On the Blaschke trihedrons of a line congruence

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1 NTMSCI 4, No. 1, (2016) 130 New Trends in Mathematical Sciences On the Blaschke trihedrons of a line congruence Sadullah Celik Emin Ozyilmaz Department of Mathematics, Faculty of Science, University of Ege, Izmir, Turkey Received: 20 October 2015, Revised: 21 October 2015, Accepted: 6 December 2015 Published online: 30 January Abstract: In this study, it is aimed to find relation between the Blaschke vectors of parameter ruled surfaces of a line congruence which are not principle ruled surfaces. By this relation, we can find some basic formulae of the line space. (e.g.the Mannheim s Liouville s formulae. Keywords: Dual Space, Blaschke Trihedron, Dual Curvature. 1 Introduction A set of one-parameter of lines is called a ruled surface. Ruled surfaces, especially, developable ruled surfaces are used applied several areas in mathematics engineering,[1]. Dual number is a useful tool for line trajectories. Indeed, lines in Euclidean 3-space can be expressed by unit dual vectors there is one to one correspondence between points of dual unit sphere S 2 lines of Euclidean 3-space due to Study theorem. For detail, see [6,7,8,9]. On the other h, if we take two parameters in unit dual vectors, we have a line congruence. In practices, the line congruence defines a family of ruled surfaces. The study of line congruence was started by E.Kummer [5],in which he gave a classificasion of those of order one. The applications of the line geometry dual number representations of line trajectories have been developed by Blaschke [4], Muller [10]. In [2] [3], Caliskan gave a formulae between the Blaschke vectors of any ruled surface R 1 the parameter ruled surfaces R 11, R 21. Here, he used the parameter ruled surfaces by choosing as principle ruled surfaces. In this study, it is aimed to find relation between the Blaschke vectors of parameter ruled surfaces of a line congruence which are not principle ruled surfaces. By this relation, we can find some basic formulae of the line space. (e.g.the Manheim s Liouville s formulae). 2 Preliminaries Let A=a+εa 0 be a dual number, A ID = {(a,a 0 ) a,a 0 IR} ID be a commutative ring with a unit element. We call dual number ε = (0,1) ID as dual unit which satisfies ε 2 = (0,0).(D 3,+) is a module on the dual number ring. We call Corresponding author ozyilmaz@ege.edu.tr

2 131 S. Celik E. Ozyilmaz: On the Blaschke trihedrons of a line congruence it ID-module, dual vectors are the elements of this modul. We denote a unit vector A as where a, a 0 R 3. A = ( a, a 0 ) = a + ε a 0,< a, a >= 1,< a, a 0 >= 0, (1) Definition 1. The scalar product of two dual vectors A = a + ε a 0 B = b + ε b 0 is given by < A, B >=< a, b > +ε(< a, b 0 > + < a 0, b >). (2) Definition 2. [4] The vectoral product between two dual vectors A = a + ε a 0 B = b + ε b 0 is defined by A B = a b + ε( a b 0 + a 0 b ). (3) 3 The ruled surface the line congruence The Blaschke trihedron { R 1, R 2, R 3 } depends on the striction point of the ruled surface R 1 (t) in dual space ID 3,[4]. According to this, the first axis R 1 (t) of the trihedron is the generator which passes from the striction point of the ruled surface, the second axis R 2 (t) is normal of the surface at this point finally the third axis R 3 (t) is the tangent of the striction line at this point. The derivative formulae of the Blaschke trihedron { R 1, R 2, R 3 } with respect to dual arc parameter S of the striction curve are written where P = R 1, Q = det (R 1,R 1,R 1 ). P 2 R 1 = P R 2, A ruled surface is given as dual vectorial function by R 2 = P R 1 + Q R 3, R 3 = Q R 2, (4) R 1 (t) = r (t) + ε r 0 (t), R 1 2 = 1. (5) Definition 3. [3] The dual spherical curvature of the ruled surface R 1 (t) is defined as follows: Σ = Q P. (6) On the other h, the line congruence in ID 3 can be represented by a unit dual vector which depends on two real parameters u v as follows: R(u,v) = r (u,v) + ε r 0 (u,v), The dual arc element of a ruled surface of the line congruence can be given as R 2 = 1. (7)

3 NTMSCI 4, No. 1, (2016) / = d R 2 = ( R u du + R v dv) 2 = Edu 2 + 2Fdudv + Gdv 2, (8) where E = e+εe 0 =< R u, R u >, F = f +ε f 0 =< R u, R v > G = g+εg 0 =< R v, R v >. Moreover, the differential form I II of the line congruence are I = edu f dudv + gdv 2 (9) respectively. Thus, we have II = e 0 du f 0 dudv + g 0 dv 2, (10) 2 = I + εii. (11) Definition 4. [4] The drall of a ruled surface of a line congruence can be written as follows: 1 d = I 2II. () 4 The relations among the magnitudes of the ruled surface R 1, R 11 R 21 Let us consider a ruled surface R= R 1 (t) of the line congruence R= R(u,v) where u v are the functions of t. Let us write the parameter ruled surfaces of the line congruence as R 1 1 = R 1 1(u,v 0 ), R 2 1 = R 2 1(u 0,v), where the ruled surfaces R 1, R 1 1 R 2 1 have common line defined as The Blaschke trihedrons of these ruled surfaces are given by hence, one can get R = 1 (13) R = 1, (14) R 0 = R(u 0,v 0 ) = R 1 1(u 0,v 0 ) = R 2 1(u 0,v 0 ) (15) { R 0, R 2, R 3 },{ R 0, R 1 2, R 1 3}, { R 0, R 2 2, R 2 3} (16) R 1 = P R 2, R 2 = P R 1 + Q R 3, R 3 = Q R 2, (17) R 1 1 = P 1 R 1 2, R 1 2 = P 1 R Q 1 R 1 3, R 1 3 = Q 1 R 1 2, (18) R 2 1 = P 2 R 2 2, R = P 2 R 21 + Q 2 R 23, R 23 = Q 2 R, (19)

4 133 S. Celik E. Ozyilmaz: On the Blaschke trihedrons of a line congruence where P 1 = Ru 2 = E, P 2 = Rv 2 = G, Q 1, Q 2 are dual curvatures of parameter ruled surfaces. The dual arc elements of these ruled surfaces are R 1, R 11 R 21 can be given respectively as = Pdt, 1 = P 1 du = Edu, 2 = P 2 dv = Gdv. (20) Moreover, the Blaschke vectors of the Blaschke trihedrons are given by B = Q R 0 + P R 3, B 1 = Q 1 R0 + P 1 R 13, B 2 = Q 2 R0 + P 2 R 23. (21) If we choose the parameter ruled surfaces as principle ruled surfaces, we may write F = R u Rv = 0, [2]. In this study we suppose that F 0. Let us consider any parameter ruled surfaces of the line congruence R= R(u,v), R 2 = 1. Remark. The second edges of the parameter ruled surfaces can be written by From (4.10), we have R 11 R = = P 1 R 0 = R u E, R u R v R u R v = R 21 R = = P 2 R v G. () R u R v (23) EG R 1 R 2 = P = E du R + G dv R. (24) On the other h, let us consider the dual angle between the edges R R as Θ, the dual angle between the edges R 2 R as Φ. If we apply dot product both sides (4.) with R R, we have R 2 R = cosφ = E du + cosθ G dv, (25) Thus, from (4.13) (4.14) R 2 R = cos(θ Φ) = cosθ E du + G dv. (26) = E du, (27) sinφ = G dv (28) are written. Finally, putting (4.15) (4.16) into (4.), we have the following equation between the second edges of the Blaschke trihedrons of the ruled surfaces R 2, R R by R 2 = R + sinφ R. (29)

5 NTMSCI 4, No. 1, (2016) / Moreover, by the definition of the angle Θ between R R we write Then, from (4.18) we have R R = cosθ. (30) = 1 = E du, (31) sinφ = 2 = G dv. (32) Corollary 1. The third elements R 3, R 13 R 23 of the Blaschke trihedrons of the ruled surfaces R 1, R 11 R 21 are lineer dependent as follows: R 3 = R 13 + sinφ R 23. (33) Proof. If we substitute the equation (4.17) in R 3 = R 0 R 2 consider the Blaschke trihedrons of the parameter ruled surfaces R 11 R 21 we obtain (4.21). Theorem 1. The third elements R 13 R 23 of the parameter ruled surfaces R 11 vectors R R 21 as follows: R 21 can be expressed by dual R 13 = 1 ( cosθ R + R ), (34) Proof. From the equations (4.4) (4.11) we write R 23 = 1 ( R + cosθ R ) (35) R R 13 = R 0, R R 23 = R 0, R R = R 0, (36) R 1 2 ( R R 13 ) = 0 R 13 R 2 2 = M R, (37) R ( R R 23 + ) = 0 R 23 + R = N R. (38) Here, M N are dual scalars. And if we apply dot product of (4.25) (4.26) by the dual vectors R R respectively, also with the relation (4.4) (4.21), we have M = cosθ N = cosθ. Finally, substuting the dual vectors M N in (4.25) (4.26), we obtain (4.) (4.23). Corollary 2. The Blaschke vectors of the ruled surfaces R 1, R 11 R 21, which are defined in (4.9), can be written according to unit dual vectors R, R R 0 as follows: B = Q R cos(θ Φ) 0 P( R cosφ R ), (39) sinhθ

6 135 S. Celik E. Ozyilmaz: On the Blaschke trihedrons of a line congruence B 1 = Q 1 R0 + P 1 ( cosθ R + R ), (40) B 2 = Q 2 R0 + P 2 ( R + cosθ R ). (41) Theorem 2. Let R 1 R 11, R 21 be a ruled surface parameter ruled surfaces of congruence R(u, v), respectively. Then, we have R R = R. R = ( E)v cosθ( G) u, (42) S 1 S 1 EG where Θ is dual angle between R R. R R = R. R = ( G)u cosθ( E) v. (43) S 2 S 2 EG Proof. Differentiating (4.10) according to the parameters v u we have Then,we get R v = R uv E ( E)v Ru E R u = R uv G ( G)u Rv G (44) (45) R R u = Ru R uv G ( G)u Rv ( ) = ( E) v cosθ( G) u, (46) E G G R R v = Rv R uv E ( E)v Ru.( ) = ( G) u cosθ( E) v. (47) G E E On the other h, considering (4.8) in (4.34) (4.35) we obtain R. R = 1 R R S 1 E u = ( E)v cosθ( G) u, (48) EG R. R = 1 R R S 2 G v = ( G)u cosθ( E) v. (49) EG Finally, differentiating dual arcs S 1 S 2, we obtain R. R = R R, (50) S 1 S 1

7 NTMSCI 4, No. 1, (2016) / R. R = R R. (51) S 2 S 2 Thus,from (4.36) (4.38),(4.30) is obtained.in a similar way one can obtain (4.31). Proposition 1. The Blaschke trihedrons { R 0, R, R 13 } { R 0, R, R 23 } of the parameter ruled surfaces of the congruence R(u,v) always coincide such that the dual unit vectors between R R are orthogonal. Moreover, the Blaschke derivative formulae are given by d R 2 = B 2 R, d R 1 = B 1 R, d R 0 = B R 0, (52) d R 1 = B 1 R, d R 2 = B 2 R. (53) Proof. If we take Θ = π 2 in (4.) (4.23), we have first assertion. Differentiating (4.) according to the dual arc paramater S 1, we get R 13 = R, R 23 = R. (54) R 13 = 1 S 1 ( cosθ R + R ), (55) S 1 S 1 Moreover, from (4.41) (4.) we know = 1 ( cosθ( B 1 R R ) + ). S 1 R 13 = B 1 R 13 = 1 B 1 ( S 1 ( cosθ( B 1 R ) + ( B 1 R ) (56) Thus, from the second relation of (4.43) (4.44), we get the second relation of (4.40). Similarly, it can be shown that other relations are satisfied. Theorem 3. The dual curvatures Q 1 Q 2 of the parameter ruled surfaces R 11 R 21 are given by 1 Q 1 = G (( E) v cosθ( G) u ), (57) Proof. From (4.36) (4.8) we have Q 2 = 1 E (( G) u cosθ( E) v ). (58) R. R = ( E)v cosθ( G) u, (59) S 1 EG

8 137 S. Celik E. Ozyilmaz: On the Blaschke trihedrons of a line congruence R = 1 R S 1 E u = 1 B1 R. (60) E Then, we obtain ( E) v cosθ( G) u = 1 R R EG E u (61) = 1 R ( B 1 R ) = R 0 B1 E E On the other h, from (4.9), taking dot product B 1 with R 0 we have R 0 B1 E = Q 1 E. (62) Thus, from(4.49) (4.50) we get Q 1. Similarly, (4.46) can be obtained. Theorem 4. Let us consider any ruled surface R 1 of a line congruence R(u,v) the dual arc elements of the ruled surfaces R 1, R 11 R as S,S 1 S 2, respectively. Let B 1 B 2 be Blaschke vectors of R 11 R 21. We also suppose that the dual angle between the line R 2 R is Φ, then we have the following relations C = 1 B sinφ B 2 (63) P 1 P 2 d R = C R, d R = C R, d R 0 = C R 0 (64) Proof. We know that the dual vector R is the function of S 1 S 2. Hence we have From the equalities (4.19), (4.20), (4.40) (4.41) we obtain dr = d R = R 1 S 1 + R 2 2. (65) 1 ( B 1 R ) + sinφ 1 ( B 2 R ) E E = ( 1 B1 + sinφ 1 B2 ) R P 1 P 2 = C R. Thus, we have the first relation of (4.52). Other assertion can be obtained.

9 NTMSCI 4, No. 1, (2016) / Theorem 5. Let us consider the ruled surfaces R 1, R 11 R which have common line R 0 on the line congruence R(u,v). Let B 1 B 2 be Blaschke vectors of R 11 R 21. We obtain following equality among the Blaschke vectors B = P( 1 B sinφ B 2 + dφ R 0 ). (66) P 1 P 2 Proof. From (4.17), we have R 2 = Then, by taking derivative with respect to dual arc S, from equation (4.17), we obtain d R 2 = d R R + sinφ R. (67) + d R sinφ (68) cos(θ Φ) dφ R + cosφ dφ R. On the other h, considering the Blaschke trihedrons { R 0, R, R 13 } { R 0, R, R 23 }, we write R 13 = R R 0 = R R R ( ) (69) = (< R, R > R < R, R > R ) = 1 (cosθ R R ) R 23 = R R 0 = R R R ( ) (70) = < R, R > R < R, R > R = 1 ( R cosθ R ). And then, substituting the equations (4.57) (4.58) in - R = R 0 R 13 R = R 0 R 23 we have (4.55) as follows d R 2 = d R +[ R 0 (cosθ R R ) cos(θ Φ) + d R sinφ + R 0 ( R cosθ R ) cosφ ]dφ. (71)

10 139 S. Celik E. Ozyilmaz: On the Blaschke trihedrons of a line congruence According to theorem 4, we have d R = C R, d R = C R, (72) By using the dual trigonometric expression, we find d R 2 C = 1 B sinφ B 2. (73) P 1 P 2 = C R + sinφ C R + (74) R 0 [ R + sinφ R ] dφ Since we have = ( C + R 0 dφ ) R 2. M = C + R dφ 0 (75) C = M R 0 dφ. (76) Thus, (4.62) is written by d R 2 = M R 2. (77) From the last equation of (4.52) we have d R 0 = C R 0 (78) = ( M R dθ 0 ) R 0 = ( M R 0 ) d R 0 = M R 0.

11 NTMSCI 4, No. 1, (2016) / By using the Blaschke vector of d R 2 (4.65) we obtain M R 2 B R 2 = 0 ( M B) R 2 = 0 M B = Λ R 2 ; ΛεD (79) M R 2 B R 0 = 0 ( M B) R 0 = 0 From (4.67) (4.68) we have M B = Ω R 0,ΩεD. (80) Λ R 2 = Ω R 0, (81) Consequently, we have obtained hence we conclude R R 11, Corollary 3. Let respectively.then, we have Λ = Ω = 0. (82) M = B. (83) B = C + R dθ 0. (84) R 21 be a ruled surface parameter ruled surface of the line congruence R(u,v), Σ = Σ 1 + Σ 2 sinhφ + dφ where Θ is dual angle between R R, Φ is dual angle between R 2 R, Σ is dual spherical curvature of the ruled surface R 1, Σ 1 is dual spherical curvature of the ruled surface R 11,Σ 2 is dual spherical curvature of the ruled surface R. Proof. If we substitute the Blaschke vectors B, B 1 B 2 in (4.54) then taking dot product of both sides by R 0, we have Q = P( Q 1 + Q 2 sinφ P 1 P 2 + dφ ). (86) (85)

12 141 S. Celik E. Ozyilmaz: On the Blaschke trihedrons of a line congruence Thus, considering (3.3) we get assertion. Theorem 6.(Mannheim s Formula) The following relation P = P 1 + P 2 sinhφ ] (87) is satisfied among the dual curvatures of ruled surfaces R 1, R 11 R 21 of the line congruence R(u,v). Proof.Substituting the Blaschke vectors B, B 1 B 2 in (4.54) then taking dot product both of sides by R 3, desired equation is obtained. Theorem 7. (Liouville s Formula) There are following relation among the dual torsions of the ruled surfaces R 1, R 11 R 21 of the spacelike line congruence R(u,v) as Q = Q 1 + Q 2 sinφ + dφ. (88) Proof. If we substitute the Blaschke vectors B, B 1 B 2 in (4.54) then taking dot product both of sides by R 0, desired equation is obtained. References [1] Ravani, B., Ku Ts.,Bertr offsets of ruled developable surface, Computer Aided Geometric Design, Elsevier, 23(2), , [2] Caliskan, A., On the Studying of a line Congruence by Choosing Parameter Ruled Surfaces as Principal Ruled Surface. Journal of Faculty Science of Ege University Series A, No. (1),1987. [3] Caliskan, A., The Relation Among Blaschke Vectors of Ruled Surfaces on a Line Congruence And Its Consequance, Commun. Fac. Sci. Univ. Ank. Series An, Number (1-2), pp ,1989. [4] W. Blaschke,Vorlesungen uber Differentialgeometrie und geometrische Grundlagen von Einsteins Relativit atstheorie, Dover Publications, New York, [5] Kummer. E., Aber die algebraischen Strahlensysteme. insbesondere uber die derersten undzweiten Ordnung. Abh. K. Preuss. Akad. Wiss. Berlin (1866), 1-0, also in E. E. Kummer. Collected Papers. Springer Verlag, [6] Gugenheimer. H. W., Differential Geometry, Graw-Hill, New York, [7] Karger, A., Novak, J., Space Kinematics Line Groups. Gordon Breach Science Publishers. New York, [8] Veldkamp, G. R., On the use of dual numbers, vectors matrices in instantaneous spatial kinematics, Mech. Mach. Theory 11 (1976), [9] Clifford, W. K., Preliminary sketch of bi-quaternions. Proc. London Math. Soc. 4 (64,65), (1873) [10] Muller, H. R., Kinematik Dersleri. Ankara University Press, 1963.