A STUDY ON A RULED SURFACE WITH LIGHTLIKE RULING FOR A NULL CURVE WITH CARTAN FRAME

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1 Bull. Korean Math. Soc. 49 (), No. 3, pp A STUDY ON A RULED SURFACE WITH LIGHTLIKE RULING FOR A NULL CURVE WITH CARTAN FRAME N ihat Ayyildiz and Tunahan Turhan Abstract. In this study, we investigate the curvature functions of ruled surface with lightlike ruling for a null curve with Cartan frame in Minkowski 3-space. Also, we give relations between the curvature functions of this ruled surface and curvature functions of central normal surface. Finally, we use the curvature theory of the ruled surface for determine differential properties of a robot end-effector motion.. Introduction In recent years a number of papers on ruled surfaces and their invariants in Minkowski 3-space have appeared ([], [9], []). Several of these have concern mainly with null scroll and B-scroll ([], [3], [4], [5], [8], []). However, there is a gap relative to the curvature theory of ruled surface, whose base curve is null, with lightlike ruling. Here we should note that the ruled surface is neither null scroll nor B-scroll. So, aim of this paper is to contribute to this theory. In Section 3, we give the curvature functions characterized the ruled surface with lightlike ruling for a Cartan framed null curve and also relations between the curvature functions of this ruled surface and curvature functions of central normal surface. Furthermore, as a rigid body moves in space lines embedded in the body trace ruled surfaces. These lines may be the axes of the joints a spatial mechanisms or manipulators or the line of action of the end-of-arm tooling of a manipulator. The curvature theory of line trajectories seeks to characterize the shape of body carrying the line that generates it ([3], []). So, in Section 4, we shall give end-effector motion in Minkowski 3-space. Received March,. Mathematics Subject Classification. 53A4, 53A5, 53A4. Key words and phrases. rigid body, null scroll, Cartan frame, Frenet formulae, striction curve. This work was supported by Süleyman Demirel University through Scientific Research Project Program (37-YL-9). 635 c The Korean Mathematical Society

2 636 NİHAT AYYILDIZ AND TUNAHAN TURHAN Let E 3. Preliminaries be Minkowski 3-space with the inner product u, v = u v + u v + u 3 v 3 and the vector product ( u v = u ) u 3 v v 3, u 3 u v 3 v, u u v v, where u = (u, u, u 3 ), v = (v, v, v 3 ) E 3. For any u, v, w and z E3, the Lagrange formula is given by, ([]), u v, w z = u, w v, z + u, z v, w. Definition. A vector u in E 3 is said to be spacelike if u, u > or u =, timelike if u, u <, lightlike or null if u, u = and u. Definition. A parametrized curve α = α(s) in Minkowski 3-space E 3 to be a null curve if its tangent vector field is null, that is, α (s), α (s) =, α (s). Definition 3 ([7]). A base F = {T, N, W } of E 3 if it satisfies the following conditions: where W = N T. T, T = N, N =, T, N =, T, W = N, W =, W, W =, is said is called a (proper) null frame Let α = α(s) be a null curve in E 3. Assume that T is a null vector field along α. So, there exists a null vector field N = N(s) along α satisfying T, N =. Here if we put W = N T, then we can obtain a (proper) null frame field F = {T, N, W } along α. In such a case, the pair (α, F) is said to be a null curve with Cartan frame if it satisfies the following, ([7]), () T = W, N = kw, W = kt N, where k = k(s) is smooth function defined by k = N, W. 3. A ruled surface with lightlike ruling Let (α, F) be a null curve with Cartan frame F = {T, N, W }. surface M may be represented by the equation X(ψ, ν) = α (ψ) + νv (ψ), A ruled where ψ and v are arbitrary real valued parameters. The curve α (ψ) is called the directrix of M and V (ψ) is a vector attached to the straight line generating

3 A STUDY ON A RULED SURFACE WITH LIGHTLIKE RULING 637 the ruled surface. The Darboux frame at α (ψ) is the basis {T, V, U} of E 3, where T is the tangent vector of the curve α, V is the unique vector obtained by, ([7]), V = { X X, T } X, X X, T T, X T α(ψ) M, X, T =, and U is the spacelike unit normal of M which is defined by U = V T. Hence, we have T, T = V, V = V, U = T, U =, T, V =, U, U =. Thus we have the following result: Theorem 3. ([6]). Let α be a null curve in M E 3. The first order variation of Darboux frame {T, V, U} is () T V U = κ g κ n κ g τ g τ g κ n where T, V = κ g, T, U = κ n and V, U = τ g. T V U These functions κ g, κ n and τ g are called the geodesic curvature, the normal curvature and the geodesic torsion of the curve α, respectively. So, a relation between a null curve with Cartan frame {T, N, W } and Darboux frame {T, V, U} of a ruled surface M is given by: Theorem 3.. Let M be a ruled surface with basis {T, V, U} in E 3 a null curve with Cartan frame {T, N, W } in M E 3. Then (3) T N = κ g κ n (+κ n ) κ g κ n T V. W U κ g κ n, and α be Proof. If we use equations (), () and the Lagrange formula we get equation (3). Using this result, it is easy to see that (4) = κ n. Substituting equation (4) into equation (3) it gives (see [6]) (5) T N = κ g κ g T V. W κ g U Now, we shall give a theorem with respect to the striction line which gives the simplest description of the positional variation of the ruled surface.

4 638 NİHAT AYYILDIZ AND TUNAHAN TURHAN Theorem 3.3. Let M be a ruled surface with basis {T, V, U} in E 3 and α be a null curve with Cartan frame {T, N, W } in M E 3. Then the location of the striction line relative to the directrix along the ruling V is ( β (s) = α (s) ) κ g dt + c V (s), where c is any constant. τ g Proof. The location of the striction line relative to the directrix along the ruling V is given by (6) β (s) = α (s) µv (s), where µ is a real valued parameter. To determine the parameter µ, if we use the definition of the striction line, we get µ = κ g τg. So, the first order positional variation of the striction line is expressed as (7) β = ΩT ΓV + U, where Ω =, Γ = µ + κ g τ and = κ g g τ g. Here, supposing β null vector and taking (7) into consideration, we have µ = κ g dt + c, τ g where c is any constant. So, from equation (6) we get ( β (s) = α (s) ) κ g dt + c V (s). τ g In this theorem, the functions Ω, Γ and characterize the ruled surface M. These functions are called the curvature functions of the ruled surface M. As the trihedron T, V, U moves along the striction curve of M the two Lorentzian vectors T and U generate ruled surfaces associated with M. Of primary importance is the ruled surface generated by U called the central normal surface of M. Now let us consider the central normal surface M which suppose β as a base curve and whose spacelike ruling is U. We can write its equation X U (s, ν) = β(s) + νu (s). Then, the location of the striction line of this surface is given by the following result:

5 A STUDY ON A RULED SURFACE WITH LIGHTLIKE RULING 639 Theorem 3.4. Let M be a central normal surface with spacelike ruling U and β be its base curve. Then the location of the striction line of the central normal surface is β U (s) = β(s) κ g κ n τ g 4τ g κ n U (s). Proof. From the definition of the striction line we can write (8) β U (s) = β(s) µ U U (s), where µ U is a real parameter. By direct computation the parameter µ U is µ U = κ g κ n τ g 4τ g κ n. When this is substituted equation (8), the proof is completed. Using this result, the first-order positional variation of the striction line of the central normal surface M can be expressed with respect to Cartan frame F = {T, N, W } as follows where β u = Ω U T + Γ U N + U W, Ω U = κ g( µ U 4 ) κ g ( µ ) + µ U U τ g + Ω, Γ U = µ U and U = µ + κ U g (µ U ). The functions Ω U, Γ U and U characterize the central normal surface M. These functions are called the curvature functions of the central normal surface M. Now, we would like to exhibit an example for these surfaces: Example. Let the ruled surface M be given by X(s, t) = ( s, cos s, sin s) + t (, cos s, sin s), where the null curve α(s) = ( s, cos s, sin s) is the directrix of the surface and V = (, cos s, sin s) is the ruling vector of the surface M (see Figure ). So, we can construct the Darboux frame of the surface M as follows: T = (, sin s, cos s), V = (, cos s, sin s) and U = (, sin s cos s, sin s + cos s). The vectors T, V and U satisfy the following conditions: T, T = V, V =, T, V = U, U =, V, U = T, U =, U = V T. Hence, we get the Cartan frame F = {T, N, W } of the null curve α as ( T = (, sin s, cos s), N =, sin s, cos s ) and W = (, cos s, sin s).

6 64 NİHAT AYYILDIZ AND TUNAHAN TURHAN Figure. The ruled surface X(s, t) with lightlike ruling V. Therefore, a Cartan framed null curve (α, F) satisfies the following Frenet equations T = W, N = W, W = T N. On the other hand, the first order variation of the Darboux frame {T, V, U} can be expressed as (9) T = T + U, V = V + U, U = T V. So, the curvature functions of the ruled surface M can be obtained as, respectively, =, Γ = and Ω =. Now, let us assume that the equation of the central normal surface M with spacelike ruling (see Figure ) is given by X U (s, t) = ( s, cos s, sin s) + t(, sin s cos s, sin s + cos s). Therefore, the geodesic curvature κ g, the normal curvature κ n and the geodesic torsion τ g of the curve α = ( s, cos s, sin s) are found as κ g =, κ n = and τ g =. Finally, the curvature functions of the central normal surface M can be obtained as follows, respectively, U = 4, Γ U = 3 4 and Ω U = 5 8.

7 A STUDY ON A RULED SURFACE WITH LIGHTLIKE RULING 64 Figure. The central normal surface X U (s, t) with spacelike ruling U. 4. END-effector motion A typical robot trajectory is shown in Figure 3. Also, there are three vector {, A, N}. The tool frame consists of these three vector. Each of the these three vector of the tool frame generates a ruled surface. It is not necessary to use all three ruled surfaces to represent a robot trajectory ([4]). One ruled surface will suffice. The ruled surface generated by the is chosen here to represent the trajectory. There are four frame of reference which are essential Figure 3. A robot trajectory. in the study of the motion of the end-effector. The four frames are: the tool frame {, A, N}, the surface frame {, }, Darboux frame {T, V, U} and Cartan frame {T, N, W } ([4]). The surface frame {, } and the tool frame {, A, N} satisfy the following, respectively, = V, S b = Y, V { Y Y, Y } Y, V V, Y T α(t) M, Y, V =,

8 64 NİHAT AYYILDIZ AND TUNAHAN TURHAN = S b, and, = S b =, = S b, =, =, =, = A, A = A, N =, N =,, A = N, N =, N = A. Now we can express relation between the tool frame {, A, N} and the surface frame {, } in matrix form as () A = A A,. N N N, S b The relation between the surface frame {, } and Darboux frame {T, V, U} is expressed in matrix form as () =, T T V. S b S b, V S b, U U Lastly, if we use equation () into equation (), we can express relation between the tool frame {, A, N} and Darboux frame {T, V, U} as () A = l l T V, N l 3 l 4 U where l = S b, A + T + A,, T, l = S b, U A,, l 3 = N + N,, T, l 4 = N,. Now we may express differential properties of the robot end-effector motion in the Darboux frame and tool frame, respectively, and where α = T (Γ µ + κ g µ)v + ( + µτ g )U α = Λ + A + Λ N, Λ = µ l 4 l 4 Γ l 4 l + l 3 l µκ g l 4 l 3 µτ g l 3 l 4 and Λ = µτ g l + l 4.

9 A STUDY ON A RULED SURFACE WITH LIGHTLIKE RULING 643 To determine the first-order variation of the tool frame, we differentiate equation () to get κ d g τ g A = l ds τ g l l κ g l κ n l + l τ T g V, N l 4 τ g l l 3 3τ g l 4 κ n l + l 4 3τ g U where l = S b, A + T + S b, A + T + ( A, + A, S n ), T + A, ( S n, T +, T ), l = S b, U + S b, U A, A, S n, l 3 = N + ( N, + N, S n ), T + N, ( S n, T +, T ), l 4 = N, N, S n. An example for the relations defined above may be given as: Example. Let the ruled surface be given by X(s, t) = ( s, cos s, sin s) + t (, cos s, sin s), where the null curve α(s) = ( s, cos s, sin s) is the directrix of the surface and V = (, cos s, sin s) is the ruling vector of the surface X(s, t) (see Figure ). So, we can construct the surface frame {, } as follows: = V = (, cos s, sin s) = ( 5, 3 sin s 4 cos s, 3 cos s 4 sin s) and = ( 3, sin s 3 cos s, 3 sin s cos s). The vectors and satisfy the following conditions:, = S b =, =, =,, = S b, =, = S b. Similarly, we can construct the tool frame {, A, N} as follows: = V = (, cos s, sin s), A = ( 3, 5 sin s cos s, 5 cos s sin s) and N = ( 5, sin s 5 cos s, cos s 5 sin s). The vectors, A and N satisfy the following conditions:, = A, A =,, A = N, N =,, N = A, N =, N = A. Hence, from equation (), we get relation between the tool frame {, A, N} and the surface frame {, } in matrix form as A =. N S b

10 644 NİHAT AYYILDIZ AND TUNAHAN TURHAN From equation (), the relation between the surface frame {, } and Darboux frame {T, V, U} is expressed in matrix form as T = 4 V. S b 4 U Lastly, if we use equation () into equation (), we can express relation between the tool frame {, A, N} and Darboux frame {T, V, U} in matrix form as (3) A N = T V U To determine the first-order variation of the tool frame, we differentiate equation (3) and use equation (9), we get d A = T V. ds N 5 6 U References [] L. J. Alias, A. Ferrandez, P. Lucas, and M. A. Moreno, On the Gauss map of B scrolls, Tsukuba J. Math. (998), no., [] N. Ayyıldız and A. Yücesan, On the scalar and dual formulations of the curvature theory of line trajectories in the Lorentzian space, J. Korean Math. Soc. 43 (6), no. 6, [3] H. Balgetir, M. Bektaş, and M. Ergüt, Null scroll in the 3-dimensional Lorentzian space, Appl. Sci. 5 (3), no., 5. [4] W. B. Bonnor, Null hypersurfaces in Minkowski space-time, Tensor (N. S.) 4 (97), [5] S. M. Choi, U. H. Ki, and Y. J. Suh, On the Gauss map of null scrolls, Tsukuba J. Math. (998), no., [6] A. C. Çöken and Ü. Çiftçi, On null curves on surfaces and null vectors in Lorentz space, SDU Fen Derg. (7), no., 6. [7] K. L. Duggal and A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Kluwer Academic Publishers, 996. [8] S. Ersoy and M. Tosun, On the trajectory null scrolls in 3-dimensional Minkowski spacetime E 3, Kyungpook Math. J. 48 (8), no., 8 9. [9] A. Ferrandez and P. Lucas, On surfaces in the 3-dimensional Lorentz-Minkowski space, Pacific J. Math. 5 (99), no., 93. [] A. Ferrandez, A. Gimenez, and P. Lucas, Null helices in Lorentzian space forms, Internat. J. Modern Phys. A. 6 (), no. 3, [] H. Liu, Ruled surfaces with lightlike ruling in 3-Minkowski space, J. Geom. Phys. 59 (9), no., [] J. M. McCarthy, On the scalar and dual formulations of the curvature theory of line trajectories, Journal of Mechanisms, Transmissions, and Automation in Design 9 (987), 6. [3] J. M. McCarthy and B. Roth, The curvature theory of line trajectories in spatial kinematics, ASME Journal of Mechanical Design 3 (98), no. 4,

11 A STUDY ON A RULED SURFACE WITH LIGHTLIKE RULING 645 [4] B. S. Ryuh, Robot trajectory planning using the curvature theory of ruled surfaces, Doctoral dissertation, Purdue University, West Lafayette, Ind, USA, 989. N ihat Ayyildiz Department of Mathematics Süleyman Demirel University 36 Isparta, Turkey address: nihatayyildiz@sdu.edu.tr, ayyildiz67@gmail.com Tunahan Turhan Seydişehir Vocational School Selçuk University 436 Konya, Turkey address: t turhan7@hotmail.com