Homothetic Bishop Motion Of Euclidean Submanifolds in Euclidean 3-Space

Transcription

1 Palestine Journal of Mathematics Vol. 016, Palestine Polytechnic University-PPU 016 Homothetic Bishop Motion Of Euclidean Submanifol in Euclidean 3-Space Yılmaz TUNÇER, Murat Kemal KARACAN and Dae Won YOON Communicated by Z. Ahsan MSC 010 Classifications: Primary 3A17, 3A0; Secondary 3A0. Keywor and phrases: Homothetic Motion, Darboux Frame, Darboux Vector, Bishop Frame. Abstract. In this study, we gave an alternative kinematic model for two smooth submanifol M and N both on another and inside of another, along given any two curves which are tangent to each other on M and N at every moment, which the motion accepted that these curves are trajectories of the instantaneous rotation centers at the contact points of these submanifol and we gave some remarks for the kinematic model at every moments by using Bishop frame. In addition, we established the relationships between Bishop curvatures of the moving and fixed pole curves. 1 Introduction R.Müller generalized 1-parameter motions in an n-dimensional Euclidean space which is given by the equation Y = AX + C and investigated axoid surfaces[7]. K. Nomizu defined the 1- parameter motion model along the pole curves on the tangent plane of the sphere, by using parallel vector fiel and obtained some results of the motion in the cases that the motion is only sliding or only rolling [9]. H.H.Hacısalihoğlu investigated 1-parameter homothetic motion and obtained some important results in an n-dimensional Euclidean space[]. B.Karakaş adapted K. Nomizu s motion model to the homothetic motion, again by defining parallel vector fiel along the curves[3]. Y. Tuncer, Y. Yaylı and M. K. Sağel showed that a smooth manifold M can be rolling, sliding and spining on or in side of another smooth manifold N along not only special curves but also any regular curves which are the pole curves of the homothetic motion on M and N by using Frenet vectors, curvatures and torsions[13]. In this study, our aim is to show that a smooth manifold M can be rolling, sliding and spining on or in side of another smooth manifold N along not only special curves but also any regular curves which are the pole curves of the homothetic motion on M and N, by using Bishop frames, curvatures and the other special orthonormal frames along these curves and obtain the equation of this motion. Consequently, we will have obtained the equation of the homothetic motion of M on N along the pole curves. The homothetic motion of the smooth submanifold M on or in side of another N in a 3-dimensional Euclidean space is generated by the transformation F : M N Xs Y s = haxs + C 1.1 where A is a proper orthogonal 3 3 matrix, X and C are 3 1 vectors and h 0 is a homothetic scale. The elements of A, C and h are continuously differentiable functions of the time-dependent parameter s and the elements of X are the coordinates of a point on M according to the Euclidean coordinate system {x 1, x, x 3 }. We take B as ha with differentiating 1.1 and we obtain dy = B dx + db X + dc where db dc X+, B dx dy and are called sliding velocity, relative velocity and absolute velocity of the point X. We called X is a center of the instantaneous rotation if its sliding velocity is 1.

2 1 Yılmaz TUNÇER, Murat Kemal KARACAN and Dae Won YOON vanished. If X is a center of the instantaneous rotation then X is a pole point at the time s of the motion F given in 1.1 [, 13, 1, 1]. Since det db 0 then every homothetic motion in E3 is a regular motion[]. Let Xs be a regular curve on M which is defined on closed interval I IR so that all of its points are the pole points. In this case, we called and Xs = [ db ] 1 [ dc ] Y s = B[ db ] 1 [ dc ] + C are the moving and fixed pole curves, respectively, where the matrix B[ db ] 1 is as follows. B [ db ] 1 = dh A + hda h 1 A 1 = dh h 1 I 3 + da }{{} A 1 }{{} We called ϕ and S are sliding part and rolling part of the motion F, respectively. For S 0, there is a uniquely determined vector W s such that SU is equal to the cross product W s U for every vector U IR 3. The vector W s is called the angular velocity vector of the point Xs at instant s. If W s is normal to N at Y s then we have a spinning at instant s. If W s is tangent to N at Y s then we say that motion is a rolling with sliding, if ϕ = 0 and S 0 then F is a pure rolling motion, if ϕ 0 and S = 0 then F is a pure sliding motion[3,, 9, 1]. Since the motion F is a homothetic motion then it contains sliding part absolutely. The ability to "ride" along a three-dimensional space curve and illustrate the properties of the curve, such as curvature and torsion, would be a great asset for mathematicians. The classic Serret-Frenet frame provides such ability,however the Serret-Frenet frame is not defined for all points along every curve. A new frame is needed for the kind of mathematical analysis that is typically done with the computer graphics. Denote by {T s, Ns, Bs} the moving Frenet-Serret frame along the curve αs in the space E 3. For an arbitrary curve αs with first and second curvature, κs and τs in the space E 3, the following Frenet-Serret formulae are given in [] written under matrix form T s N s B s = 0 κ s 0 κ s 0 τ s 0 τ s 0 ϕ T s N s B s S where Here, curvature functions are defined by T, N = T, B = N, B = 0, T, T = N, N = B, B = 1. κ s = α s, and τ s = detα s, α s, α s α s. The Relatively Parallel Adapted Frame or Bishop Frame could provide the desired means to ride along any given space curve.the Bishop Frame has many properties that make it ideal for mathematical research. Another area about interested in the Bishop Frame is so-called Normal Development, or the graph of the twisting motion of the Bishop Frame. This information with the initial position and the orientation of the the Bishop Frame provide all of the information which is necessary to define the curve. The Bishop frame may have the applications in the area of Biology and Computer Graphics. For example, it may be possible to compute the information about the shape of the sequences of DNA using a curve defined by the Bishop frame. The Bishop frame may also provide a new way to control virtual cameras in computer animations[, 10, 11].

3 Homothetic Bishop Motion Of Euclidean Submanifol 1 The Bishop frame or parallel transport frame is an alternative approach to define a moving frame that is well defined even when the curve is vanished the second derivative. We can transport by parallel an orthonormal frame along a curve simply by parallel transporting each component of the frame. The parallel transport frame is based on the observation that, while T s for a given curve model is unique, we may choose any convenient arbitrary basis {N 1 s, N s} for the remainder of the frame, so long as it is in the normal plane perpendicular to T s at each point. If the derivatives of {N 1 s, N s} depend only on T s and not each other, we can make N 1 s and N s vary smoothly throughout the path regardless of the curvature. In addition, suppose that the curve α is an arclength-parametrized C curve and we have C 1 unit vector fiel N 1 and N = T N 1 along the curve α so that T, N 1 = T, N = N 1, N = 0, i.e., T, N 1, N will be a smoothly varying right-handed orthonormal frame as we move along the curve to this point, the Frenet frame would work just fine if the curve were C 3 with κ 0. But now we want to impose the extra condition that N 1, N = 0.We say that the unit first normal vector field N 1 is parallel along the curve α. This means that the change of N 1 is only in the direction of T. A Bishop frame can be defined even when a Frenet frame can not e.g., when there are points with κ = 0. Therefore, we have the alternative frame equations T s N 1 s N s = 0 k 1 s k s k 1 s 0 0 k s 0 0 T s N 1 s N s 1.3 where κ s = k 1 + k, δ s = arctan k k 1, τ s = dδs so that k 1 s and k s effectively correspond to a cartesian coordinate system for the polar coordinates κ s,δ s, with δ s = τ s. The orientation of the parallel transport frame includes an arbitrary choice of the integration constant δ 0, which disappears from τ and hence from the Frenet frame due to the differentiation [, 10]. Let us consider the smooth manifol M and N which are tangent inside or outside to each other, Xs on M and Y s on N be the moving and fixed regular pole curves and the tangent planes of M and N along Xs and Y s coincide at the contact points. We shall take a rectangular coordinate system in E 3. Let e 1, e and e 3 be the unit vectors1, 0, 0, 0, 1, 0 and 0, 0, 1 respectively. We denote ξ = ξs and η = ηs as the normal vector fiel of M and N along the curves Xs and Y s, respectively. In addition, we denote the systems {T, N 1, N } and { } T, N 1, N as the Bishop vector fiel of the curves Xs and Y s, respectively. Since the homothetic motion F : M N consists of rolling then W s is tangent to both Xs on M and Y s on N at every moments[11]. Since ξ and η have same or opposite directions depending on the orientation of M and N, we have Bξ = ɛhη at the contact points, where ɛ is the sign such that; if ɛ = +1 then M moves inside of N along the pole curves, if ɛ = 1 then M moves out side of N along the pole curves. Suppose that {b 1 = b 1 s, b = b s} and {a 1 = a 1 s, a = a s} be orthonormal systems along the regular pole curves Xs and Y s respectively, and let b 1, b and a 1, a transform to each another as b 1 = hb 1 a 1 and b = hb 1 a, respectively. Hence {b 1, b, ξ} and {a 1, a, η} will be the moving and fixed orthonormal systems for X = Xs and Y = Y s, respectively. Since X is the pole curve, we can write the equation dy = B dx by using 1.. Let the parameter s be arc-length parameter for the curve X. Thus we can write dy = hat and then we obtain h = dy furthermore the tangent vector of Y will be T = 1 dy h. On the other hand, since ξ Sp {N 1, N } then we can write ξs = cos ψsn 1 s + sin ψsn s 1.

4 16 Yılmaz TUNÇER, Murat Kemal KARACAN and Dae Won YOON We must construct the frames {b 1, b, ξ} and {a 1, a, η} for determining the orthogonal matrix A in 1.1. During this operations, we used the frames {T, ξλt, ξ} and { T, ηλt, η } which are called Darboux frames along X and Y at contact points on M and N, respectively. We can easily find the orthogonal matrices Q, P and R which transform {T, N 1, N } to {T, ξλt, ξ}, {e 1, e, e 3 } to {T, N 1, N } and {T, ξλt, ξ} to {b 1, b, ξ} by using 1., respectively. The matrix A 1 = P T Q T R T transforms b 1 to e 1, b to e and ξ to e 3. We obtain that the skew symmetric matrix w 1 = dat 1 A 1 is w 1 = 0 ρ ɛk 1 sin ψ +ɛk cos ψ ɛk 1 cos ρ cos ψ +ɛk sin ψ cos ρ +ψ sin ρ ρ + ɛk 1 sin ψ ɛk cos ψ 0 ɛk 1 sin ρ cos ψ ɛk sin ψ sin ρ +ψ cos ρ ɛk 1 cos ρ cos ψ +ɛk sin ψ cos ρ +ψ sin ρ ɛk 1 sin ρ cos ψ ɛk sin ψ sin ρ +ψ cos ρ 0 1. where k 1 = k 1 s and k = k s are the Bishop curvatures of the pole curve X and ρ = ρ s is the rotation angle of {b 1, b } according to {T, ξλt }.. Corollary 1.1. The vector fiel b 1 and b are the parallel vector fiel along curve X according to the connection of M if and only if is satisfied. ρ + ɛ k 1 sin ψ k cos ψ = 0 Proof. Let be Levi Civita connection and S M be the shape operator of M. We can write b 1 as follows by using the matrices R and P. Using the Gauss equation and after routine calculations, we obtain b 1 = cos ρt + sin ρ sin ψn 1 sin ρ cos ψn T b 1 = T b 1 + S M T, b 1 ξ T b 1 = {ρ + k 1 sin ψ k cos ψ} {sin ρt sin ψ cos ρn 1 + cos ψ cos ρn } It is easily to see that T b 1 = 0 if and only if ρ + k 1 sin ψ k cos ψ = 0. Hence, b 1 is a parallel vector field along curve X according to the connection of M if and only if ρ + k 1 sin ψ k cos ψ = 0 is satisfied. Similarly, we can easily proof that b is a parallel vector field along curve X according to the connection of M if and only if ρ + k 1 sin ψ k cos ψ = 0 is satisfied, too. On the other hand, since ηs = cos ψ s N 1 s + sin ψ t N s 1.6 then we can easily find the orthogonal matrices Q, P and R by using 1.6 which transform { T, N 1, N } to { T, ηλt, η }, {e1, e, e 3 } to { T, N 1, N } and { T, ηλt, η } to {a1, a, η}, respectively. The matrix A = P T Q T R T transforms a 1 to e 1, a to e and η to e 3. We obtain that

5 Homothetic Bishop Motion Of Euclidean Submanifol 17 the skew symmetric matrix w = dat A is w = 0 ρ k 1 sin ψ +k cos ψ k 1 cos ρ cos ψ +k sin ψ cos ρ +ψ sin ρ ρ + k 1 sin ψ k cos ψ 0 k 1 sin ρ cos ψ k sin ψ sin ρ +ψ cos ρ k 1 cos ρ cos ψ +k sin ψ cos ρ +ψ sin ρ k 1 sin ρ cos ψ k sin ψ sin ρ +ψ cos ρ where k 1 = k 1 s and k = k s are the Bishop curvatures of the pole curve Y and ρ = ρ s is the rotation angle of {a 1, a } according to { T, ηλt }. Corollary 1.. The vector fiel a 1 and a are the parallel vector fiel along curve Y according to the connection of N if and only if is satisfied. ρ + k 1 sin ψ k cos ψ = 0 Proof. We can proof similarly to corollary 1.1. Therefore, we obtain the matrix A using A 1 and A as A = A A T 1 so that A transforms b 1 to a 1, b to a and ξ to ɛη, respectively. The skew-symmetric matrix S = da AT is an instantaneous rotation matrix and S represents a linear ishomorphism as T Y t N Sp {η}. We can find the matrix S by using 1. and 1.7 as S = A w + w 1 A T. Consequently the matrix S determines an unique vector w Sp {a 1, a, η} as follows. where w = u 1 a 1 + u a + u 3 η 1. u 1 = k 1 cos ψ + k sin ψ sin ρ + ψ cos ρ + {ɛ k 1 cos ψ + k sin ψ sin ρ ψ cos ρ} u = k 1 cos ψ + k sin ψ cos ρ ψ sin ρ + {ɛ k 1 cos ψ + k sin ψ cos ρ + ψ sin ρ} u 3 = ρ + k 1 sin ψ k cos ψ ρ ɛk 1 sin ψ + ɛk cos ψ Thus, we obtained the main condition for two moving smooth submanifol on or inside of another, along the regular pole curves. So, we prove the following theorem. Theorem 1.3. F is rolling with sliding motion defined as hab 1 = a 1, hab = a and haξ = ɛη along the regular pole curves if and only if ρ + k 1 sin ψ k cos ψ ρ ɛk 1 sin ψ + ɛk cos ψ = 0 This condition shows that any smooth submanifol can be rolling with sliding, pure sliding or sliding with spining on or inside of another along the pole curves which are tangent to each other at every moment. This is possible by choosing one of ρ and ρ as a constant even if we face hard integrals. In addition, ρ and ρ show that how we must define the vector fiel a 1, a and b 1, b along the pole curves according to what we desire a homothetic motion. We can also find the geodesic and normal curvatures and geodesic torsions of M and N in the Bishop means, along the curves X and Y as follows. The curvatures of M along X are κ g = k 1 sin ψ k cos ψ, κ ξ = k 1 cos ψ + k sin ψ, τ g = ψ 1.9 and the curvatures of N along Y are κ g = k 1 sin ψ k cos ψ, κ η = k 1 cos ψ + k sin ψ, τ g = ψ 1.10

6 1 Yılmaz TUNÇER, Murat Kemal KARACAN and Dae Won YOON Hence we restore 1. as follows. u 1 = κ η sin ρ + τ g cos ρ + ɛκ ξ sin ρ τ g cos ρ 1.11 u = κ η cos ρ τ g sin ρ + ɛκ ξ cos ρ + τ g sin ρ u 3 = ρ ρ + κ g ɛκ g If M is rolling on or inside of N along the curves X and Y then ρ ρ + κ g ɛκ g = 0. If b 1, b, a 1 and a are the parallel vector fiel then the motion is rolling with sliding automatically. In the same conditions, the following equalities are satisfied at the points that the motion is pure sliding. κ ξ = ɛκ η cos τ g = κ η sin ɛκ g κ g + c + ɛτ g sin ɛκ g κ g + c + τ g cos ɛκ g κ g + c ɛκ g κ g + c where c is a constant. In this case, u 1 = u = u 3 = 0. In the case, b 1, b, a 1, a are not the parallel vector fiel and κ ξ + τ g 0 and κ η + τ g 0 then ρ ρ = arccos ɛκ ξ κ η + τ g τ g κ η + τ g or ρ ρ = arcsin ɛκ ξ τ g κ η τ g κ η + τ g ρ ρ + κ g ɛκ g 0 and κ ξ + τ g = κ η + τ g are satisfied at the points that the motion is sliding with spining. If the curves X and Y are both the principal curves and geodesics of M and N then τ g = τ g = κ g = κ g = 0 and also ψ and ψ are constants. If M is any manifold in E 3 and N is a plane then angular velocity vector at the contact points will be as follows w = { k 1 cos ψ + k sin ψ sin ρ + {ɛ k 1 cos ψ + k sin ψ sin ρ ψ cos ρ} } a 1 { k 1 cos ψ + k sin ψ cos ρ {ɛ k 1 cos ψ + k sin ψ cos ρ + ψ sin ρ} } a + { ρ + k 1 sin ψ k cos ψ ρ ɛk 1 sin ψ + ɛk cos ψ } η In this case, F is a rolling with sliding if and only if ρ ρ = k cos ψ k 1 sin ψ s + ɛ k 1 sin ψ k cos ψ + c is satisfied, where c, ψ, k 1 and k are constants. We can restate?? as follows by using 1.9 and w = { κ η sin ρ + ɛκ ξ sin ρ τ g cos ρ} a 1 + { κ η cos ρ + ɛκ ξ cos ρ + τ g sin ρ} a + { ρ ρ + κ g ɛκ g } η Thus, F is a rolling with sliding if and only if ρ ρ = sκ g ɛ κ g + c is satisfied, where c, and κ g are constants, too. Corollary 1.. If X and Y are geodesics of M and N, respectively, then F is a rolling with sliding motion if and only if ρ ρ =constant.

7 Homothetic Bishop Motion Of Euclidean Submanifol 19 Theorem 1.. Let M and N be two submanifol and X and Y be the smooth curves on M and N, respectively, which are satisfied given condition in theorem 1.3 and be tangent to each other at the contact points. Then we can find a unique homothetic motion F of M on or inside of N along the pole curves X and Y. Theorem 1.6. Let S M and S N be the shape operators of M and N along the curves X and Y respectively. If dx dy h 1 S M = S N then F is sliding motion without rolling. Proof. We can write the following equations along the curves X and Y, respectively. dx S M = dξ dy and S N = dη By differentiating 1. and by using 1.3, we obtain dξ = {ɛκ ξ cos ρ + τ g sin ρ} b 1 { ɛκ ξ sin ρ + τ g cos ρ} b since b 1 = hb 1 a 1, b = hb 1 a and ξ = ɛhb 1 η, dξ h 1 B = {ɛκ ξ cos ρ + τ g sin ρ} a 1 { ɛκ ξ sin ρ + τ g cos ρ} a by differentiating 1.6 and by using 1.3, we obtain Since h 1 B and dη = { κ η cos ρ + τ g sin ρ} a 1 { κ η sin ρ + τ g cos ρ} a = dη, we can write dξ ɛκ ξ cos ρ + τ g sin ρ = κ η cos ρ + τ g sin ρ 1.1 ɛκ ξ sin ρ + τ g cos ρ = κ η sin ρ + τ g cos ρ we substitute 1.1 and 1.13 in 1.11 and from 1., we obtain that F is sliding motion without rolling. Corollary 1.7. If F is rolling with sliding motion then the shape operators of M and N satisfy the following inequality. dx dy h 1 S M S N Corollary 1.. Let X and Y be the smooth curves on M and N such the curves not passing through the flat points of M and N. In this case M is sliding and rolling on or inside of N along these curves. M is sliding without rolling or inside of N at the flat-contact points. All of the corollaries, theorems and the things we said in this study are consistent with[3] and [13]. If h = 1 then this study gives us a one parameter kinematic model for the smooth submanifol in Euclidean 3-space. In this case, the notions rolling with sliding and sliding with spining transform to pure rolling and pure spining, respectively. Example 1.9. For ɛ = 1: Let Xs = sins, 0, coss, s [0, 1] be a unit speed curve on φu, v = sin v sin u, sin v cos u, cos v and Y s = sins, s, coss is any curve on x + z + = 1. The Bishop trihedron of the curve X is T = coss, 0, sins, N 1 = sins, 0, coss, N = 0, 1, 0

8 190 Yılmaz TUNÇER, Murat Kemal KARACAN and Dae Won YOON and since δ = 0 then the curvatures of the curve X are the unit normal vector field of sphere is k 1 = 1, k = 0, ξs = sins, 0, coss with the angle ψ = π. The Bishop trihedron of the curve Y is 1 T = coss, 1, 1 sins N 1 = N = { 1 sin 1 s coss cos 1 s sins { 1 cos 1 s coss + sin 1 s sins and the curvatures of the curve Y are } },, { sin s sin 1, s sins cos 1 s coss { sin 1 s coss 1 1 cos s, k 1 = 1 cos1 s, k = 1 sin1 s with δ = 1 s. The unit normal vector field of cyclinder is 1 cos 1 s sins ηs = sins, 0, coss with the angle ψ = 1 s. Since dy = then the homothetic scale is h = and we calculate the orthogonal matrix A = [a ij ] and so the matrix B in 1.1 is B = A where a ij are a 11 = cos3s + coss + coss + + a 1 = coss + a 13 = sin3s sins + sins a 1 = coss coss a = coss a 3 = sins sins a 31 = sin3s sins sins a 3 = sins a 33 = cos3s coss coss + and the matrix C is 1 sins + sins C = sins s 1 cos s + coss 3 } }

9 Homothetic Bishop Motion Of Euclidean Submanifol 191 Figure 1. Sphere is rolling without sliding on the cylinder along the curves X and Y. Since X is the solution of the equation d BX + d C = 0 then X is a pole curve as a moving curve and Y is a fixed pole curve on the sphere and the cylinder x +z + = 1, respectively. Unit normal vectors ξ and η are the opposite direction and linear dependent at the contact points, thus the signature is ɛ = 1. The components of the anti-symmetric matrix S = [s ij ] are s 11 = s = s 33 = 0 s 1 = s 1 = sins sins s 31 = s 13 = coss + s 3 = s 3 = coss + coss 1 with respect to the standart base of IR 3 and so the angular velocity vector is W = 1 a a with respect to the base {a 1, a, η}, where the vector fiel a 1 and a are a 1 = coss,, sins a = coss +, coss, sins Since h is a constant and W lies on the tangent plane at the contact points then the sphere is rolling without sliding on the cylinder along the curves X and Y. Example For ɛ = 1: Let Xs = sins, 0, coss 1, s [0, π] is the unit speed curve on φu, v = sin v sin u, sin v cos u, cos v 1 and Y s = sins, s, coss is any curve on x + z + =. The Bishop trihedron of the curve X is T = sins, 0, coss, N 1 = sins, 0, coss, N = 0, 1, 0 and since δ = 0 then the curvatures of the curve X are the unit normal vector field of sphere is k 1 = 1, k = 0, ξs = sins, 0, coss

10 19 Yılmaz TUNÇER, Murat Kemal KARACAN and Dae Won YOON with the angle ψ = π. The Bishop trihedron of the curve Y is T = coss,, sins N 1 = N = sin cos cos + sin s s s s coss sins coss sins,, and the curvatures of the curve Y are k 1 = cos sin cos s s,, s, k = sin with δ = s. The unit normal vector field of cyclinder is sins coss coss sins s s sin cos s sin cos s ηs = sins, 0, coss with the angle ψ = s. Since dy = then the homothetic scale is h = and we calculate the orthogonal matrix A = [a ij ] and so the matrix B in 1.1 is B = A where a ij are a 11 = s cos s cos sin s s + 10 sin sins a 1 = s s cos coss sin sins s 1 s a 13 = sin cos s cos + sins { } s a 1 = sin sins coss a = s cos { } s a 3 = sin coss + sins s a 31 = sin sin s + 1 s cos sins s s a 3 = cos sins sin coss a 33 = s s sin s cos cos s 10 sin sins and the matrix C is C = 1 coss { cos s + coss 1 sin s 1 coss { 10 + cos s s sins sin s sins s coss + sin s } coss } sins

11 Homothetic Bishop Motion Of Euclidean Submanifol 193 Figure. Sphere is rolling without sliding inside of the cylinder along the curves X and Y. Since X is the solution of the equation d BX + d C = 0 then X is a pole curve as a moving curve and Y is a fixed pole curve on the sphere φu, v and the cylinder x + z + = 1, respectively. The unit normal vectors ξ and η are the same direction and linear dependent at the contact points, thus the signature is ɛ = 1. The components of the anti-symmetric matrix S = [s ij ] are s 11 = s = s 33 = 0 s 1 = s 1 = 3 sin s s 31 = s 13 = s 3 = s 3 = coss + s sins 1 cos s cos s coss cos coss s cos coss + s sin sins with respect to the standart base of IR 3 and so the angular velocity vector is W = a 1 + a with respect to the base {a 1, a, η}, where the vector fiel a 1 and a are a 1 = coss,, sins a = coss cos s sins sin s s, cos, sins cos s sin s coss Since h is a constant and W lies on the tangent plane at the contact points then the sphere is rolling without sliding inside of the cylinder along the curves X and Y. References [1] A. Karger and J. Novak, Space Kinematics And Lie Grups,Gordon and Breach Science Publishers, Prague, Czechoslovakia, 197. [] B. Bükçü and M. K. Karacan, Special Bishop Motion and Bishop Darboux Rotation Axis of The Space Curve, Journal of Dynamical Systems and Geometric Theories, 61, [3] B. Karakaş, On Differential Geometry and Kinematics of Submanifol, Phd Thesis, Atatürk University, Ankara, Turkey, 19. [] Do Carmo, M. P., Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliffs, NJ, 1976.

12 19 Yılmaz TUNÇER, Murat Kemal KARACAN and Dae Won YOON [] H. H. Hacısalihoğlu, On The Rolling Of One Curve or manifold Upon Another, Proceedings Of The Royal Irish Academy, Sec. A,-Dublin, 71, [6] H. H. Hacısalihoğlu, Diferensiyel Geometri, Ankara Üniversitesi Fen Fakültesi, Cilt., Ankara, Turkey, [7] H. R. Müller, Zur Bewegunssgeometrie In Raumen Höherer Dimension, Mh. Math. 70, 1, [] J.H. Andrew and Hui Ma, Parallel Transport Approach To Curve Framing, Indiana University, Techreports- TR, January 11,199. [9] K. Nomizu, Kinematics And Differential Geometry Of Submanifol, Tohoku Math. Journ., 30, [10] L. R. Bishop, There Is More Than One Way To Frame A Curve, Amer. Math. Monthly, 3, [11] M.G. Cheng, Hypersurfaces in Euclidean spaces, Proc. Fifth Pacific Rim Geom. Conference, Tohoku University, Sensai, Japan. Tohoku Math. Publ. 0, [1] P. Appell, Traite de Mecanique Rationnelle, Tome I, Gauthiers-Villars, Paris, [13] Y. Tunçer, Y. Yayli and M. K. Sağel, On Differential Geometry and Kinematics of Euclidean Submanifol, Balkan Journal of Geometry and Its Applications, 13, [1] Y. Yaylı, Hamilton Motions and Lie Grups, Phd Thesis, Gazi University Science Institute, Ankara, Turkey, 19. [1] W. Clifford and J.J. McMahon, The Rolling Of One Curve or Manifold Upon Another, Am. Math. Mon. 6, 3A 13, Author information Yılmaz TUNÇER, Murat Kemal KARACAN and Dae Won YOON, Uşak University, Science And Faculty, Math. Dept. 1 Eylul Campus-Uşak, TURKEY. yilmaz.tuncer@usak.edu.tr Received: November, 01. Accepted: July 1, 01.