr. Ali A. Said, Assistant Lecturer, Faculty of Science, Assiut University, for helping to draw curves’ frames.

References

K. Nakayama, H. Segur, and M. Wadati, “Integrability and the motion of curves,” Physical Review Letters, vol. 69, no. 18, pp. 2603–2606, 1992.

K. Nakayama, H. Segur, and M. Wadati, “A discrete curve-shortening equation,” Methods and Applications of Analysis, vol. 4, no. 2, pp. 162–172, 1997.

M. Hisakado and M. Wadati, “Moving discrete curve and geometric phase,” Physics Letters A, vol. 214, no. 5-6, pp. 252–258, 1996.

P. Pelce, Dynamics of Curved Fronts, Academic Press, New York, NY, USA, 1988.

J. S. Langer, “Instabilities and pattern formation in crystal growth,” Reviews of Modern Physics, vol. 52, no. 1, pp. 1–28, 1980.

R. C. Brower, D. A. Kessler, J. Koplik, and H. Levine, “Geometrical models of interface evolution,” Physical Review A, vol. 29, no. 3, pp. 1335–1342, 1984.

H. Hasimoto, “A soliton on a vortex filament,” Journal of Fluid Mechanics, vol. 51, no. 3, pp. 477–485, 1972.

P. G. Saffman and G. I. Taylor, “The penetration of a fluid in a porous medium or Hele-Shaw cell containing a viscous liquid,” Proceedings of the Royal Society of London A, vol. 245, pp. 312–329, 1958.

T. Tsurumi, H. Segur, K. Nakayama, and M. Wadati, “Motion of curves specified by accelerations,” Physics Letters, Section A: General, Atomic and Solid State Physics, vol. 224, no. 4-5, pp. 253–263, 1997.

K. Nakayama, “Motion of curves in hyperboloid in the Minkowski space,” Journal of the Physical Society of Japan, vol. 67, no. 9, pp. 3031–3037, 1998.

K. Nakayama and M. Wadati, “The motion of surfaces,” Journal of the Physical Society of Japan, vol. 62, no. 6, pp. 1895–1901, 1993.

S. Morigi, “Geometric surface evolution with tangential contribution,” Journal of Computational and Applied Mathematics, vol. 233, no. 5, pp. 1277–1287, 2010.

N. H. Abdel-All and M. T. Al-dossary, “Motion of curves specified by acceleration field in in ,” Applied Mathematical Sciences, vol. 7, no. 69, pp. 3403–3418, 2013.

N. H. Abdel-All and M. T. Al-Dossary, “Motion of hypersurfaces,” Assiut University Journal Mathematics and Computer Science, vol. 40, no. 1, pp. 91–104, 2011.

N. H. Abdel-All, R. A. Hussien, and T. Youssef, “Evolution of curves via the velocities of the moving frame,” Journal of Mathematical and Computational Science, vol. 2, no. 5, pp. 1170–1185, 2012.

N. H. Abdel-All, M. A. Abdel-Razek, H. S. Abdel-Aziz, and A. A. Khalil, “Geometry of evolving plane curves problem via lie group analysis,” Studies in Mathematical Sciences, vol. 2, no. 1, pp. 51–62, 2011.

N. H. Abdel-All, M. A. Abdel-Razek, H. S. Abdel-Aziz, and A. A. Khalil, “Evolution of a helix curve by observing its velocity,” Life Science Journal, vol. 11, no. 5, pp. 41–47, 2014.

N. H. Abdel-All, H. S. Abdel-Aziz, M. A. Abdel-Razek, and A. A. Khalil, “Evolution of a space curve observing by it's frame,” Science International, vol. 26, no. 3, pp. 965–969, 2014.

N. H. Abdel-All, H. N. Abd-Ellah, H. S. Abdel-Aziz, M. A. Abdel-Razek, and A. A. Khalil, “Evolution of a helix curve by observing its acceleration,” Wulfenia Journal, vol. 21, no. 11, pp. 382–395, 2014.

N. H. Abdel-All, M. A. Hamad, M. A. Abdel-Razek, and A. A. Khalil, “Computation of some geometric properties for new nonlinear PDE models,” Applied Mathematics, vol. 2, no. 6, pp. 666–675, 2011.

H. N. Abd-Ellah, “Motion of tubular surfaces,” Italian Journal of Pure and Applied Mathematics, vol. 22, pp. 161–176, 2007.

H. N. Abd-Ellah, “Evolution of translation surfaces in Euclidean 3-space E3,” Applied Mathematics & Information Sciences, vol. 9, no. 2, pp. 661–668, 2015.

E. Ozyilmaz, “Classical differential geometry of curves according to type-2 Bishop trihedra,” Mathematical & Computational Applications, vol. 16, no. 4, pp. 858–867, 2011.

R. L. Bishop, “There is more than one way to frame a curve,” The American Mathematical Monthly, vol. 82, no. 3, pp. 246–251, 1975.

M. Petrović, J. Verstraelen, and L. Verstraelen, “Principal normal spectral variations of space curves,” Proyecciones, vol. 19, no. 2, pp. 141–155, 2000.

B. Bükcü and M. K. Karacan, “Special Bishop motion and Bishop Darboux rotation axis of the space curve,” The Journal of Dynamical Systems and Geometric Theories, vol. 6, pp. 27–34, 2008.

B. Bükcü and M. K. Karacan, “The slant helices according to Bishop frame,” International Journal of Computational and Mathematical Sciences, vol. 3, no. 2, pp. 67–70, 2009.

B. Bükcü and M. K. Karacan, “On the slant helices according to bishop frame of the timelike curve in Lorentzian space,” Tamkang Journal of Mathematics, vol. 39, no. 3, pp. 255–262, 2008.

Y. Süha, “Bishop spherical images of a spacelike curve in Minkowski 3-space,” International Journal of Physical Sciences, vol. 5, no. 6, pp. 898–905, 2010.

M. K. Karacan, B. Bükcü, and N. Yuksel, “On the dual Bishop Darboux rotation axis of the dual space curve,” Applied Sciences, vol. 10, pp. 115–120, 2008.

S. Yılmaz and M. Turgut, “A new version of Bishop frame and an application to spherical images,” Journal of Mathematical Analysis and Applications, vol. 371, no. 2, pp. 764–776, 2010.

M. K. Karacan and B. Bukcu, “An alternative moving frame for tubular surfaces around timelike curves in the Minkowski 3-space,” Balkan Journal of Geometry and its Applications, vol. 12, no. 2, pp. 73–80, 2007.

M. K. Karacan and B. Bükcü, “An alternative moving frame for a tubular surface around a spacelike curve with a spacelike normal in Minkowski 3-space,” Rendiconti del Circolo Matematico di Palermo, vol. 57, no. 2, pp. 193–201, 2008.

J. Bertrand, “Memoire sur la theorie des courbes a double courbure,” Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, vol. 31, pp. 623–624, 1850.

H. Liu and F. Wang, “Mannheim partner curves in 3-space,” Journal of Geometry, vol. 88, no. 1-2, pp. 120–126, 2008.

F. Wang and H. Liu, “Mannheim partner curves in 3-Euclidean space,” Mathematics in Practice and Theory, vol. 37, no. 1, pp. 141–143, 2007.

B. Ravani and T. S. Ku, “Bertrand offsets of ruled and developable surfaces,” Computer-Aided Design, vol. 23, no. 2, pp. 145–152, 1991.

E. Kasap, K. Orbay, and I. Aydemir, “Mannheim offsets of ruled surfaces,” Mathematical Problems in Engineering, vol. 2009, Article ID 160917, 9 pages, 2009.

H. Pottmann, W. Lü, and B. Ravani, “Rational ruled surfaces and their offsets,” Graphical Models and Image Processing, vol. 58, no. 6, pp. 544–552, 1996.

M. A. Soliman, N. H. Abdel-All, R. A. Hussien, and A. A. Said, “Geometric properties and invariants of Mannheim offsets of timelike ruled surface with timelike ruling,” Mitteilungen Klosterneuburg, vol. 65, pp. 285–299, 2015.

S. Kiziltuğ, “On characterization of inextensible flows of curves according to type-2 Bishop frame ,” Mathematical and Computational Applications, vol. 19, no. 1, pp. 69–77, 2014.

T. Shifrin, Differential Geometry: A First Course in Curves and Surfaces, Preliminary Version, Springer, 2015.