Motion of Bishop Frenet Offsets of Ruled Surfaces in
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.
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 . 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 . 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 .
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 . 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 . Recently, the authors in  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 , 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.  presented classical and circular offsets of rational ruled surfaces. More recently, Soliman et al.  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 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 asHere, the type 2 Bishop curvatures are defined byIt can be also deduced asThe 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 fieldswhere , , 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 bywhereUsing type 2 Bishop frame, Kiziltuğ  considered the flow of the curve as the following:and in view of type 2 Bishop formulas (2), Kiziltuğ  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 vectorwhere 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 byrespectively. Thus, the metric and the coefficients of the second fundamental form are given bywhere is the Euclidean inner product.
Thus, the Gaussian curvature and the mean curvature are given byrespectively, 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 asFrom 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 bywhere and are the tangential and the normal velocities, 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 aswhere 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 haveThus, 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 :where , , and .
3.1. Curvatures of
Using (13), the unit normal vector field to the surface is given byThis leads to the coefficients of the second fundamental form of given byThus, using (15) one can see that the Gaussian and mean curvature functions of are given, respectively, by the following.
Lemma 1. ConsiderFrom (17) one can see that the Christoffel symbols of are given by the following.
Lemma 2. Considerand 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.
Based on the above results, we have the following.