RELATION BETWEEN THIRD ORDER FINITE SCREW SYSTEMS AND LINE CONGRUENCE

Transcription

1 INTERNATIONAL JOURNAL OF GEOETRY Vol. 3 (014), No., 5-34 RELATION BETWEEN THIRD ORDER FINITE SCREW SYSTES AND LINE CONGRUENCE ŞER IFE NUR YALÇIN, AL I ÇALIŞKAN Abstract. In this paper, by using Dimentberg s de nition of pitch, a screw system which consists of displacement of a line is considered. After forming this third order system, by considering three di erent systems; the external bisectors which are the elements of these screw systems are obtained. It is demonstrated that these external bisectors belong to the same recticongruence. Then the axis of this recticongruence is found. Finally, it is concluded that the angles between the rst, second, third screws and the axis of recticongruence are the same. In this way, it is shown that these screws belong to the same line congruence. 1. Introduction In the last decade, researchers have paid much attention to line segments and point-lines as opposed to in nite lines, [1]. Having viewed a point-line as an embedded part of a rigid body, some researchers used screws and screw systems to describe a point-line displacement, [6]. Parkin, [10], modeled the nite point-line displacement as a -system by assigning a new de nition to the pitch. By following Parkin s de nition of a pitch, Huang and Roth [5], Hunt and Parkin, [7], and Zhang and Ting, [1, 13], studied di erent nite displacement of various geometrical elements. With the conventional pitch de nition, it was found that the linear properties, [1], in in nitesimal twists do not exist in nite cases. This problem was resolved by Parkin who rede ned the pitch to be the ratio of one-half the translation to the tangent of one-half rotation, [10]. Parkin showed that all possible nite twists to displace a point-line from one position to another could be presented with a -system, [7]. Keywords and phrases: Line geometry, Screw system, Congruence (010)athematics Subject Classi cation: 53A17, 53A5 Received: In revised form: Accepted:

2 6 Şerife Nur Yalç n, Ali Çal şkan Even though various nite screw systems have been discovered by using Parkin s de nition of pitch [10], the displacement of a line was an exceptional case. It was known the nite screws for displacing a line had not form a screw system, [5, 7], since Huang and Wang noticed Dimentberg s de nition of pitch, [4]. Upon this, they adopt this de nition and show that all possible screws for displacing a line can form a screw system of the third order, [6]. In this paper, by using Dimentberg s de nition of pitch three screw systems of the third order are formed. Then the relation between these screw systems and congruence concept are studied.. Formation of screw systems A screw can be de ned as 6-dimensional vector composed of a pair of 3-dimensional vectors or as a dual vector as below (.1) b V = ( v ; v0 ) = v + " v 0 : where " = 0. And also a screw can be de ned, geometrically, as a straight line with which a de nite linear magnitude termed the pitch is associated, [1]. The pitch of a screw can be determined by (.) p = v v0 v v : The Plücker coordinates of a line are as below (.3) E b = a ; b a = b a + " a. where a denotes the direction cosine vector of the line and b is the position vector of a point on the line. A screw and its pitch can also be de ned as below, respectively, (.4) b V = _ c b E = (c + "c0 ) b E; (.5) p = c 0 c where _ c is a dual number and b E is a line vector. In kinematics, screws are used to describe displacements of a rigid body and these displacement screws are named as twists. The pitch of a twist is de ned as (.6) p = t ;

3 Relation between third order nite screw systems and line congruence 7 where and t are the rotation and translation parameters of the twist, respectively. On the other hand, Dimentberg [4] presented a novel de niton of pitch in formulation the resultant displacement of two successive displacement screws. He showed that if a nite twist of a rigid body from position i to position j is de ned as (.7) T = tan _ b E ; where _ = + "t is the dual angle of the twist, then the resultant nite twist (T 13 ) of two successive nite twists ( T 1 and T 3 ) are related by as below, [4]. (.8) T 13 = T 1 + T 3 T 1 T 3 1 T 1 T 3 : The tangent of the dual angle of the twist in Eq. (:7) is (.9) tan _ = tan + "t 1 + tan. From this point of view with the help of Eq. (:5), the pitch of the nite twist de ned by Dimentberg, [4], is (.10) p = t tan 1 + tan = t : sin The twist can also be de ned as the product of an amplitude and a unit screw, where its pitch is de ned by Eq. (:10), as below (.11) T = tan b V. To determine the position of a rigid body requires six independent parameters, four independent parameters to describe the position of a line and two free parameters to choose a screw for displacing a line from one position to another. That is important if these screws can form a screw system of the third order. To check it we need to know if these screws provide the representation in (.1) V = f 1 b V1 + f b V + f 3 b V3 ; where f 1 ; f and f 3 are scalar functions of the free parameters and b V 1 ; b V ; bv 3 are linearly independent screws, [6].

4 8 Şerife Nur Yalç n, Ali Çal şkan Tsai and Roth, [11], formulated an incompletely speci ed displacement as computing the resultant of two successive displacement screws. The twist V for displacing a particular element of a body can be thought of as the resultant of a twist K which containing free parameters and a twist L which virtually displaces the element. In the case of displacing a line, the axis of K is taken to be initial position of the line. On the other hand, d and which are the translation and rotation parameters of K, respectively, do not displace the position of the line because of this they are free parameters. The axis of L is selected to be the line perpendicular to both the initial and nal positions of the line and the rotation and translation parameters of L are the angle and distance between the two positions of the line, respectively. Figure 1 Let the Plücker coordinates of two positions of a line as shown in Fig.1, [6], be (.13) ba 1 = ( a 1 ; a 01 ) = a1 ; b 1 a 1 ; ba = ( a ; a 0 ) = a ; b a : The unit screw of K, whose axis coincides with b A 1, and the pitch of K named as p K are given as below, respectively, (.14) b K = k ; k0 (.15) p K = d sin. The unit screw of L is (.16) b L = l ; l0 = = ( a 1 ; a 01 + p K a1 ) ; l ; bl l + p L l ;

5 Relation between third order nite screw systems and line congruence 9 where (.17) b L = 1 j a 1 a j (.18) p L = t sin ; hl a0 a1 l i a01 a ; j (.19) = tan 1 a1 a j a1 ; a b (.0) t = b1 l ; (.1) l = ( a 1 a ) j a 1 a j. Since the resultant of K and L is V, according to Eqs. (:8) and (:11), (.) tan V b = tan b K + tan b L tan tan b K L b 1 tan tan b K L b ; where is the rotation parameter of V. Then because of K and L intersect each other perpendicularly, (.3) b K b L = 0: From Eq. (:3), Eq. (:), (.4) tan b V = tan b K + tan b L tan tan b K b L is obtained. So, the cross product of K and L can be simpli ed as " # bk L b k l = ( b 1 : l ): k + ( k + l 0 ) + p K ( k l ) " # k l (.5) = ( b 1 : l ): k + ( k + l 0 ) + 0 p K ( k l ) Substituting the above equation into Eq. (:4) gives

6 30 Şerife Nur Yalç n, Ali Çal şkan (.6) " l # tan V b = tan l 0 + tan k tan k 4 l b1 k tan b1 : l k k tan l0 " # d cos k tan k l = tan b V 1 + tan b V + d 1 + cos b V Since that the above equation is in the form of Eq. (:1), all possible screws for displacing a line form a three system as illustrated in Fig., [6]. As illustrated in Fig, b V1 is the screw b L, b V is a pure rotation screw whose axis is the internal bisector of b A1 and b A, b V 3 is a screw parallel to bv : In order to displace the speci ed line properly, it can be shown that b V 3 must intersect b D, which is external bisector of b A 1 and b A ; perpendicularly, [3]. Figure 3. The recticongruence which includes external bisectors Let; lines A 1, A are denoted with b A 1, b A dual unit vectors. And b V, b D dual unit vectors are bisectors of A 1, A, respectively, where the dual angle between b A 1 and b A is _ 1 = 1 + "t 1, (3.1) bv = ba 1 + b A s ; D b = A1 b b s A : 1+cos _ 1 1 cos _ 1

7 Relation between third order nite screw systems and line congruence 31 So from equations in following; bv b D = 0; V b ba 1 A b A b 1 A b = 0; D b ba 1 A b A b 1 A b = 0; we can say that V b ; D b and perpendicular bisector of A b 1 and A b vectors are formed a perpendicular trihedron. On the other hand, if dual angle between ba 1 and V b is indicated with = + " and with the aid of equation as below; s (3.) b A1 b V = 1 + cos _ s 1 1 = + cos _ cos _ 1 = cos _ 1 = cos ; (3.3) = 1 ; = t 1 are found, [9]. In Eq. (3:3), as the denominators are equal to two, the top of perpendicular trihedron is in the middle of the perpendicular bisector. And each b V and bd vectors are made the same angle between b A 1 and b A as illustrated in Fig. Now consider dual unit vectors b A 1 ; b A ; b A 3 which are not parallel to each other and dual unit vectors b D; b E; b F which are dual external bisectors of ba 1 ; b A and b A 3 as below; (3.4) A1 b A b = cos _ 1 ; A b A b 3 = cos _ 3 ; A3 b A b 1 = cos _ 31 ; (3.5) ba 1 A b bd = s 1 ; b ba A3 b E = s cos _ 1 1 ; b ba 3 A1 b F = s cos _ 3 1 : cos _ 31 Then we can say that b D; b E; b F belong to the same recticongruence because these bisectors are linearly dependent. To prove this, let show that there are dual numbers x; y; z which are not pure dual and provide the following equation: (3.6) x b D + y b E + z b F = 0: When Eq. (3:5) and (3:6) are considered; an equation system is obtained. To show vectors b D; b E; b F are linearly dependent, let us proof that this equation system have a solution system which are not consist of pure dual

8 3 Şerife Nur Yalç n, Ali Çal şkan numbers. Namely, let us proof that the determinant of the coe cient of this equation system given as (3.7) = s 8 1 cos _ 1 1 cos _ 3 1 cos _ 31 is equal to zero. Then because of A 1 ; A ; A 3 lines are not parallel to each other, is equal to zero. And thus x; y; z dual numbers which are not pure dual can be found. On the other hand, let show that the perpendicular bisectors of D; E; F two by two is the same line. (3.8) D b E b A1 b A b + A b A b 3 + A b 3 A b 1 = A b 1 A b + A b A b 3 + A b 3 A b 1 is the perpendicular bisector of D; E. Then because of b D; b E; b F bisectors are linearly dependent we can write (3.9) F b D b E b D b E b = 0. Because of this, b D; b E; b F external bisectors belong to the same recticongruence such that the axes of recticongruence is (3.10) b N = c ; where c = A b 1 A b + A b A b 3 + A b 3 A b 1, = c : 4. The Congruence Which Includes b A 1 ; b A and b A 3 Let us consider A 1 ; A ; A 3 lines and with the aid of Eq. (3:10); Eq. (4:1), (4:) and (4:3) are found as (4.1) ba 1 b N = b A 1 c h A1 b ba1 A b + = b A b A 3 + b A 3 b A 1 i ba1 ; b A ; b A 3 = ba1 ; b A ; b A 3 ; (4.) b A b N = ;

9 Relation between third order nite screw systems and line congruence 33 ba1 ; b A ; b A 3 (4.3) b A3 b N = And thus let us say ba1 ; b A ; b A 3 : (4.4) So = cos : (1) b A1 b N = b A b N = b A 3 b N = cos % is obtained. As a result, any three lines such as A 1 ; A ; A 3 are in the same line congruence, which have a constant dual slope, whose axis is the same with the axis of the recticongruence which dual external bisectors of A 1 ; A ; A 3 lines are belong and nally which is consist of the lines which made the constant dual angle given in (4:5) with the axis of recticongruence. 5. Conclusion However, previous researchers had deduced that the screws for displacing a line did not form a screw system by using di erent de nitions of pitch, Huang and Wang [6] proved that the displacement screws can generated a screw system of the third order by using Dimentberg s de nition of pitch. In this study; rst, the formation of screw system with the help of Dimentberg s de nition of pitch and the concept of a screw triangle was given. Second, the external bisectors of the lines which form the screw system and the recticongruence to which these bisectors were belong was determined. Finally, it was shown that the lines were in the same line congruence. We hope that our study may contribute to the relation between theoretical and practical kinematics and also the relation between concepts of the screw system of a displacing line and the line congruence. References [1] Ball, R.S., A Treatise of the Theory of Screws, Cambridge University Press, Cambridge, UK, [] Blashke, W., Vorlesungen uber Di erential Geometrie (Lectures on Di rential Geometry), Bd 1, Dover Pubblications, New York, [3] Bottema, O. and Roth, B., Theoretical Kinematics, North-Holland Publishing Company, New York, [4] Dimetberg, F.., The Screw Calculus and Its Application in echanics (in Russian), oscow, [5] Huang, C. and Roth, B., Analytic Expressions for the Finite Screw Systems, ech. ach. Theory, 9()(1994) 07-. [6] Huang, C. and Wang, J-C., The Finite Screw System Associated With the Displacement of a Line, Journal of echanical Design, 15(003)

10 34 Şerife Nur Yalç n, Ali Çal şkan [7] Hunt, K.H. and Parkin, I.A., Finite Displacement of Points, Planes and Lines via Screw Theory, ech. ach. Theory, 30()(1995) [8] Hunt, K.H., Kinematics Geometry of echanisms, Clarendon Press, Oxford, [9] Kula., Dual Bisectors of Two Lines and Line Congruences About The Bisectors (in Turkish), E.Ü. Fen Fak. Der; S. A, C.II., S.(1978) 155. [10] Parkin, I.A., A Third Conformation with the Screw Systems: Finite Twist Displacements of a Directed Line and Point, ech. ach. Theory, 7()(199) [11] Tsai, L.W. and Roth, B., Incompletely Speci ed Displacement: Geometry and Spatial Linkage Synthesis, ASE J. Eng. Ind., 95B()(1973) [1] Zhang, Y. and Ting, K-L., On Point-Line Geometry and Displacement, ech. ach. Theory, 39(004) [13] Zhang, Y. and Ting, K-L., Point-Line Distance Under Riemannian etrics, Journal of echanical Design, 130(008) 1-7. DEPARTENT OF ATHEATICS EGE UNIVERSITY FACULTY OF SCIENCE IZIR, TURKEY address: serife.nur.yalcin@ege.edu.tr DEPARTENT OF ATHEATICS EGE UNIVERSITY FACULTY OF SCIENCE IZIR, TURKEY address: ali.caliskan@ege.edu.tr