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INTERNATIONAL JOURNAL OF COMPUTATIONAL COGNITION (HTTP://WWW.YANGSKY.COM/YANGIJCC.HTM), VOL. 3, NO. 3, SEPTEMBER 2005

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A Note on the Blaschke Invariants of the Generalized Ruled Surfaces Under the Symmetric Homothetic Motion Murat Tosun, Soley Ersoy and Mehmet Ali Güngör

Abstract— In this work, we give the relations between the blaschke invariants and principal blaschke invariants of the pair of generalized ruled surfaces under symmetric homothetic motion of order k in n−dimensional Euclidean space E n . In addition to this we express the blaschke invariants in terms of principal distribution parameter and total distribution parameter. Furthermore, as a special case the relation of the blaschke invariants are given in two-dimensional ruled surface in E n . c 2004-2005 Yang’s Scientific Research Institute, LLC. Copyright ° All rights reserved. Index Terms— Ruled surface, symmetric homothetic motion, blaschke invariant.

I. I NTRODUCTION N Euclidean space E n of n−dimensional, William Clifford and James J. McMohan [11] give a treatment of the rolling of one curve or surface upon another during the rigid body’s motion generated by the most general one parameter affine transformation. The results are: if n is even, the motion may be generated by the rolling of one curve upon another. The case of odd n, however, needs some physical conditions to interpret the motion as the rolling of one surface upon another. Another treatment has been given by H. R. Müller, [7] on the same kind of the motion. The results are: in the case n is even, the rolling curves upon each other are the polar curves of the motion like planar case; in the case is n odd, the rolling surfaces upon each other are the axoid surfaces which correspond to the viration motion of three dimensional space In [10] S. Keles and R. Aslaner have generalized the blaschke invariant to (k + 1)−ruled surfaces in E n that it has been defined for 2−ruled surfaces by H. Frank and O. Giering in [5]. Symmetrical homothetic motions were first studied by I. Aydemir and it has been shown that symmetric homothetic motion of order k in E n is a reflection with respect to a (n − k − m)−dimensional subspace of E n , [8]. In this work, taking into consideration of the studies by [1], [8], we have elaborated the blaschke invariants under the symmetrical homothetic motions in E n .

I

Manuscript received July 25, 2004; revised September 16, 2004. Murat Tosun, Soley Ersoy, Mehmet Ali Güngör, Department of Mathematics, Faculty of Arts and Sciences, Sakarya University, Sakarya, TURKEY. Email: [email protected](M. Tosun), [email protected](S. Ersoy), [email protected](M.A. Güngör). Publisher Item Identifier S 1542-5908(05)10309-1/$20.00 c Copyright °2004-2005 Yang’s Scientific Research Institute, LLC. All rights reserved. The online version posted on September 26, 2004 at http://www.YangSky.com/ijcc33.htm

II. H OMOTHETIC M OTION IN E n The homothetic motion of a body in n−dimensional Euclidean space is generated by the transformation; x = Sx + C, S = hA,

AAT = I

(2.1)

where AT is transposed of orthogonal matrix A, h is homothetic scale and A : J → O(n), C : J → IRn , h : J → IR

(2.2)

are the functions of differentiablity class C r (r ≥ 3) on real interval J. x ¯ and x correspond to the position vectors of the same point with respect to rectangular coordinate systems of the moving space E and the fixed space E, respectively. At the initial time t = t0 , we consider the coordinate systems of ¯ and E are coincident, [9]. E P (t) is the center of the intantaneous rotation of the motion t ∈ J and called the pole of the motion. At the pole P , the velocity vector vanishes by the equation (2.4). If |B| does not vanish on J, by considering the regularity condition of the motion we get a differentiable curve P : J → E of poles in the fixed space E, called the fixed pole curve. By (2.1) there is uniquely determined the moving pole curve P : J → E from the fixed pole curve point to point on J : S(t)P (t) + C. For the purpose of this paper we first summarize the basic properties of the generalized ruled surface from the paper, [3], [4]. In any k-dimensional generator Ek (t) of a (k + 1)dimensional generalized ruled surface φ ⊂ E n there exist a maximal linear subspace Kk−m (t) ⊂ Ek (t) of dimension k − m with property that in every point of Kk−m no tangent space of φ is determined (Kk−m contains all singularities of φ in Ek (t)) or there exist a maximal linear subspace Zk−m (t) ⊂ Ek (t) of dimension k − m with the property that in every point of Zk−m the tangent space of φ is orthogonal to asymptotic bundle of the tangent spaces in the points of infinity of Ek (t)(all points of Zk−m (t) have the same tangent of φ). We call Kk−m the edge space in Ek (t) ⊂ φ and Zk−m the central space in Ek (t) ⊂ φ. A point of Zk−m is called the central point. If φ possesses generators all of the same type the edge spaces respectively the central spaces generate a generalized ruled surface contained in φ which call the edge ruled surface respectively the central ruled surface. For m = k the edge ruled surface degenerates in the edge of φ, the central ruled surface in the line of striction so the ruled surface with edge ruled generalize the tangent surfaces of E 3 , the ruled

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surface with central ruled surface generalize the ruled surface with line of striction of E 3 . For the analytical representation of a (k + 1)-dimensional ruled surface φ we choose a leading curve α in the edge respectively central ruled surface Ω ⊂ φ transversal to the generators. In [3] it is shown that there exist a distinguished moving orthonormal frame (ONF) of φ {e1 , ..., ek } with the properties: (i) {e1 , ..., ek } is an ONF of Ek (t) ⊂ φ, (ii){em+1 , ..., ek } is an ONF of the Kk−m resp Zk−m (t) ⊂ Ek (t), k P (iii) e˙ i = αij ej + Ki ak+i , 1 ≤ i ≤ m, j=1

e˙ m+ρ = (2.6)

m P i=1

α(m+ρ)i ei , with Ki > 0, αij = −αji ,

α(m+ρ)(m+κ) = 0 , 1 ≤ ρ, κ ≤ k − m, (iv) {e1 , ..., ek , ak+1 , ..., ak+m } is an ONF. A moving ONF of φ with properties (i)-(iv) is called a principal frame of φ. If K1 > K2 > ... > Kk > 0, the principal frame of φ is determined up to signs. By a given principal frame vectors ak+1 , ..., ak+m are well defined. A leading curve α of (k + 1)-dimensional ruled surface φ is a leading curve of the edge respectively central surface Ω ⊂ φ too iff its tangent vector has the form α˙ (t) =

k X

ζi ei + ηm+1 ak+m+1

is called the blaschke invariant of φ, [10]. In this case m = k the central ruled surfaces Ω ⊂ φ degenerate in the line of striction . Thus the blaschke invariant of the 2-ruled surface φ generate by 1-dimensional subspace E (t) = Sp {e (t)} ⊂ Ek (t) can be given by k P

ζν cos θν ν=1 b= v "µ # u ¶2 k k uP P 2 t cos θν ανµ + (cos θµ κµ ) ν=1

µ=1

where e (t) = [10].

k P ν=1

cos θν eν (t) , θν = constant and kek = 1,

III. T HE B LASCHKE I NVARIANTS OF THE PAIR OF G ENERALIZED RULED S URFACE U NDER S YMMETRIC H OMOTHETIC M OTION Let α ⊂ E and α ⊂ E be moving and fixed pole curves, respectively. Suppose that {e1 (t) , ..., ek (t)} is an orthonormal vector field system at α (t) and let E k (t) = Sp {e1 (t) , ..., ek (t)} generate a (k + 1)-dimensional ruled surface with the leading curve α in the moving space E which called the moving ruled surface φ. φ has the following parameter representation

(2.7) φ (t, u1 , ..., uk ) = α (t) +

i=1

where ηm+1 6= 0, is a unit vector well defined up to the sign with the property that {e1 , ..., ek , ak+1 , ..., ak+m , ak+m+1 } is an ONF of the tangential bundle of φ. One shows: ηm+1 = 0, in t ∈ J iff generator Ek (t) ⊂ φ contains the edge space Kk−m (t). If ηm+1 6= 0, we call m−magnitudes ηm+1 ρi = , 1≤i≤m (2.8) κi the principal parameters of distribution, [3]. Moreover, the total parameter of distribution of a generalized surface φ is given in [2] by m Y D= ρi (2.9) i=1

Now we can give the defination of the blaschke invariant of 2-ruled surface in E n . Let α (t) and Ek (t) = Sp {e (t)} be leading curve and generating space of 2-ruled surface in E n , respectively. Then magnitude ζ (2.10) b= κ is called the blaschke invariant of the φ where κ and ζ are given by (2.6) and (2.7), [5]. Let φ be a (k + 1)-ruled surface in E n The dimension of the asymptotic bundle of φ being k + m, m > 0 the magnitudes ζi , 1≤i≤m (2.11) bi = κi are called the principal blaschke invariants of φ and p (2.12) B = m |b1 ...bm |

(2.13)

k X

ui ei , t ∈ J, ui ∈ IR.

(3.1)

i=1

Let {ε1 (t) , ..., εk (t)} be an orthogonal vector field system satisfying the following equations at the point α (t) in the fixed space E: ¡

S (ei ) = −εi , 1 ≤ i ≤ k ¢ S.S −1 .εi = 0, 1≤i≤k

(3.2) (3.3)

If we denote Ek (t) = {ε1 (t) , ..., εk (t)}, Ek (t) generates a (k + 1)-dimensional ruled surface with leading curve α given by (2.1) in the fixed space E, which is called the fixed ruled surface and denoted by φ. If a homothetic motion given by (2.1) satisfies the equations (3.2) and (3.3), then this homothetic motion is called a symmetric motion of order k. Considering equation (3.3) we obtain from equation (3.2) by differentiation ³. ´ S ei = −˙εi , 1 ≤ i ≤ k (3.4) T (t) and T (t) being two tangential bundles which are correspond to each other under the symmetric homothetic motion of order k. Let {e1 , ...ek , ak+1 , ..., ak+m , ak+m+1 , ..., an } and {e1 , ...ek , ak+1 , ..., ak+m , ak+m+1 , ..., an } are two orthonormal frame of E n with respect to T (t) and T (t), respectively. Then we have following equations; Sei = −hei

,

1≤i≤k

(3.5)

¨ BLASCHKE INVARIANTS OF THE GENERALIZED RULED SURFACES UNDER THE SYMMETRIC HOMOTHETIC MOTION ¨ TOSUN, ERSOY, & GUNG OR,

Sak+j Sak+m+λ

= =

−hak+j , hak+m+λ ,

1≤j≤m 1≤λ≤n−k−m

It is clear that the symmetrical homothetic motion of order k of E n is reflection with respect to the subspace Sp {ak+m+1 , ..., an } of dimension (n − k − m), [8]. Since {e1 (t) , ..., ek (t)} and {e1 (t) , ..., ek (t)} are the principal orthonormal frame of φ and φ, respectively, we have the equation (2.6). k k . P P . For ei (t) = αij ej + κi ak+i and ei (t) = αij ej + j=1

j=1

κi ak+i , (1 ≤ i ≤ m) we have the following results which are given in [8], Ã ! h˙ αii = + αii , i = j, 1 ≤ i ≤ m (3.6) h αij

= αij ,

κi

= κi ,

.

For α (t) =

k P

i 6= j,

1≤i≤k 1≤i≤m

ζ i ei + η m+1 ak+m+1 and α˙ (t) =

i=1

ηm+1 ak+m+1 and we have ζi = −hζ i , ηm+1

1 ≤ i ≤ k,

=

k P

81

ρi and ρi , respectively, under symmetric homothetic motion of order k. There are the relations v¯ ¯ u¯ m ¯ u¯Y 1 m t¯ ζ .ρ ¯¯ B= (3.12) i i ¯ ¯ η m+1 i=1 and B=

1 ηm+1

v¯ ¯ u¯ m ¯ u¯Y m t¯ ζi .ρi ¯¯ ¯ ¯

(3.13)

i=1

between the blaschke invariants and principal distribution parameters. Proof: ¿From (2.8) we can write for the moving ruled surface φ η ρi = m+1 , 1 ≤ i ≤ m κi ¿From the last equation we get write η κi = m+1 , 1 ≤ i ≤ m. ρi If this equation substitutes into equation (3.9), then we find

ζi ei +

bi =

i=1

(3.7)

hη m+1

.

where α and α˙ are the velocity vectors of α and α, respectively. Theorem 1: Let φ and φ be the (k + 1) −dimensional moving and fixed generalized ruled surfaces which correspond to each other under the symmetric homothetic motion with principal blaschke invariants bi and bi , respectively, then we have following result

ζ i .ρi , η m+1

1 ≤ i ≤ m.

Considering the equation (2.12), we reach the relation between principal distribution parameter and blaschke invariant of moving ruled surface φ to be v¯ ¯ u¯ m ¯ u¯Y 1 m t¯ ζ .ρ ¯¯ B= i i ¯ ¯ η m+1 i=1

In a similar way, we can do this for fixed ruled surface φ and find that v¯ ¯ u¯ m ¯ u¯ Y 1 m ¯ t B= ¯ ζi .ρi ¯. ¯ ¯ ηm+1 i=1

bi = −hbi . (3.8) Proof: ¿From the definition of principal blaschke invariants of φ and φ, we have bi =

ζi , κi

1≤i≤m

bi =

ζi , κi

1 ≤ i ≤ m.

and

(3.9)

Using (3.6) and (3.7) we get bi = −hbi ,

1 ≤ i ≤ m.

(3.10)

Considering the equation (2.9) and the last theorem, we can give the following corollary. Corollary 4: Let φ and φ be the moving and fixed generalized ruled surfaces with the total distribution parameters D and D, resp., under symmetric homothetic motion of order k. Then there are the relations v¯ u¯ m ¯¯ u¯ Y ¯ 1 m t¯D. ζ ¯ B= (3.14) i ¯ ¯ η m+1 i=1 and B=

If we consider the equations (2.12) and (3.8), then we can give the following corollary. Corollary 2: If the blaschke invariants of φ and φ are B and B, respectively, then there exist the relation between invariants; (3.11) B = hB. Theorem 3: Let φ and φ be the moving and fixed generalized ruled surfaces with the principal distribution parameters

1 ηm+1

v¯ u¯ m ¯¯ u¯ Y ¯ m t¯D. ζi ¯. ¯ ¯

(3.15)

i=1

between the blaschke invariants and total distribution parameters. Now we would like to give the relation between the blaschke invariants of ψ ⊂ φ and ψ ⊂ φ 2-ruled surfaces corresponding to each other under symmetric homothetic motion of order k. Theorem 5: We have the following relation between the blaschke invariant of ψ ⊂ φ and ψ ⊂ φ 2-ruled surfaces

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corresponding to each other under symmetric homothetic motion of order in n−dimensional Euclidean space E n

Proof: Let e=

k P ν=1

b = −hb. k P e= cos θν eν ν=1

cos θν eν

¡

(3.16) ¢ θν = constant and

(θν = constant) be the tangential vectors

of E k (t) and Ek (t), respectively. If we consider equation (3.2), then we obtain cos θν = cos θν , 1 ≤ ν ≤ k

(3.17)

By substituting equations (3.6), (3.7) and (3.17) into equation (2.13) we get −h.

k P

ζ ν cos θν ν=1 b= v "µ #. u ¶2 k k uP ¢2 ¡ P t cos θν ανµ + cos θµ κµ µ=1

ν=1

Using (2.13) and (3.8) the theorem is proved. Remark 6: In this paper blaschke invariants of twodimensional ruled surfaces and the pair of generalized ruled surfaces under the homothetic motion have been investigated in E n when compared the results gained in this paper with the results by [1]. It can be seen that principal blaschke invariants are equal to each other with the change of sign, whereas blaschke invariants are equal to each other without any change of sign. R EFERENCES [1] A. I. Sivridag, S. Keles, On the Blaschke Invariants the Pair of the Generalized Ruled Surfaces Under the Homothetic Motions, Commun. Fac. Sci. Univ. Ank. Series A1 v.41 (1992) pp105-110. [2] C. Thas, Een(lokale) Studie Van de (m + 1)−dimensonale Varieteiten, Van de n-dimensionale Euclidshe Ruimte Beschereven door Een Eendimesionale Familie Van m−dimensionale Lineare Ruiten. Paleis Der Academien-Herttogsstraat, I Brussel, (1974). [3] H. Frank, O. Giering, Verallgemeinerte Regelflachen Math. z.150 (1976), 261-271 [4] H. Frank, On the kinematics of the n-dimensional Euclidean Space Contribution to Geometry, Proceedings of the Geometry Symposium in Siegen 1978. [5] H. Frank, O. Giering, Verallgemeinerte Regelflachen in Grossen II. Journal of Geo. Vol.23 (1984). [6] H. H. Hacısalihoglu, On the rolling of one curve on surface upon another, Proceedings of Royal Irish Academy, vol.71 sec A. number 2 (1971) 1318. [7] H. R. Müller, Zur Bewegungsgeometrie in Raumen höherer Dimension, Mh. Math. 70 Band, 1 heft. pp. 47-57, 1966. [8] I. Aydemir, N. Kuruoglu, M. Çaliskan, On the Pair of Generalized Ruled Surfaces Under the Symmetric Homothetic Motions of Order k In the Euclidean Space , Commun. Fac. Sci. Univ. Ank. Series A1 v.41 (1992) pp13-26. [9] K. Nomizu, Fundamentals of Linear Algebra p269, New York, McGraw Hill Book Company, (1966). [10] S. Keles, R. Aslaner, E n de -regle Yüzeylerin Blaschke Invaryantlari ¨ ¨ Uzerine, Erciyes Universitesi, Fen Bilimleri Dergisi, (1990) 928-935. [11] W. Clifford and J. J. McMahon , The rolling of one curve or surface upon another. Am. Math. Mon. 68, 338-341, 23 A 2134, (1961).