Keywords: Kinematics, overconstrained linkages, Darboux motions, motions. Abstract: Given two Darboux motions 1 n 0 and 2 n 0 in Euclidean 3-

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1 Mathematica Pannonica 7/2 (996), 29 { 3 LINKED DARBOUX MOTIONS Otto Roschel Institute of Geometry, TU Graz, Kopernikusgasse 24, A-8 Graz, Austria Received: February 996 MSC 99: 53 A 7 Keywords: Kinematics, overconstrained linkages, Darboux motions, motions with many spherical point paths. Abstract: Given two Darboux motions n and 2 n in Euclidean 3- space, which are linked by common angle of rotation as time parameter. Then we areable to show, that the relative motion 2 n is line-symmetricinthe sense of J. Krames with a special ruled surface of degree 4 as basic surface. Further in 2 in the general case there exists at least a two-parametric family of real points, which are moved on spheres under 2 n. Examples nish the paper.. In the last years a great variety ofoverconstrained linkages has been studied, which are gained by linking systems with spherical 2R-links (see [4], [5], [7], [8], [9], [2], [2]). The most famous of these linkages seem to bethe socalled Heureka-polyhedron (see [8], [2] and the references given there). It has been shown, that most of these models consist of relative motions gained by combining an axial Darboux motion with aninverse of another (in most cases congruent) Darboux motion both parametrized with respect to their angle of rotation. In generalisation to these facts we here start with not necessarily axial Darboux motions.. In the 3-dimensional Euclidean space E 3 we use cartesian frames fo i x i y i z i g (i = 2) to describe points of given systems i (i = = 2) by their position vectors. There and 2 are moved by

2 292 O. Roschel Darboux motions n and 2 n with respect to the xed frame. Both one-parameter-motions shall be parametrized by the angle of rotation denoted by t. We will call such Darboux motions linked Darboux motions. The Darboux motions are cylindrical and therefore have xed directions. In ( 2 )the z -(z 2 -)axis shall denote this xed direction. Then a parametrization of such a motion may be given by ;! x (t ;! x i ):= a i c i A + ;a i + x i ;c i + y i A cos t+ e i () + z i ;e i + b i ; y i A sin t (t 2 [ 2]) d i + x i f i with real constants a i ::: f i (i = 2 cf. [, p. 36]). Rotation about the x -axis through angle ' i gives representations of n and 2 n in the form ;! x (t ;! a x i ):= i c i ; z i sin ' i A + ;a i + x i ;c i + y i cos ' i A cos t+ e i + z i cos ' i ;e i + y i sin ' i (2) + b i ; y i d i + x i cos ' i A sin t (t 2 [ 2]) f i + x i sin ' i (i = 2) with other real constants a i := a i, b i := b i, c i := c i cos ' i ; ; e i sin' i, d i := d i cos ' i ; f i sin' i, e i := c i sin' i + e i cos ' i, f i := d i sin' i +f i cos ' i.in the z -axis shall be a lineofsymmetry for the two xed directions (, ; sin', cos ') and (, sin ', cos '). Therefore we have ' = ;' = ' The relative motion 2 n. By taking the inverse motion of n and then combining itwith 2 n we get the following representation of the point paths of 2 n (3) ;!x (t ;! x 2 )= = + A cos t + C sin t cos ' ; E sin t sin ' ;A sin t + C cos t cos ' ; E cos t sin ' C sin ' + E cos ' B cos t + D sin t cos ' ; F sin t sin ' ;B sin t + D cos t cos ' ; F cos t sin ' D sin ' + F cos ' A ( ; cos t)+ A sin t+ A x 2 y 2 A : z 2 + cos2 t +sin 2 t cos 2' ; sin t cos t( ; cos 2') ; sin t sin 2' ; sin t cos t( ; cos 2') sin 2 t + cos 2 t cos 2' ; cos t sin 2' sin t sin 2' cos t sin 2' cos 2' There we have usedabbreviations A := a 2 ; a ::: F := f 2 ; f. By substituting u := tan t=2 we see,that this motion has a rational

3 Linked Darboux motions 293 parametrisation of degree 4. It is a special motion of those considered by the author in [3] there motions with this property were called \rational one-parameter motions of degree 4". Therefore we have Theorem. The relative motion 2 n of two linked Darboux motions n and 2 n (in general) is a special rational motion of degree 4. All point paths allow a rational parametrisation of degree 4 by a common parameter. Remark. Formula (3) shows that our relative motion 2 n does not change its representation, if we xthe constants A B ::: F and change our input data a i ::: f i according tothe xed values A = a 2 ; ;a ::: F = f 2 ;f.sowe are allowed to take avery special Darboux motion to bethe rst generatingmotion n.the second generating Darboux motion then has to be constructed as mentioned above. 3. The relative motion 2 n is line-symmetric in the sense of J. Krames. We now are able to prove the following Theorem 2. The relative motion 2 n of two linked Darboux motions n and 2 n (in general) is a special line-symmetric motion of degree 4 in the sense of J. Krames [5]{[]. The basic surface of these line-symmetric motions is generated as path of a straight line under the inverse motion n or n 2 of one of the generating Darboux motions. These relative motions already have been studied by R.Bricard[3] and J. Krames []. Proof. A) In [3] I proved that rational motions of degree 4 may have real points with plane paths. We look for these points now: We determine (4) _ ;!x (t ;! x 2 )= A sin t + ;A = ;C cos ' + E sin ' + z 2 sin 2' C cos ' ; E sin ' ; z 2 sin 2' ;A A cos t+ C sin ' + E cos ' ; y 2 sin 2' D sin ' + F cos ' + x 2 sin 2' + A + D cos ' ; F sin ' ; x 2( ; cos 2') ;B + C cos ' ; E sin ' + y 2 ( ; cos 2') A sin 2t+ + B ; C cos ' + E sin ' ; y 2( ; cos 2') A + D cos ' ; F sin ' ; x 2 ( ; cos 2') A cos 2t: For the special points

4 294 O. Roschel A + D cos ' ; F sin ' ; x 2 ( ; cos 2') = (5) g 2 ::: B ; C cos ' + E sin ' ; y 2 ( ; cos 2') = these derivation vectors are parallel to a plane andtherefore the corresponding points of 2 describe planepaths in.ifwehave sin2' 6= (the generating Darboux motions then have dierent xed directions our relativemotion is not cylindrical the case sin2' = will be treated seperatly in chapter 5) we have a straight line g 2 xed in 2 with plane point paths (these paths in general are ellipses see (4)). But inthe case sin2' 6= the points of the straight line (6) h 2 ::: ;D sin ' ; F cos ' = x 2 sin 2' C sin ' + E cos ' = y 2 sin 2' describe plane paths, too see (4). The planes containing the paths then are parallel. Summing up we at least have two straight lines xed in 2 with plane point-paths under the relative motion 2 n. B) The straight line g 2 may be moved into the z 2 -axis of 2.Our new origin may be moved onto this axis, too. We put (7) x 2 := A + D cos ' ; F sin ' ; x 2 ( ; cos 2') ( ; cos 2') y 2 := B ; C cos ' + F sin ' ; y 2 ( ; cos 2') ( ; cos 2') z 2 := C cos ' ; E sin ' ; z 2 sin 2') sin 2': For symmetry reasons we change the position of the origin of the rst system, too: (8) x := ; A ; D cos ' ; F sin ' ; x ( ; cos 2') ( ; cos 2') y := ; B + C cos ' + E sin ' ; y ( ; cos 2') ( ; cos 2') z := C cos ' + E sin ' ; z sin 2') sin 2': We substitute according to (7) and (8) and compute the vectors (9) = ;! d (t ;! x ;! x 2 ):= ;! x (t ;! x 2 ) ; ;! x (t ;! x )= A (z + z 2 )sin' ;(z 2 ; z )cos' A + cos t +sint ;G cos ' + y 2 ; y H ; (x 2 ; x ) cos ' A : ;(x + x 2 )sin' For further simplication we have put () H cos ' ; x 2 + x G ; (y 2 ; y ) cos ' A + ;(y + y 2 ) sin ' G := ; ; C + B cos ' sin 2 ' H := ; D + A cos ' sin 2 ': Using ;! d (t ;! x ;! x 2 )= ;! o we determine ;! x. This gives the new representation of the relative motion 2 n. Formula (9) shows, and

5 Linked Darboux motions 295 that 2 n may be gained by starting with the generating Darboux motions n and 2 n parametrized by ;! x (t ;! x ):= ;A=2 ;z sin ' ;z cos ' A + + :5G cos ' + y ;H=2 ; x cos ' A sin t x sin ' () and ;! x (t ;! x 2 ):= + A=2 z 2 sin ' ;z 2 cos ' A + ;:5G cos ' + y 2 H=2 ; x 2 cos ' ;x 2 sin ' ;:5H cos ' ; x ;G=2 ; y cos ' A cos t+ y sin ' (t 2 [ 2]) :5H cos ' ; x 2 G=2 ; y 2 cos ' A cos t+ ;y 2 sin ' A sin t (t 2 [ 2]) instead of (2). If wenow reect and 2 () with respect to the z -(z 2 -)axis into e, e 2, resp., the motion 2 n obtains the representation (2) e 2 =e :::g;! x (t g;! x 2 )= ;A=2 ;ez 2 sin ' ;ez 2 cos ' A + + :5G cos ' + ey 2 ;H=2 ; ex 2 cos ' A sin t: ex 2 sin ' ;ex 2 ; H=2 cos ' ;G=2 ; ey 2 cos ' A cos t+ ey 2 sin ' This is exactly the representation () of the generating Darboux motion n if we put x = ex, y = ey, z = ez and x = ex 2, y = ey 2, z = ez 2. Therefore we have: n may be gained by reecting the generating Darboux motion 2 n with respect to a straight line xed in.thus our relative motion is line-symmetric inthe sense of J. Krames [5]{[]. The basic surface of this motion is generated as path surface ; of the z -axis under the inverse motion n.as n is the inverse of a Darboux motion, this surface ; in the general case is algebraic of degree 4 it rst was used by J.Krames [] in order to dene special line-symmetric motions. For special cases (see our examples) the degree of ; may be lower.

6 296 O. Roschel 4. Spherical paths under therelative motion. In following weuse the representation () of our generating Darboux motions. We omit the \" inthe following. J. Krames [] has shown, that at least a twoparametric family of points of 2 under the relative motion 2 n is moved on spheres centered in. As our concept allows a(new and quick) determination of these points we will give ithere: Corresponding points ;! x and ;! x 2, xed in the moving frames and 2, resp., whichin hold xed distances during the two Darboux motions (we may say their distances are xed under the relative motion 2 n ), have the following property: The vectors ;! d (t ;! x ;! x 2 ):= = ;! a + ;! b cos t + ;! c sint (9) with (3) ;! a := A (z + z 2 )sin' ;(z 2 ; z ) cos ' A ;! b := ;! c := ;G cos ' + y 2 ; y H ; (x 2 ; x ) cos ' ;(x + x 2 ) sin ' H cos ' ; x 2 + x G ; (y 2 ; y ) cos ' A ;(y + y 2 ) sin ' have constant length, i the identity ;! d (t ;! x ;! x 2 ): t;! d (t ;! x ;! x 2 )= =holds for all t 2 [ 2]. The vectors ;! a, ;! b, ;! c only depend ofthe constants of the twomotions andonthe choice of themoving points ;! x, ;! x 2 resp. they do not depend onthe time parameter t. Therefore ;! d (t ;! x ;! x 2 ) t;! d (t ;! x ;! x 2 ) = is equivalent to =( ;! a: ;! c )cost +( ;! a: ;! b ) sin t +( ;! b: ;! c ) cos 2t; (4) ; :5( ;! c 2 ; ;! b 2 ) sin 2t for all t 2 [ 2]: As fcos t sint cos 2t sin2tg are linearly independent, the above equation yields 4 characteristic equations for the coordinates of corresponding points ;! x, ;! x 2 whose distance remains xed under the relative motion 2 n : ;! ;! ;! a: b = a: ;! c = (5) ;! b: ;! ;! 2 c = b = ;! c 2 : Computation of these equations yields (6) ;Ax2 ; y 2 z sin 2' + z 2 (G sin ' + y sin 2') =;A(H cos ' + x ) ; Gz sin ' ;x 2 z sin 2' + Ay 2 + z 2 (H sin ' + x sin 2') =A(G cos ' + x ) ; Hy sin ' x 2 y + y 2 x = ;HG=2 ;x 2 x + y 2 y =(H 2 ; G 2 )=4: The correspondence between \linked points" of and 2 in the rst two coordinates reads A

7 Linked Darboux motions 297 (7) x 2 = R2 (;x cos 2 + y sin 2) x 2 + y 2 = R2 (x sin 2 + y cos 2) y2 x 2 + y2 with abbreviations H := 2R cos, G := ;2R sin and certain reals R and. Thus we have: If it is possible to link points of and 2 by sticks with constant length, the ground projection of corresponding points of and 2 may be gained by aninversion with respect to a circle, followed by a displacement (see also the results of R. Bricard [3] and J. Krames []). The relation between the z-coordinates of corresponding points reads (8) z 2 = A(2R cos cos ' + x ; x 2 )+(;2R sin sin ' ; y 2 sin 2')z ;2R sin sin ' + y sin 2' where x 2 and y 2 have tobetaken from (7). In (6) there remains an additional equation for z,which gives one constraint tothe linked points in the rst system. After some algebra we get (9) (2y + G cos ') 2 +(2x + H cos ') 2 +(G 2 + H 2 ) sin 2 ' A(y G + x H +2(x 2 + y 2 ) cos ') ; z (Hy ; Gx ) sin 2') =: Therefore the points of, which may be linked with points of 2 (via (7), (8)) are situated on an algebraic surface (9) of degree 4: But our surface splits into two parts: A nonreal rotational cylinder and ahyperboloidofonesheet containingthe z -axis.asthe planes z = = const give circular intersections, this hyperboloid is called orthogonal (see J. Krames []). Change of indices (and signs) gives the corresponding points in 2 on another degenerating algebraic surface of order 4: Its equation reads (2) (2y2 ; G cos ') 2 +(2x 2 ; H cos ') 2 +(G 2 + H 2 ) sin 2 ' A(y 2 G + x 2 H ; 2(x y 2 2)cos') ; z 2 (Hy 2 ; Gx 2 ) sin 2') =: This surface splits into two parts, too and is congruent to (9). The hyperboloidofonesheet belongingtothe points of will be called 2. Similar results have been gained by J. Krames in []. We sum up in Theorem 3. Given two linked Darboux motions n and 2 n in a xed space, both parametrized with respect to the angle of rotation. Then in the general case the relative motion 2 n moves a two-parametric family of points of 2 on spheres centered in. The corresponding points are situated on algebraic surfaces of order 4 in both systems.

8 298 O. Roschel Remarks. ) It may happen that the two linked surfaces do not contain a twoparametric family of real points. Our result is an algebraic one and therefore makes use of the complex embedding of the real space. 2) The straight lines g 2, h 2 (5) and (6) are situated on the hyperboloid 2. The points of h 2 via (7), (8) correspond tothe point at innity onthe z -axis. The point-paths under 2 n therefore are rational quartics and situated in planes (like discussed earlier). These planes have equations z = const. An analogous result holds for the inverse motion. 3) The points on the circles of constant height z = const on via (7), (8) belongto points on hyperbolas on 2 they are situated in planes parallel to the z 2 -axis. Points on generators of meeting the z -axis via (7), (8) correspond to points on generators of 2 meeting the straight line g 2 = z 2 -axis. 5. The special case sin2' =. Here we have two possibilities: A) ' =: Then (3) shows, that the relative motion 2 n in this case is a translation with rational trajectories of degree 4. B) ' = =2: Then the relative motion 2 n has a xed direction and therefore is cylindrical. In (7) and (8) the transformation of the last coordinates shall be of the shape z = C=2 ; z, z 2 = C=2 ; z 2.Then all calculations proceed till formulas (9), (2) with ' = =2. In the general case the real part of surface 2 having spherical paths under 2 n splits into a plane (parallel to the xed direction) and the plane at innity. In the very special case C = D =wehave G = H = in (9) and (2) hence in this special case all points of 2 under 2 n move onspheres. This gives a rational type of the well-known Bricard-motion [3], which was seen to beline-symmetric byj. Krames in [6]. 6. Examples. We study examples generated by two (congruent) axial Darboux motions () with constants a = ;A=2, a 2 = A=2, b = b 2 = c = c 2 =,d = A=2, d 2 = ;A=2, e = e 2 := " and f = = f 2 = with real constants A, ". The axesofthe two motions shall be orthogonal: ' = =4. Following (2), (3) and () we get constants A B =, C = ;" p 2, D = ;:5A p 2, E = F =and G =2" p 2, H =. The transformations (7) and (8) give new coordinates (2) x 2 = A=2 ; x 2 y 2 = " ; y 2 z 2 = ;" ; z 2 x = ;A=2 ; x y = ;" ; y z = ;" ; z : Omitting the \" according to (9) the points of the algebraic surface

9 (22) Linked Darboux motions 299 (y + ") 2 + x 2 + " 2 A(2"y + x 2 + y 2 )+2"x z = are moved on spheres under n 2. The corresponding points in the system 2 belong to a surface congruent to (22). For corresponding points formula (7) now reads (23) x 2 = 2"2 x x 2 + y2 y 2 = ;2"2 y x 2 + : y2 As the constant A has no inuence on (23) the ground-projection of our relationship may be studied in the case A) A =, " 6= : The axes of the generating axial Darboux motions meet at a point. Then the real part (second factor in (22)) splits into two orthogonal planes. Points of the plane z = may be linked with points of the plane z 2 = via the inversion (23). From another starting point this situation was reached in the paper [4]. There two of the neighbour facets of a moveable model of a cube have been linked by4sticks of constant length. Fig. shows this situation it shows parts of the planes z =and z 2 =,the centers I 2 2, I and circles k 2 2, k of inversion, too. Here 6 connecting sticks have Figure. been drawn. B) A 6=," = : Here the generating Darboux-motions are pure rotations. Their axes do not meet. This is a very special situation of Bennett's isogram. The surface (22) splits into complex planes as J. Krames has shown in [9]. In our special case the basic surface ; of this line-symmetric motion is gained by rotating a straight line round a xed axis therefore ; is a rotational hyperboloid ofonesheet.

10 3 O. Roschel Figure 2. C) A 6=," 6= : This case gives a partial motion of themechanism treated in [6]. Fig. 2 shows the situation: One hyperboloid and some possible links are shown in two dierent views. There the sticks connect points of two circles of thehyperboloid with correspondingpoints on hyperbolas situated on the second hyperboloid. This second hyperboloid is not shown here in order to give a more instructive picture. References [] BOTTEMA, O., undroth, B.: Theoretical kinematics, North-Holland Series, Amsterdam, 979. [2] BOREL, E.: Memoire sur les deplacements a trajectoires spheriques, Mem. Acad. Sciences 33(2) (98), {28. [3] BRICARD, M.: Memoire sur les deplacements a trajectoires spheriques, Journ. de l' Ecole Polytechnique 2() (96), {93. [4] GIERING, O.: Vorlesung uber hohere Geometrie, Vieweg, Braunschweig{ Wiesbaden 982.

11 Linked Darboux motions 3 [5] KRAMES, J.: Uber Fupunktkurven von Regelachen und eine besondere Klasse von Raumbewegungen ( Uber symmetrische Schrotungen I), Monatsh. Math. 45 (937), 394{46. [6] KRAMES, J.: Zur Bricardschen Bewegung, deren samtliche Bahnkurven auf Kugeln liegen ( Uber symmetrische Schrotungen II), Monatsh. Math. 45 (937), 47{47. [7] KRAMES, J.: Zur aufrechten Ellipsenbewegung des Raumes (Uber symmetrische Schrotungen III), Monatsh. Math. 46 (937), 48{5. [8] KRAMES, J.: Zur kubischen Kreisbewegung des Raumes (Uber symmetrische Schrotungen IV), Sb.d. Osterr. Akad. d.wiss. 46 (937), 45{58. [9] KRAMES, J.: Zur Geometrie des Bennettschen Mechanismus ( Uber symmetrische Schrotungen V), Sb.d. Osterr. Akad. d.wiss. 46 (937), 59{73. [] KRAMES, J.: Die Borel{Bricard{Bewegung mit punktweise gekoppelten orthogonalen Hyperboloiden ( Uber symmetrische Schrotungen VI), Monatsh. Math. 46 (937), 72{95. [] KRAMES, J.: Uber eine konoidale Regelache funften Grades und die darauf gegrundete symmetrische Schrotung, Sb.d. Osterr. Akad. d.wiss. 9 (98), 22{23. [2] M ULLER, E. und KRAMES, J.: Konstruktive Behandlung der Regelachen (Vorlesungen ber Darstellende Geometrie III), Leipzig/Wien, 93. [3] R OSCHEL, O.: Rationale raumliche Zwanglaufe vierter Ordnung, Sb. d. Osterr. Akad. d.wiss. 94 (985), 85{22. [4] R OSCHEL, O.: Zwanglaug bewegliche Polyedermodelle I, Math. Pann. 6/ (995), 267{284. [5] R OSCHEL, O.: Zwanglaug bewegliche Polyedermodelle II, Studia Sci. Math. Hung. (to appear). [6] R OSCHEL, O.: A Remarkable Class of Overconstrained Linkages (To be published). [7] STACHEL, H.: Zwei bemerkenswerte bewegliche Strukturen, Journal of Geometry 43 (992), 4{2. [8] STACHEL, H.: The HEUREKA-Polyhedron, Proc. Conf. Intuitive Geometry, Szeged Coll. Math. Soc. J. Bolyai (99), 447{459. [9] VERHEYEN, H. F.: The complete set of Jitterbug transformers and the analysis of their motion, Computers Math. Appl. 7 (989), 23{25. [2] WOHLHART, K.: Dynamic of the \Turning Tower", Ber. d. IV. Ogolnopolska Konf. Maszyn Wlokienniczych i Dzwigowych (993), 325{332. [2] WOHLHART, K.: Heureka Octahedron And Brussels Folding Cube As Special Cases Of The Turning Tower, Syrom 93 II (993), 33{32.