SOME RESULTS ON THE GEOMETRY OF TRAJECTORY SURFACES. Basibuyuk, 34857, Maltepe, Istanbul, Turkey,

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1 SOME RESULTS ON THE GEOMETRY OF TRAJECTORY SURFACES Osman Gürsoy 1 Ahmet Küçük Muhsin ncesu 3 1 Maltepe University, Faculty of Science and Art, Department of Mathematics, Basibuyuk, 34857, Maltepe, stanbul, Turkey, osmang@maltepe.edu.tr Kocaeli University, Faculty of Education, Kocaeli, Turkey,akucuk@kou.edu.tr 3 Karadeniz Technical University, Faculty of Science and Art, Department of Mathematics, Trabzon,Turkey, mincesu@ktu.edu.tr Abstract-n this study, the relationships between the invariants of trajectory ruled surfaces generated by the oriented lines xed, in a moving body, in E 3 are investigated. Some new results on the pitches and the angle of pitches of the trajectory surfaces generated by the Steiner and area vectors are obtained and new comments are given. Also, the area of projections of spherical closed images of these surfaces are studied. 1.ntroduction The geometry of path trajectory ruled surfaces, generated by the oriented lines, xed in a moving rigid body is important in the study of rational design problem of spatial mechanism. An x-closed trajectory ruled surface (x-c.t.s.) is characterized by two real integral invariants, the pitch `x and the angle of pitch x. Using the integral invariants, the closed trajectory surfaces have been studied in some papers [1], [], [3]. n this study, based on [4], introducing a relationship between the dual integral invariant, x, and the dual area vector, V x, of the spherical image of an x-c.t.s., new results on the feature of the trajectory surfaces are investigated. And also, since the dual angle of pitch, de ned in [5], of an x-c.t.s. is a useful dual apparate in the study of line geometry, we use the dual representations of the trajectory surfaces with their dual angle of pitches. Therefore, besides the results on the real angle of pitches, that some of them given [4] many other results on the pitches of closed trajectory ruled surfaces are obtained. And some relationships between the other invariants are given. Also, using the some other methods, the area of projections of spherical closed images of the trajectory surfaces are studied. t is hoped that the ndings will contribute to the geometry of trajectory surfaces, so the rational design of spatial mechanisms.. Basic Concepts Let a moving orthonormal trihedron {v 1, v, v 3 } make a spatial motion along a closed space curve r = r(t), t R, in E 3. n this motion, an oriented 1

2 line xed in the moving system generates a closed trajectory surface in E 3. A parametric equation of a closed trajectory surface generated by v 1 -axis of the moving system is x(t; ) = r(t)+v 1 (t); x(t+; ) = x(t; ); t; R (1) and denoted by v 1 (t)- c.t.s. Consider the moving orthonormal system fv 1 ; v = v1= 0 kv1k 0 ; v 3 = v 1 ^ v g, then the axes of the trihedron intersect at the striction point of v 1 -generator of v 1 -c.t.s. and v and v 3 are the normal and tangent to the surface, at the striction point, respectively. The structural equations of this motion are 3X dv i =! j i v j;! j i (s) =!i j (s); s R; i; j = 1; ; 3 () j=1 db ds = cos v 1+sin v 3 (3) b = b(s) is the striction line of v 1 -c.t.s. and the di erential forms! 1;! 3 and are the natural curvature, the natural torsion and the striction of v 1 -c.t.s., respectively. Here, the striction is restricted as < < for the orientation on v 1 -c.t.s., and s is the arc-length of the striction line. The pole vector and the Steiner vector are given by p = k k ; d = (4) spectively, where =! 3 v 1 +! 1v 3 is instantaneous Pfa an vector of the motion. The pitch (Ö nungsstrecke) of v 1 -c.t.s. is de ned by `v1 := d = hdr; v 1 i (5) The angle of pitch (Ö nungswinkel) of v 1 -c.t.s. and is given by one of the forms v1 := d = hdv ; v 3 i = hv 1 ; di = a v1 = (6) a v1 and g v1 are the measures of the spherical surface area bounded by the spherical image of v 1 -c.t. surface and the geodesic curvature of this image, respectively. The pitch and the angle of pitch are well-known integral invariants of a closed trajectory surface, [1], [3], [6]. The real area vector of an x-closed space curve in E 3 is given by v x = x^dx (7) g v1 [7]. And in a spatial closed motion, the area of projection of x-closed spherical image along any y-closed trajectory surface is de ned by

3 f x;y = hv x ; yi (8) x and y are the unit vectors in the moving system, [4]. According to E. Study s transference principle, a unit dual vector x = x+"x corresponds to only one oriented line, in E 3, where the real part x shows the direction of this line and the dual part x shows the vectorial moment of the unit vector x with respect to the origin, in E 3, [8]. Let K be a moving dual unit sphere generated by a dual orthonormal system V 1 ; V = V 0 1 kv1k ; V 0 3 = V 1 ^ V ; V i = v i +"vi ; i = 1; ; 3: (9) K 0 be a xed dual unit sphere with the same center. Then, the derivative equations of the dual spherical closed motion of K with respect to K 0 are given as dv i = X j i V j; j i (t) =!j i (t) + "!j i (t); j i = i j ; t R, i = 1; ; 3: (10) The dual Steiner vector of the closed motion is de ned by D = ; = k k P; = +" (11) = 3 V 1 + 1V 3 and P are the instantaneous Pfa an vector and the dual pole vector of the motion, respectively. As known from the E.Study s transference principle, the dual equations (10) correspond to the real equations () and (3) of a closed spatial motion, in E 3. n this sense, the di erentiable dual closed curve, V 1 = V 1 (t); t R, is considered as a closed trajectory ruled surface in E 3 and denoted by v 1 (t)-c.t.s.. A dual integral invariant which is called the dual angle of pitch of a v 1 -c.t.s. is given by ^v1 = hdv ; V 3 i = hv 1 ; Di = A v1 = G v1 = v1 "`v1 (1) [5], [6], where D = d + "d ; A v1 = a v1 + "a v 1 and G v1 = g v1 + "gv 1 are the dual Steiner vector of the motion, the dual spherical surface area and the dual geodesic curvature of spherical image of v 1 -c.t.s., respectively. 3. The Relationships and Results Consider the di erentiable unit dual spherical closed curve X = X(t); X(t+) = X(t); kxk = 1; t R. (13) We know from E.Study s transference principle that the dual corresponds to an x-closed trajectory surface generated by an x-oriented line xed in a moving rigid body, in E 3. The dual area vector of an x-dual closed spherical curve can be de ned by V x = X^dX (14) 3

4 an analogy to the de nition, in [7], where dx = ^X is the di erential velocity of a dual point,x; xed in the moving sphere K. From (4) and (14), the dual area vector may be developed as V x = X ^ ( ^ X) = (hx; Xi hx; i X) = X; X = D hx; Di X (15) with the aid of (1) V x = D+^xX: (16) The statement shows that there is a relationship between the dual angle of pitch of an x-c.t.s. and the dual area vector of x-closed spherical image of this surface. From (1) and (16), we may write hv x ; Di = hd; Di+^x hx; Di D E V kv x k x kv ; D xk = kdk ^x ^x kv x k ^vx = kdk (17) ^vx is the dual angle of pitch of v x -trajectory surface generated by the area vector of x-closed spherical image of x-c.t.s.. On the other hand, from (1), the dual angle of pitch of d-c.t.s. generated by the Steiner vector D ofethe motion is D ^d = kdk ; D = kdk : (18) Since D =, the dual angle of pitch, ^d, gives the total dual spherical rotation in the interval with one period. Seperating (18) into real and dual parts, we may give the following theorem. Theorem 1: The angle of pitch and the pitch of d-c.t.s. give the total rotation and the total translation of the closed spatial motion, i.e.; d = kdk ; `d = hd;d i kdk (19) spectively. Therefore, with the aid of (1) the following results may be given. Result 1: There is the relationship a d = +kdk (0) 4

5 tween the spherical surface area bounded by the spherical image of d-c.t.s. and the total rotation of the motion. Result : The total geodesic curvature of the spherical image of d-c.t.s. is equal to the total rotation of the motion, i.e.; g d = kdk : (1) d On the other hand, from (19); we may write d`d = hd ; di D E = kd d k kd k ; d = kd k d is the angle of pitch of d -moment trajectory surface. Thus, the following result may be given. Result 3: There is the relationship d`d = kd k d () tween the invariants of d- and d -c.t.surfaces. From (1), (16) and (18), it follows that kv x k = p hd + ^xx; D + ^xi = p hd; Di + ^x hx; Di + ^x hx; Xi = p^d ^x: (3) Thus, with the aid of (17) and (3) the dual angle of pitch of v x - trajectory surface is obtained as ^vx = p^d ^x (4) there is the relationship ^v x = ^d ^x (5) tween the dual angle of pitches of d-steiner and an x-closed trajectory surfaces. From (5), we may give the following theorem. Theorem : Global invariants of the line surfaces generating by x; v x ; and d satisfy the following relations in E 3 ; v x = d x (6) and `vx = x`x p d`d. (7) d x f `vx = 0 then v x -closed trajectory surface is a cone. n this case, the relation (7) comes to x`x = d`d. (8) Thus, the following result may be given. Result 4: f v x -closed trajectory surface is a cone, then there are the relationships 5

6 `x `d = d x = a d a x = g d g x (9) tween the global invariants of d-steiner and x-closed trajectory surfaces and the spherical areas and the total geodesic curvatures of d and x-spherical images [6]. From (1) and (5), it follows that ( A vx ) = ( A d ) ( A x ) A v x +A x A d = 4(A v x +A x A d ) (30) A vx = a vx + "a x; A x = a x + "a x; A d = a d + "a d : And from (30) a v x +a x a d = 4(a v x +a x a d ) (31) a vx a v x +a x a x a d a d = (a v x +a x a d ) (3) obtained. Therefore, the following result may be given. Result 5: There are the relationships (30)-(3) between the measures of the spherical surface areas bounded by the spherical images of v x -, d-, and x-closed trajectory surfaces. n case of the axes of V x and D are perpendicular to each other, with the aid of (1), D (17) and E(18) V x kv ; D xk kdk = 0 () 1 kdk v x = 0 () vx = 0 () vx = 0; `vx = 0 () a vx = ; a v x = 0 () x = d; x`x = d`d obtained. Thus, the following result may be given. Result 6: n a closed spatial motion, the axes of V x -area vector and D-Steiner vector,d 6= 0; are perpendicular, vx = 0, v x -closed trajectory surface is a cone, i.e., vx = 0; `vx = 0,The spherical image of v x -c.t.s. divides measure of the spherical surface area into two equal parts. ()The relationships x = d; x`x = d`d are satis ed by the invariants of x- and d-c.t. surfaces. As it s known the angle of pitch of the closed trajectory surface generated by the second axis of the moving trihedron is zero, i.e. v = 0. This means that the spherical image of the second trajectory surface of the moving trihedron divides the measure of the spherical surface area into two equal parts. 6

7 n a spatial motion, the dual area vectors of v 1 ; v and v 3 -dual closed spherical images, drawn by the axes of the moving system are given with the aid of (16) as V v1 = D + v1 V 1 = D V v V v3 = D+ v3 V 3 (33) respectively. From (1), (18), (3) and (33) hv v1 ; Di = hd + v1 V 1 ; Di, kv v1 k V v1 kv v1 k ; D = hd; Di+ v1 hv 1 ; Di, kv v1 k vv1 = kdk v 1, vv1 = v 1 d p d v 1 v v1 = d v 1 (34) obtained. Seperating (34) into real and dual parts vv1 "`vv1 = (d "`d) ( v1 "`v1 ) v v1 " vv1 `vv1 = d " d`d v 1 + " v1`v1 v v1 = d v 1 ; vv1 `vv1 = d`d v1`v1 (35) gained. These are the relationships between the integral invariants of d-, v 1 -, and v v1 -trajectory surfaces. Similar statements on the invariants of v v - and v v3 - trajectory surfaces may be given. From (1) and (18) vv = d (36) vv = d ; `vv = `d (37) we nd that v v3 = d v 3. (38) From (38) vv3 "`vv3 = (d "`d) ( v3 "`v3 ) v v3 = d v 3 ; vv3 `vv3 = d`d v3`v3 (39) obtained. Thus, the following theorem can be stated. Theorem 3: There are the relationships (35)-(39) between the real and dual integral invariants of area vector-closed trajectory surfaces and the corresponding axestrajectory surfaces generated by the axes of the moving trihedron. 7

8 n a closed spatial motion, let X and Y be unit dual vectors xed in the moving system, then the dual area of projection of x-dual closed spherical image of x-closed trajectory surface, in direction y-generator of y-closed trajectory surface is de ned by F x;y = hv x ; Y i (40) is an analogy to the de nition, given in R 3 ; [4]. With the aid of (1), (16) and (40) we may write F x;y = hv x ; Y i = hd + x X; Y i = hd; Y i + x hx; Y i = y + x cos (41) = '+"' ; 0 ' ; ' R, is a constant dual angle between X and Y -unit dual vectors and F x;y = f x;y + "fx;y: Here, ' and ' are the real angle and the real distance between x and y-generators of x and y-c.t. surfaces, respectively. Considering F x;y = f x;y + "fx;y, we may give the following theorem. Theorem 4: The relationships f x;y = y + x cos ' (4) fx;y = `y x ' sin ' `x cos ' (43) satis ed between the area of projection of x-closed spherical image and the invariants of x and y-closed trajectory surfaces. Some special cases in (4) and (43) may be given as following; (i) ' = 0; ' = 0, x y, x = y ; `x = `y ) f x;y = 0; fx;y = 0 (44) (ii) ' = 0; ' 6= 0, x==y, x = y ; `x = `y ) f x;y = 0; fx;y = 0 (45) (iii) ' = ; ' = 0, x?y ) f x;y = y ; fx;y = `y (46) (iv) ' = ; ' 6= 0, x?y ) f x;y = y ; fx;y = `y x ' (47) (v) ' = ; ' = 0, x y, f x;y = y + y ( 1) = y + y = 0; fx;y = `y ` y( 1) = `y `y = 0 (48) (vi) ' = ; ' 6= 0, x==y; x = y, f x;y = 0; fx;y = 0 (49) Thus, from (44)-(49) we may give the following results; Result 7: The area of projection of an x-closed spherical image of x- closed trajectory surface in direction the same x or ( x)-generator is zero, i.e. f x;x = fx;x = 0 (50) and f x; x = fx; x = 0: (51) Result 8: f x and y intersect each other perpendicularly, then f x;y = y ; fx;y = `y (5) if x and y are perpendicular but they don t intersect each other, then f x;y = y ; f x;y = `y x '. (53) 8

9 Similarly, the dual area of projection of an x-dual closed spherical curve in direction of the dual unit Steiner vector of the motion is D E D F x;d = V x ; kdk = 1 kdk hv x; Di = 1 kdk hd + xx; Di = 1 kdk (hd; Di + x hx; Di) kdk from (18) = 1 kdk x F x;d = x d. (54) d Seperating the relation (54) into real and dual parts, the following theorem may be given. Theorem 5: n a closed spatial motion, the relationships, f x;d = x d d ; (55) fx;d = 1 d d`d x`x + x `d d (56) tween the area of projection of an x-closed spherical curve in direction d- generator of d-steiner trajectory surface and the invariants of x and d-c.t. surfaces are satis ed. From (54), the dual area of projections of v i -dual closed spherical images (indicatrices) in direction d-generator are d F vi;d = v i d ; i = 1; ; 3:: (57) From (57), the relationships d d f vi;d = v i d ; (58) fv = 1 i;d d d`d vi`vi + v i d `d ; i = 1; ; 3: (59) gained. Since v = 0, it follows, from (36) and (57) that F v ; d = vv = d. (60) Thus, the following result may be given. Result 9: n a closed spatial motion, the angle of pitches and the pitches of v v -, and d-trajectory surfaces can be stated in terms of the area of projections as vv = d = f v ; d (61) and `vv = `d = fv. (6) ;d From (1), (33) and (40), the area of projections of v i -dual spherical closed images in directions V j -generators are F vi;v j = hv vi ; V j i = hd + vi V i ; V j i = vj + vi ij ; i = 1; ; 3: d 9

10 ij is the Kronecker delta. Hence, we may write F vi;v j = f vj ; i 6= j (63) 0; i j So, the following result may be given. Result 10: The area of projections of v i -closed spherical images of v i -c.t. surfaces in direction v j -generators of v j -c.t. surfaces are given by f vi;v j = f vj ; i 6= j (64) 0; i = j fv i;v j = f `v j ; i 6= j 0; i = j. (65) From (1) and (63), the following dual quantities may be obtained F v;v 1 = F v3;v 1 = v1 = A v1 = G v1 (66) F v1;v = F v3;v = v = A v = G v (67) F v1;v 3 = F v;v 3 = v3 = A v3 = G v3. (68) Therefore, separating the formulas above the following relations between the invariants of closed trajectory surfaces can be given. t follows from (66), that f v;v 1 = f v3;v 1 = v1 = a v1 = g v1 (69) f v ;v 1 = f v3;v 1 = `v1 = a v 1 = g v 1 (70) from (68) f v1;v 3 = f v;v 3 = v3 = a v3 = f v 1;v 3 = f v ;v 3 = `v3 = a v 3 = g v3 (71) g v 3 (7) obtained. So, the following result may be given. Result 11: The relations (69)-(7) are satis ed by the invariants of corresponding closed trajectory surfaces. On the other hand, since v = 0; from (67) the relations F v1;v = F v3;v = 0, v = 0, A v =, G v = 0 (73) given. With the aid of (73) the following result on the invariants of trajectory surfaces may be given. Result 1: f v1;v = f v3;v = 0; f v 1;v = f v 3;v = 0 10

11 , v = 0; `v = 0, a v = ; a v = 0, g v = 0; gv = 0, v -c.t.s. is a cone and divides the measure of unit spherical surface area into two equal parts. REFERENCES [1] J.Hoschek, ntegralinyarianten von Regel achen. Arch. Math. VoL.XXV. pp.18-4,1973. [] A.T. Yang, Y. Kirson and B. Roth, On a Kinematics Theory For Rued Surface. Proceedings Of Fourth World Congress On The Theory Of Machines And Mechanisms. Newcastle Upon Tyne, England, Sep.1975, pp [3] H.H. Hac salio¼glu, On the pitch of a Closed Rued Surface. Mech. Mach. Theory Vol.7, pp , 197. [4] O. Gürsoy and A.Küçük, On the nyariants of Trajectory Surfaces. Mechanisms and Machine Theory, [5] O. Gürsoy, The Dual Angle of Pitch of a Closed Ruled Surface. Mech. Mach. Theory, Vol. 5, No., pp , [6] O. Gürsoy, On the ntegral nyariants of a Closed Ruled Surface. Journal of Geometry, Vo.39, pp.80-91, [7] HR. Müller, Erweiterung des Satzes yon Hoditch für Geschlossene. Raumkurven. Abhandl. Braunsch. Wiss. Ges , [8] G.R. Weldkamp, On the Use of Dua Numbers, vectors, and Matrices in nstantaneous, Spatial Kinematics. Mechanisms and Machine Theory, Vol. pp ,