RATIONAL MOTIONS OF
SPECIAL SPATIAL FOUR-BARS
RATIONAL MOTIONS OF SPECIAL SPATIAL FOUR-BARS
BIBLIOTHEEK TU Delft P 1888 2401
RATIONAL MOTIONS OF
SPECIAL SPATIAL FOUR-BARS
PROEFSCHRIFT
ter verkrijging van de graad van doctor in de technische wetenschappen aan de Technische Hogeschool Delft,
op gezag van de rector magnificus Ir. H. R. van Nauta Lemke, hoogleraar in de afdeling der elektrotechniek,
voor een commissie aangewezen door het college van dekanen te verdedigen op woensdag 9 mei 1973 te 16.00 uur
door
D E R K JAN SMEENK doctorandus wis- en natuurkunde
geboren te Gorssel / (S^ <S^ P 3. V O /
1973
KONINKLIJKE DRUKKERIJ VAN DE GARDE ZALTBOMMEL
r
Dit proefschrift is goedgekeurd door de promotor PROF. DR. o . BOTTEMA
Aan mijn vrouw Aan mijn kinderen
CONTENTS
CHAPTER I. INTRODUCTION I I
1.1 Indication of the subject of this thesis 11
1.2 Description of the mechanism 11
1.3 Some notations 12 1.4 A survey of recent literature 12
1.5 The subject of this thesis 14
CHAPTER II. THE FUNDAMENTAL RELATION 16
2.1 Derivation of the fundamental relation 16 2.2 Preliminary discussion of the fundamental relation and the
position curve 18 2.3 Further discussion of the fundamental relation and the
po-sition curve 19 2.4 The parameter representation of a rational motion 20
CHAPTER III. THE COUPLER CURVE. THE RULED SURFACE GENERATED BY
THE COUPLER 21
3.1 The parameter representation of the coupler curve 21
3.2 Some properties of the coupler curve 21 3.3 The double points of the coupler curve if /i = 0 23
3.4 The ruled surface generated by the coupler 23
3.5 Applications 24
CHAPTER IV. AN EXAMPLE OF A REDUCIBLE RATIONAL POSITION CURVE 2 6
4.1 A rational motion with c?i(/2 ?^ 0 26 4.2 A particular subcase: ^ 2 = 0 28 4.3 The kinematical meaning 28 4.4 The coupler curve and the ruled surface 31
CHAPTER V. AN EXAMPLE OF AN IRREDUCIBLE RATIONAL POSITION CURVE 3 2
5.2 The condition for a rational motion 32
5.3 The case P^ =0 32 5.4 The case P^ # 0, 9P2^ - leP^P^ = 0 33
5.5 The third double point 34 5.6 The case ^ s i n a —7?cosa = (i?^—/z^)* 35
5.7 The parameter representation of the position curve 36
5.8 The coupler curve and the ruled surface 37 5.9 The case dsinoi-Rcosa = -{R^-h^)^ 38 5.10 The case J sin a-I-7? cos a = (i?^-/!^)^ 40
5.11 The types 40 5.12 The case of intersecting rotation axes 40
5.13 The case A = 7? 41 5.14 The case of four equal links 41
5.15 Planar motions 41
CHAPTER VI. THE SIMPLY SKEWED FOUR-BAR 4 2
6.1 The fundamental relation of the simply skewed four-bar 42
6.2 The case PQ = 0 43 6.3 The case ^4 = 0 44 6.4 The case 9/^2 ^ --Po^4 = 0 44
6.5 The eight cases of single stretchability 44 6.6 First case of single stretchability 45
6.7 First subcase 46 6.8 The coupler curve and the ruled surface in the first subcase 47
6.9 Second subcase 48 6.10 Third subcase 49 6.11 Second case of single stretchability 49
6.12 Third case of single stretchability 50 6.13 The coupler curve and the ruled surface in the third case 51
6.14 Fourth case of single stretchability 52 6.15 The coupler curve and the ruled surface in the fourth case 52
6.16 Fifth case of single stretchability 52
6.17 First subcase 53 6.18 The coupler curve and the ruled surface in the first subcase 54
6.19 Second subcase 55 6.20 Third subcase 56 6.21 Sixth case of single stretchability 56
6.22 The coupler curve and the ruled surface in the sixth case 57
6.23 Seventh case of single stretchability 58 6.24 Eighth case of single stretchability 58 6.25 The coupler curve and the ruled surface in the eighth case 59
6.26 Double stretchability 60 6.27 First case of double stretchability 60
6.28 The coupler curve and the ruled surface in the first case 61
6.29 Second case of single stretchability 61
6.30 Threefold stretchability 62 6.31 The coupler curve and the ruled surface in the case of
three-fold stretchability 62 6.32 A non stretchable simply skewed four-bar with a rational
motion 62 6.33 The coupler curve and the ruled surface 64
6.34 The case a = 0 64
CHAPTER VII. THE SPATIAL FOUR-BAR WITH PARALLEL ROTATION AXES 6 5
7.1 The mechanism 65 7.2 The ruled surface generated by the coupler 66
7.3 The locus of the double points 69
7.4 The locus of the foci 69
CHAPTER VIII. PLANAR POSITIONS OF THE FOUR-BAR 7 0
8.1 The problem 70 8.2 Planar positions of the skew parallelogram 71
SAMENVATTING 7 2
LIST OF REFERENCES 7 5
CHAPTER I
I N T R O D U C T I O N
1.1 INDICATION OF THE SUBJECT OF THIS THESIS
In planar kinematics the four-bar is a well-known subject of study. During the last century many publications have come out dealing with its mobility, the properties of the coupler curve, etc.. R. KANAYAMA [9]* and j . T.
GROEN-MAN [6] have given a survey of the Uterature issued in this domain, the former till 1930, the latter since 1920.
During the last few decennia a number of authors has investigated the skewed four-bar. This mechanism is important in mechanical engineering and is inter-esting as well, seen from the mathematical point of view. In this thesis we will give a contribution to the knowledge of this three-dimensional four-bar.
1.2 DESCRIPTION OF THE MECHANISM
We consider the four-bar M,M2^2^i with the fixed link M1M2 = g, the cranks MiAi = Ri rotating around their fixed axes a, (/=1,2), and the coupler A1A2 = b with spherical joints at A^ and Aj. The four-bar has two spherical pairs and two revolute ones (at M^ and Mj) and therefore we speak of a (/?/?SS)-linkage.
We notice that the length of no link is longer than the sum of the other link lengths, otherwise the quadrilateral cannot be assembled.
Assume that aj and «2 are not parallel. Then the cranks move in two inter-secting planes P , and P2; their line of intersection is c.
In order to investigate if the mechanism is movable we apply Grüblers formula. It says that a rigid body has six degrees of freedom, and that the total number of freedoms must be reduced by the number of constraints. The number of constraints is five for a revolute pair and three for a spherical one. So our linkage has 3 x 6 — 2 x 5 — 2 x 3 = 2 degrees of freedom. But as we are not interested in the rotation of the coupler around its own axis it must be reduced by one, and we find that the four-bar can perform a motion with one degree^of freedom.
1.3 SOME NOTATIONS
The common perpendicular of a, and «2 is denoted by Z),Z)2; ö j is lying on fli, D2 on 02- The lengths of the line segments D1D2, M^Di and M2D2 are /;,
di and ^2 respectively. We assume g>0, Ri>0, R2>0,b>0,h^0,di^0, di ^ 0. Unless mentioned otherwise we suppose h > 0, (/, > 0, 6^2 > 0.
As the common perpendicular of «i and 02 'S parallel to c, DiD2 is in one of the four dihedral angles formed by Pj and Pi- We denote it by a = ip. ( O ^ a ^ r r ) . Fig. 1.
When di = d2 = 0 D1D2 coincides with c, Z), with M^, D2 with M2. More-over h= g. We speak of a simply skewed four-bar [8]. In that case we choose a non obtuse a.
1.4 A SURVEY OF RECENT LITERATURE
The mechanism described in section 1.2 is used in mechanical engineering to transmit angular motion between two skewed shafts. Important is the be-haviour of the output crank R2 if the input crank /?, is made to perform a complete revolution. R2 can make a complete revolution as well, or it can pass through one circular arc or through two separated ones. Which of these possibiUties occurs depends on the parameters mentioned in section 1.3. We will return to that problem later on. In planar kinematics we determine the type with the aid of the Grashof rule which indicates the conditions the para-meters b, g, Ri and /?2 must satisfy so that the four-bar is a double crank, a double rocker or a crank rocker. In the spatial case the type depends on b, g, Ri, R2, h, rf,, d2 and a. Here the Grashof rule cannot be applied [3].
During the last few decennia many authors have investigated the (RRSS)-linkage. We mention L. HARRISBERGER [7, 8]; M. SKREINER [16]; K. OGAWA, H. FUNABASHI and o. HAYAKAWA [11]; F. FREUDENSTEIN and i. F. KISS [4]; H.
NOLLE [10]; O. BOTTEMA [3].
HARRISBERGER [7] stated that a limit position of/?2 occurs when /?,, b and a, are coplanar. (HARRISBERGER gives no proof of this theorem. We observe that the condition mentioned is necessary but not sufficient.) The author applied this criterion to the simply skewed four-bar. If, in our terminology,
/_M2M^Ay = (pi and ZMiM2'^2 = 'Pi' HARRISBERGER derives an equation for the limit position angle (/)2 of cpi- He finds
Jcos^(/)2-l-Ncos(/)2 + X = 0, (1.4.1) where
H = b^-g^ + R,^-R2\ N = 4gHR2,
K = H^-4b^Ri^ + 4Ri^R2^ sin^a.
By interchanging Ri and R2, (pi and 4>2, an analogous equation is found for the limit position angle (pi of (p^. Equation (1.4.1) defines the limit positions of the output crank, indicating that there are at the most four limit positions defining two symmetrically located ranges of positions of the crank.
When the input crank is at a hmit position the output crank is at a "dead centre". As the input crank moves from its limit position the output crank can move in either direction from dead centre. That is the reason why dead centres should be avoided. They do not occur when the mechanism has a rotating input crank. In practice that is usually the case. Making use of the property that R2, b and ^2 are coplanar when R2 is at dead centre the author derives an expression for the dead centre value </)2,D of Vi- He obtains
tan</>2,D = ^1 cosasin(/>i(g—7?i cos</)])~\ (1-4.2) With the aid of the preceding formulas, and assuming that in a practical
situ-ation the mobility ranges of the cranks and a are prescribed it is possible to determine the combination of g, R^, R2 and b which will produce a mecha-nism satisfying the design requirement. The following graphical method leads to success: let g = 1, and give b and a certain values. Draw for the input crank a graph of the limit positions versus R1/R2, and for the output crank a graph of the hmit positions and of the dead centre positions versus R1/R2. Any horizontal hne on these "Mobihty Charts" defines a value of R1IR2 and represents a particular mechanism configuration. Seen from the mathematical point of view the problem even of the simply skewed four-bar is rather complicated. We have discussed HARRISBERGER'S paper in some detail as it is an example of practical approach that will do in engineering, OGAWA,
FUNA-BASHI and HAYAKAWA solve the problem in an analogous way [U].
SKREINER [16] has developed a method to identify the mobihty regions of a general spatial four-bar which in fact is based on the problem of the extreme distances of two circles in space. An exact discussion produces an equation of the eighth order and seems unsuitable for practical application. The author has designed mobihty charts according to a method founded on an iterative solution of that equation.
FREUDENSTEIN and KISS [4] approach the problem in a similar way. Wrongly they conclude that for a simply skewed four-bar the determination of type is identical to that of a planar four-bar with the same hnk lengths.
After these authors NOLLE [10] has reported about the general skewed four-bar. He does not use the Harrisberger criterion for the limit positions. In our
notation he finds the relation
sin(Pi = F2{(p2)cos (pi+F,{(p2). (1.4.3) When the input link Ri reaches a limit position the relation (1.4.3), regarded
as an equation in (pi, has two coinciding roots, and this happens only if ^+F2'{<P2)~F,'i(P2) = 0,
or, if cos (p2 = X,
AX* + BX^ + CX^ + DX + EI=0. (1.4.4)
This equation is similar to (1.4.1) but it is more comphcated as Ö?IÖ?2 ¥" 0. It is solved by Cardano's method, and the results are applied to mobihty charts with the aid of a digital computer. The planar, the spherical and the three-dimensional linkage with a^ jj «2 are considered as special cases of the general spatial Unkage.
The authors cited before have studied the four-bar using more or less different methods with the same purpose: determining the limits of the ranges of its motion without direct interest in the motion itself of the coupler.
o. BOTTEMA [3] has given a complete discussion of the simply skewed four-bar in terms of g, i?i, R2, b and a, mapping the positions of the coupler on the points of a plane quartic with two double points. He found that for the mobility range of some types of the simply skewed four-bar the value of a is essential, contrary to a statement of FREUDENSTEIN and KISS [4]. After PRIM-ROSE and FREUDENSTEIN [13] BOTTEMA investigates the coupler curve of an arbitrary point of the coupler. It proves to be a space curve of the eighth order with four double points at infinity, coinciding with the isotropic points of Py and P2.
The same author fixes the attention on two subjects which are of great im-portance for this thesis. First the complexity of the algebra when his method is applied to the general four-bar, confirming with this the statement of NOLLE [10].
Secondly BOTTEMA remarks that there exist simply skewed four-bars which can have a position in which all sides are in hne with M1M2.
In that case the quartic on which the positions of the coupler are mapped has a third double point, and it is therefore rational. Then we shall speak of a rational motion [2].
1.5 THE SUBJECT OF THIS THESIS
In this thesis we continue some aspects of the Bottema paper. We investigate whether the general skewed four-bar can produce a rational motion. This
appears to be possible. Because of the complexity (already mentioned) of the algebra we restrict ourselves, however, to two particular cases in chapter IV and Chapter V. In chapter VI a complete survey is given of the rational motions of the simply skewed four-bar. It is proved, e.g., that for any non stretchable simply skewed four-bar two values of a can be found so that the motion is rational. In the cases dealt with attention is given to the coupler curve and to the ruled surface, generated by the coupler during its motion.
CHAPTER II
THE FUNDAMENTAL RELATION
2.1 DERIVATION OF THE FUNDAMENTAL RELATION
Choose the origin O of a Cartesian frame in the midpoint of D^D2, the 7-axis along DyD2, the XOZ-plane perpendicular to D^Di, ihtXOY- and the FOZ-plane parallel to the bisecting planes of P^ and P j . For geometrical rea-sons it is clear that the K-axis is parallel to the hne of intersection c of P , and P2. The angle between M-^A^ and the K-axis is denoted by (p^. Then the coordi-nates of M l and M2 are respectively
( - ^1 sin j8, \h, di cos P), ( - ^2 sin P, - i/z, - fi?2 cos P), (2.1.1) hence
M i M 2 ^ = g 2 = h'- + d^^ + d2^ + 2did2Co&a.. (2.1.2) Those of ^ 1 and A 2 are
Xi = i?i sin^Ji cosjS—(^1 s i n ^ , ^2 = ^ 2 sin<?52 cos)S—<^2 sinj?,
ƒ , = ^ / i - i ? i coscpi, ^2 = -i/j-l--R2 cos ()52, (2.1.3) Zi = i?i sin(pi sinjS-l-t^i cosj3, Z2 = — i?2 sin (^2 sinjS — (^2 c o s ^ ,
and
b^ = ^ i / l 2 ^ = /?i^ + / ? 2 ^ + ^ ^ - 2 / ! ( / ? i C0S(Pi-t-i?2C0S(?)2)
-|-+ 2i?i/?2(cos^i COS (^2 —COS a sin ^ 1 sin<p2) -|-+
+ lüna{d2R\ sincpj-l-J,^2 sin(P2)- (2.1.4)
See fig. 1; in fig. 2 the four-bar is projected o n the XOZ-plane. We define
A = Ri^ + R2^+g^-b^ + 2hRi+2hR2 + 2RiR2, B = 4 Ê ? I ^ 2 sin a,
C = 4^/2^1 sin a,
D= Ri^ + R2^+g^-b^ + 2hRi-2hR2~2R^R2, (2.1.5)
£ = R{^ + R2^ +g^ -b^ -2hR^+2hR2-2RiR2,
F = - 8 . R i i ? 2 C o s a ,
Fig.1
Introducing tan^(pi = u^ we have sin (p, = 2Mj(I-l-Mi^)"\
(2.1.6) C0S(pi = (l-Ui')il+U,Y'.
We exclude the isotropic positions for which (H-MI^)(1-I-M2^) = 0. Then (2.1.4) becomes
Aui^U2^ + Bui^U2 + CUiU2^ + Dui^ + Eu2'^ + FuiU2 + Cui + Bu2 + G = 0. (2.1.7) This is the relation between the positions of the rotating Hnks. Therefore it informs us about the possible positions of the coupler. We call it the funda-mental relation and denote it from now on by FR. It is of the fourth degree and quadratic in both MJ and «2, as could be expected. For the sphere (Ai,b) has two intersections, real or imaginary, with the circle (M2, R2) in P2. So to any position of ^1 no more than two positions of A2 are conjugate, and conversely.
In the (MI, «2) plane the FR represents a quartic curve. We call it the position curve and denote it by PC.
2.2 PRELIMINARY DISCUSSION OF THE FR AND THE PC
We pass on to homogeneous coordinates ( « / , «2', M3') with M; = u/ju^' (i = 1,2). Now the points with u^' = 0 are joint to the PC. The FR becomes
AUi'^U2'^+BUi'^U2'U3' + CMI'M2'^W3' +^«l'^«3'^ +£W2'^«3'^ +
+ FUi'U2'U3'^ + CUi'U3'^+BU2'U3'^ + GU3'^ = 0. (2.2.1) As no confusion is possible we omit from now on the accents.
Obviously the PC passes through 5i(l,0,0) and ^2(0,1,0). Suppose ^ # 0. We intersect the PC with an arbitrary hne Xu2 = nu^ through By, and find two coinciding points of intersection at By because terms w / and Uy^ are missing. Therefore By is a double point of any PC and so is 82- In general the points By and B2 do not correspond to linkage positions. For By corresponds to q)y = n, (p2 = 0. Substituting this in (2.1.4) we find Ry^ + R2^+g^-b^ +
+ 2hRy — 2hR2 — 2RyR2 = 0, and in general this relation is not satisfied. We notice that it can be read as D = 0. In the same way we conclude that B2 corresponds to a coupler position only if £ = 0.
The points of the PC outside By and B2 correspond to linkage positions. However we must distinguish the analytical and the kinematical motion.
Suppose the range of A y consists of two separated arcs. Seen from the ana-lytical point of view ^1 can pass through both arcs. Seen from the kinematical point of view /4i cannot leave the arc on which it is at a certain moment unless we disconnect the linkage.
2.3 FURTHER DISCUSSION OF THE FR AND THE PC
Provided that ^ # 0 the line u^ = 0 has no points in common with the PC outside By and B2. We investigate if there are more double points, either coinciding with 5 , or B2, or not coinciding with any of these double points. Therefore we write (2.1.7) as
Uy^(Au2^ + Bu2 + D) + Uy(Cu2^ + Fu2 + C) + Eu2^ + Bu2 + G = 0. (2.3.1) Considering this as a quadratic equation with the unknown Uy we find for its discriminant
A = {Cu2^ +Fu2 + C)^ -4iAu2^ +Bu2+D)(Eu2^ + Bu2 + G), or
A = P^U2'^ + 4P3U2^ + 6P2U2^ + 4PyU2 + Po, (2.3.2) with P4 = C^-4AE, 4P3 = - 4 ^ i S - 4 5 £ + 2C£, 6P2 = -4AG-4B^ + 2C^-4DE + F^, (2.3.3) 4Pi = -4BD-4BG + 2CF, Po = C^-4DG.
The double points present themselves if the equation A =0 has two coin-ciding roots. The necessary and suflicient condition for this is that the dis-criminant i? of ^ vanishes.
i = 2(PoP4-4PiP3 + 3P2^), (2.3.4) J = 6 ( P o P 2 P 4 - P l ' ^ 4 - P o ^ 3 ' - i ' 2 ' + 2 P i P 2 P 3 ) .
The equation i? = 0 is of the sixth order in P; and of the twelfth order in A, B, ..., G. Its discussion is a difficult problem. In the following chapters we limit ourselves to special cases.
Resuming we state that the PC has double points at By and at 82- In general it has no other double points, and then it has genus one. When the PC has a third double point (that may coincide with ^1 or with B2) it is rational. The
third double point can be found by applying to the PC the characteristic mentioned in this section. This method will be followed in this thesis.
2.4 THE PARAMETER REPRESENTATION OF A RATIONAL MOTION In section 1.5 we announced the aim of this thesis: the tracing of rational motions. In section 2.3 we explained how we intend to do that. Now we shall sketch how to find the parameter representation.
An irreducible curve of order n has at most ^(n—!)(«-2) double points [12]. So the PC, if irreducible, has no more than three double points, and if it has this maximum number it is rational. If it has more than three double points it is rational as well, though degenerate.
The coordinates Uy, «2 and u^ of a rational PC can be expressed as rational functions of a parameter t. For this purpose we intersect the rational PC with a pencil of conic sections, taking care that any conic section of the pencil has seven fixed points in common with the PC. To make sure of that we let the conic sections pass through the three double points of the PC and an arbitra-ry fixed fourth point. The latter may coincide with a double point. This case arises if the PC and any conic section of the pencil have a tangent in common at one of the double points. As a quartic and a conic section have eight points of intersection, and as seven of them coincide with the three double points and the fixed fourth point respectively the coordinates of the eighth point are found as rational functions of the parameter of the pencil.
CHAPTER HI
THE COUPLER CURVE. THE RULED SURFACE,
GENERATED BY THE COUPLER
3.1 THE PARAMETER REPRESENTATION OF THE COUPLER CURVE Any point E(x, y, z) on the coupler AyA2 is given by
(>ll"r/2J'^ ^^^ ^iXy'T- A2X2,
{l,-\-l2)y = Xyyy+X2y2, (3.1.1) iXy+X2)Z= XyZy+X2Z2,
wherein (x,-, _Vi, Z;) are the coordinates of ^4,- (/' = 1,2).
Passing on to homogeneous coordinates and making use of (2.1.3) and (2.1.6) we find
x = 2cos P[XyRyUy{\ +U2^)->r X2R2Uz(\ + Ui^)] + -%mp{Xydy+X2d2){\ + Uy^){\+U2\ y = -XyRy{\-Uy^){\+U2') + X2R2{\+Uy\\-U2^) + + \h{Xy-X2){\+Uy^){\+U2\ (3.1.2) Z = 2 sin P[XyRyUy{\ + M2^) - ^2^2"2(1 + " l ^)] + + C0SP{Xydy-X2d2){\+Uy^){\+U2\ W = {Xy+X2){l+Uy'-){\+U2^).
These equations give a parameter representation of the path F of E, provided that Uy and «2 are related by (2.1.7).
3.2 SOME PROPERTIES OF THE COUPLER CURVE
If we want to determine the order of F we must find the number of its inter-sections with an arbitrary plane QiX+Q2y+Q^z+Q^w = 0. Substitution of (3.1.2) produces another relation of the shape
LyUy^Ui^ +L2Uy^U2+L2,UyU2^ +L^Uy^ +
+LsU2^+L(,Uy+LjU2+Ls = 0. (3.2.1)
It has a double point at fi, and at B2. Uy and «2 are related by the FR, representing the PC. The latter has double points at By and at B2 as well. C4 and the PC have 16 points in common. Four of them coincide with By, four with B2. Generally these points do not correspond to linkage positions. They must be cancelled and we conclude that F is of the order 1 6 - 8 = 8.
We put the question whether there is a point which every coupler curve passes through. Then the coordinates of this point must be independent of the ratio of Xy and X2. We state that this condition is fulfilled if (1-I-Mi^)(l-I-W2^) = 0, provided that AiA2#0. First we investigate the case 1 -fMi^ = 1 +«2^ = 0. Substituting this in the FR we find a = 0. We exclude the planar four-bar and we assume 1 +Wi^ = 0, I +«2^ 5^ 0- If "i = ', we find the point ly (icosp, — 1,/sin^, 0). This is an isotropic point of Py. Its coordinates are independent of «2 as well, apart from the condition that Uy and «2 must satisfy the FR. To any value of Uy two values of «2 are conjugate. So the coupler can take up two positions in which the coupler curve of any of its points passes through ly. Therefore ly is a double point of any coupler curve. We conclude that the isotropic points of Py and P2 are double points of any coupler curve.
The reasoning is not vaHd if A1A2 = 0. If, e.g., X2 = 0, the coupler curve is the path of .4i, and this is the circle (M,, /?,) in Pi, counted twice. Substituting into (3.1.2) A2 = 0 and I-I-M,^ = 0 we find the isotropic points of Pi. Substi-tuting Aj = 0 and 1 + «2^ = 0 we obtain x = y==z = w = Q, which can be in-terpreted as "any point of the space". As for any A1A2 # 0 any coupler curve has double points at the isotropic points of Pi and P2 we can understand, for reasons of continuity, that the path of ^1 passes through the isotropic points of P2 and that any point of the coupler generates a curve of the eighth order.
Now we will prove the theorem: if the coupler cannot take up a double position and if /; # 0 the coupler curve has no real double points.
Assume the path of a certain point Dofy4i/42(/)#v4i,Z>#/42) has a double point. Then there are two positions ^1^2 and Ay'A2' of the coupler so that DAy =DAy' and DA2 = DA2'. The triangles DAyAy' and DA2A2' are co-planar and both isosceles. Hence ^ 1 ^ 1 ' // ^2^2'- ^ i ^ i ' 'i^s in Py, ^2^2' in P2. So both are parallel to c, the intersection of Pi and P2. The plane through D perpendicular to c passes through the midpoints of AyAy' and A2A2' and therefore also through My and M2. This conclusion is incompatible with the assumption A # 0.
3.3 THE DOUBLE POINTS OF THE COUPLER IF h = 0
When /; = 0 the points My and M2 are both lying in the XOZ-plane and the configuration assumed in the preceding section is possible now. Suppose that in a position of the coupler, indicated by certain values of (py and <P2, F> coincides with a point of the XOZ-plane. Then it will coincide with the same point when we replace (py byn — cpy, and (p2 hyn — (p2. So the path of Z) does have a double point.
This statement can be verified in the following way. In view of (2.1.3) we find for the equations of AyA2:
(x-Xy){X2-Xiy^ = (y-yy)(y2-yi)~^ = (z-Zi)(Z2-Zi)"^
Intersection with the plane y = 0 gives the coordinates (x, y, z) of D. We find x{Ry cos(pi-l-7?2Cos{p2) = Pi cos(pi(P2 ^^^92 cosj?—fi?2 s,mP) +
+ R2Cos(p2{Ri sincpy cosp—dy sinP), y = o,
z(Ry COS (py + R2 COS cp2) = —Ry COS (py{R2 sin(p2 sinp + d2 cosP) + + R2Coscp2(Ry sintpi cosp + dy cosP).
Indeed, when we replace cpi by n — (Pi we find the same coordinates. Making use of (2.1.6) and passing on to homogeneous coordinates we have
X = 2RyR2COS P[Uy(l -U2^) + U2il -Uy^)] +
-sinP[R2dy(l+Uy^)(l-U2^) + Rid2il-Uy^)(l+U2^)], y = 0, (3.3.1) Z = 2Pii?2Sini?[Mi(l-M2 ) - " 2 ( l - W i )] + + COSp[R2dy(l+Uy^)(l-U2^)-Ryd2{l-Uy^)il+U2^)], W=Ryil-Uy'){l+U2') + R2il+Uy')il-U2').
To investigate the order of the curve represented by (3.3.1) we apply the method used in section 3.2 and we conclude then that it is ofthe eighth order. But as it is the locus of the double points we must have a quartic, counted twice.
Anticipating to section 3.4 we state that it is the intersection with the XOZ-plane of the ruled surface generated by the coupler.
3.4 THE RULED SURFACE GENERATED BY THE COUPLER
We suppose the four-bar has a fixed position. Then the coordinates of ^ j are fixed and the equations ofthe line A1A2 are
The expressions for X;, j , and z, are given by (2.1.3). We denote ix-Xy)(X2-Xyy^ = (>'->'i)(>'2->'l)"^ = (z-Z,)(Z2-Zi)-l = X.
(3.4.2) If (pi and (p2 vary and X is fixed, (3.4.2) gives the equations of a coupler curve. For if we substitute X = X2{Xy+X2)'^ we find the equations (3.1.1). From (3.1.1) we derived (3.1.2) wherein the homogeneous parameters Xy and X2 are supposed to have a fixed ratio. If this ratio varies (3.1.2) represents the ruled surface, generated by the coupler, and described as the locus of coupler curves. We denote it from now on by RS.
In order to find the degree of the RS we investigate its intersection with the plane at infinity w = 0. As for the method we refer to section 3.2. Weconclude that it is of the eighth order.
To affirm this we inquire into the intersection of the RS and P i . We suppose the circle (M2, R2) in P2 intersects Pi at two points F and G on c. Moreover we assume that the circles (F, b) and (G, b) in Pi intersect circle (Mi, Ry) at H and K and at L and iV respectively. Then during its motion the coupler can coincide with each of the lines FH, FK, GL and GN. As we noticed before the circle (M,, Ry) is part of the intersection. It must be counted twice. We conclude that the intersection ofthe RS and Pi is a (degenerate) curve of the eighth order.
3.5 APPLICATIONS
In this chapter we noticed that the equations of the coupler curve and those ofthe RS are united in (3.1.2). In the chapters IV, V and VI we will derive the equations ofthe PC, the coupler curve and the RS for all motions dealt with. Each time the parameter representation of the PC will be derived in homo-geneous coordinates Uy, «2 and M3. The expressions for M; are substituted in (3.1.2) and then we have a parameter representation of the rational coupler curve and the rational RS. Therefore we rewrite (3.1.2) in homogeneous coordinates.
X = 2cOSi?[AiPiMiM3(M2^-|-W3^)-f-A2-'?2"2"3(Wl^ + "3^)] + - sin P(Xydy + X2d2){uy ^ + M3^)(«2^ + "3^).
;; = 1 I P , ( « I ^ - W 3 ^ ) ( M 2 ' + « 3 ' ) - A 2 « 2 ( W I ' + W 3 ' ) ( W 2 ' - « 3 ' ) +
+ ^h{Xy-X2){Uy^ + Ui'){U2^ + U,\ (3.5.1) Z = 2sin^[AiPiMiM3(M2^ + "3^) —'^2^2«2"3("l^ + «3^)] +
+ COSp{Xydy-X2d2){Uy^+Ui'){U2^ + Ui'), W = ( A I + A 2 ) ( M , ' +
M3')(W2'+W3')-In particular we shall give attention to the double points ofthe PC. If A: double points of the PC correspond to linkage positions they indicate double po-sitions. Then the coupler is a double generator ofthe RS, and any point ofthe coupler is at a double point of its path. The locus of the double points of the coupler curves consists of k generators. Herewith we do not take into con-sideration the four isotropic double points.
CHAPTER IV
AN EXAMPLE OF A REDUCIBLE RATIONAL POSITION CURVE
4.1 A RATIONAL MOTION WITH didz it 0
If a # ITT the line segment M iDi or its extension intersects P2 at P. We suppose P to lie on the extension of MiD, (fig. 3). If P is lying on the segment MyDy an analogous reasoning is vahd.
In the assumed situation PMi =dy+d2lcos a. and PM2' = 6?2 tan a. Sup-pose that P is lying on circle (M2, ^2)- Then PM2 = R2 and the parameters of the four-bar satisfy two conditions
(M2M2')^ + (^M2')' = ^ 2 ^ 2 ' , or h^ + d2^tsin^d= Ri^ (4.1.1) and
PMy^ + AyMy^ = AyP\ OT {dy+d2JC0Sa.f + Ry^ = b\ (4.1.2) The coupler AyA2 can rotate around PMi, generating a quadratic cone,
while (p2 is constant. The PC is degenerated into a straight line («2 = con-stant) and a cubic.
As the conditions (4.1.1) and (4.1.2) are difficult to introduce into the FR we approach the problem in the following manner.
In the "cone position" of the coupler we have
cosipj = hR2~\ or «2 = (R2-h)i(R2+h)~'' = p. (4.1.3) From (2.3.1) and (4.1.3) it follows
Uy^(Ap^ + Bp + D) + Ui(Cp^ + Fp+C) + Ep^ + Bp + G = 0. (4.1.4) As (4.1.4) must be true for any Uy we find
Ap^ + Bp + D = 0, (4.1.5) Cp^ + Fp + C =0, (4.1.6) Ep^ + Bp + G = 0. (4.1.7) By direct computation we verify that Ap^ + Dis equivalent to Ep^ + G. So we
can restrict ourselves to the conditions (4.1.5) and (4.1.6). We shall prove that they are equivalent to (4.1.1) and (4.1.2). For from (2.1.5) and (4.1.6) it follows
4d2RiSmoi[(R2-h){R2+h)- ^ + l]-8RiR2(R2-h)HR2 + h)~^ cos(x -= 0, or
d2^tiin^a = R2^-h\ which is the same as (4.1.1).
In the same way we prove that (4.1.2) and (4.1.5) are equivalent.
We return to the FR. From (4.1.5), (4.1.6) and (4.1.7) we derive, supposing that i ? 2 ^ ^ , sopy^O:
B= -Ap^-Bp, F = -C(p' + l)p-\ G = -Ep^-Bp. The Pi? can be written as
Auy^U2^ + Buy^U2 + CM,«2^ - iAp^ + Bp)Uy^ + Eu2^ +
-C{p^ + \)p-^UiU2 + Cuy+Bu2-Ep^-Bp = 0. (4.1.8) We know that (4.1.8) is reducible. One factor on the left hand side is U2—p.
We find for the other
Auy^U2 + (Ap + B)ui^ + CuyU2-Cp'^Uy+Eu2 + B+Ep = 0. (4.1.9) As terms with «2^ are missing this represents a rational cubic with a double point at B2.
Its parameter representation is Uy = t(At^ + Ct + E),
U2= -iAp + B)t^ + Cp-^t-B-Ep, (4.1.10) «3 = At^ + Ct + E.
The values of t for the double point are the roots of At^ + Ct + E = 0.
The double point is real with two diff"erent tangents if C^ — 4AE > 0. It is a cusp \fC^-4AE = 0, and isolated if C^ - 4AE < 0.
The PC, regarded as a degenerate quartic, has three other double points, namely the intersections ofthe cubic and the line «2 =P- One of them is By with parameter value t = oo, the others are found from (4.1.10) and «2 =P> or
(2Ap + B)t^ + {C-Cp-^)t + 2Ep + B = 0. (4.1.11) To make the situation clearer we pass on to a subcase.
4.2 A PARTICULAR SUBCASE: c/^ = 0
If i/2 = 0 we find C = 0. From (4.1.1) it follows h = ^2^ Hence p = 0. From (4.1.5) and (4.1.7) we obtain D = G = 0.
The FR is
U2{Aui^U2 + Buy^ + Eu2 + Fui+ B) = 0, (4.2.1) mthA = Ri+R2, B = dy sin a, E = R2-Ri, F = -2/?iC0sa.
We investigate the rational cubic
Auy^U2 + Buy^ + Eu2 + Fuy+B = 0. (4.2.2) Its parameter representation is
«1 = t{At^ + E),
U2= -(Bt^ + Ft+B), (4.2.3) «3 = At^ + E.
The values of t for its double point B2 are the roots of At^ + E = 0, or t^ = iRy-R2)iRy+R2y^.
We see that there are three possibilities according as Ry > R2, Ry = ^2» or P l < P 2
-4.3 THE KINEMATICAL MEANING
In section (4.1) we found that the coupler can move in two ways: it generates a quadratic cone while M2A2 is at rest («2 = P), or M2^2 is moving, and then
the motion of the coupler can be mapped on a cubic, which, in our case, is represented by (4.2.2). The transition of one kind of motion into the other can occur in the positions corresponding to the points of intersection of the hne « 2 = 0 and the cubic. One intersection, independent ofthe parameters of the four-bar, is By, indicating the position <pi =n, (p2 = 0. We find the other transition positions from Buy^ + Fuy+B = 0.
They are real if F^-4B^ ^ 0, or Pi ^ dy tan a. For kinematical reasons there are nine possibilities:
I. P i > P 2 . ^1 >di tana.
The range of A, is a complete circle, that of A2 is a circular arc. When A2 coincides with P the coupler can change its kind of motion in the positions corresponding to the points Q, R and By. Fig. 4.
IL P i > P 2 . ^1 =di tana. Q and R coincide.
III. P i > P2. ^1 < dy tana. Q and R are imaginary.
The motion can be changed only in the position (py =n, (p2=0. IV. P i =P2> ^1 > 0^1 tan a.
V, Pi = P2, P = dy tana. Q and R coincide.
VI. Pi = P2, Pi < dy tana. Q and R are imaginary. VII. P i < R2, Ry > dy tana. Fig. 5.
VIII. Pi < P2, Pi = dy tana. Q and P coincide.
IX. Pi < P2, Ry < dy tana. Q and P are imaginary.
The discussion of IV, V and VI is similar to that of I, II and III, and so is that of VII, VIII and IX.
We notice that the condition Pi = dy tan a has a geometrical meaning: when A2 coincides with P, and (py = ^n it is easy to verify that the coupler is perpendicular to P2.
As for the type: in all cases we have a crank rocker.
4.4 THE COUPLER CURVE AND THE RS
We investigate the coupler curve and the RS corresponding to the cubic (4.2.3).
We denote / i =At^ + E,
f2=-{Bt^ + Ft + B).
Then the parameter representation of the cubic is "i =fit,
« 2 = ƒ2.
« 3 = / l
-The equations of the coupler curve and the RS are X = 2C0SP[XyRyt(fy'+f2') + X2R2fj2{t'+l)] + -SmP.Xydyify'+f2'){t'+l), y = hRiifi'+f2')it'-n+^2R2if,'-f2')it'+\)+ + mXy-X2)if^'+f2')it'+l), z = 2smP[XyRyt(fy'+f2')-X2R2fj2(t' + m + + COSP.Xydy(Jy^+f2^)it'+l), W=iXy+X2)(fy^+f2')(t'+l).
We see that the coupler curve and the RS are of the sixth order, as could be expected. For when M2A2 is at rest the coupler generates a quadratic cone K which is part of the degenerate RS ofthe eighth order. The other part must be a surface S of the sixth order. And any coupler curve is degenerated into a circle (in a plane perpendicular to M 1^2. centre on MyAz) and a sextic. K and S have one or three real generators in common. Two of them can coincide. They are indicated by the transition positions of the coupler. It is evident that the circle and the sextic have one or three real points in common. We refer to section 4.3.
In the case (py = n, (P2 = 0 the common generator of K and S has the equations
CHAPTER V
AN EXAMPLE OF AN I R R E D U C I B L E RATIONAL POSITION C U R V E
5.1 THE SPECIAL FOUR-BAR
We consider a four-bar with Ri=R2 = R, b=g, dy = d^O, ^2 = 0. 0 < a g i7i. From (2.1.5) it follows
A = R + h,B = Jsina, C = D = £ = 0,
£ = - 2 P c o s a , G = P - / i . (5.1.1) The FR is
(P -I- /!)MI^«2^ -t- <i sin a. «1 ^«2 - 2P cos a. «1 «2 -f
+ dsm(x.U2 + R-h = 0. (5.1.2)
5.2 THE CONDITION FOR A RATIONAL MOTION From (2.3.3) and (5.1.1) it follows
Po = P4 = 0, Pi = -BG,
6P2 = -4AG-4B^ + F\ (5.2.1)
P3 = -AB.
(2.3.2) becomes 2P3U2^ + 3P2«2^ + 2PIM2 = 0. (5.2.2)
As a ?^ 0 we have P3 # 0.
The equation (5.2.2) has two coinciding roots « 2 = 0 if Pi = 0. If P i # 0 it has two coinciding roots «2 # 0 if 9P2^ — I6P1P3 = 0. We investigate both cases.
5.3 THE CASE ^, = 0
Then BG = 0. If P = 0 we find d = 0, and we have a simply skewed four-bar which is dealt with in chapter VI. If G = 0, P 7^ 0, we obtain R = h, and the £ P i s
M2(2P«i^«2-l-(^sina.Mi^-2Pcosa.Mi-l-É?sina) = 0. (5.3.1) For the discussion of this case we refer to section 4.2.
5.4 THE CASE Pi ^ 0, 9?^^ - I6P1P3 = 0
Substituting in the equation 9P2^ - I6P1P3 = 0 the expressions for Pj found in (5.2.1) we obtain
{4AG-4B^-F^f = 16P^£^ or
[(2P-l-£)^-4/4G][(2P-£)2-4^G] = 0. (5.4.1) With the aid of (5.1.1) we find
[(rfsina-Pcosa)^-P^-l-/7^][(rfsina-l-Pcosa)^-P^-|-/!^] = 0, (5.4.2) which can be reduced to the following four relations.
fi? sin a - P cos a = (P^ - Z/^)*, (5.4.3) < / s i n a - P c o s a = - ( P ^ - / ; ^ ) ^ (5.4.4) rfsina-fPcosa = (P^-/2^)% (5.4.5) «/ sin a -I- P cos a = - (P^ - h^. (5.4.6) As 0 < a ^ JTT (5.4.6) is disregarded. Consider the remaining three relations as
equations in a. That means: if there is an angle a,, 0 < ai ^ \n, satisfying one of these three equations, the motion is rational if a = ai.
A necessary condition for the solvability of the equations is h^R. For the time being we suppose h < R.
Denoting tan ^a = y we find 0 < y g I. Moreover we introduce the angles y and Ö with
rf= A cot y and P sin ^ = /!. (0 < y ^ ^TT, 0 <Ö^ in) (5.4.7) (5.4.3) can now be written as
tani(5.t)^-2coty.!;- cot^^ = 0. (5.4.8) Its solution is
Vy = tan^ai = tan ^y cot i^, i;2 = tania2 = — cot ^y cot ^«5. As 0 < y ^ 1 i;2 is disregarded.
We have to investigate if 0 < ^i ^ 1. This proves to be true if d^'^R^ — h^, and this condition can be fulfilled.
In the same way we can replace (5.4.4) by
coti(5. t)^ -I- 2 cot y. y - tan iS = 0, (5.4.9) with Vi = tan iy tan ^ö, V2= — cot \y tan \d.
V2 does not satisfy, and 0 < DI ^ 1 for any d, h and P. Hence 0 < a ^ ^TI.
The third equation to be investigated is (5.4.5) .We obtain
c o t i ö . t ; ^ - 2 c o t y . ! ; - t a n i ^ = 0, (5.4.10) with ^1 = cotiytani(5, V2 = — tan^ytan^^.
V2 must be cancelled and 0 < ^i ^ 1 if d^ ^ R^-h^.
The auxihary angles y and 5 have a geometrical meaning. That of y is simple. S will prove to be the complement of | (^2! in the double position. Fig. 6 and 8. In the next sections we discuss the preceding three cases.
5.5 THE THIRD DOUBLE POINT
In section 5.4 we found that the PC has a third double point if (5.4.3), (5.4.4) or (5.4.5) is vahd. In order to find this double point we write the FR in homo-geneous coordinates.
/ ( « I , «2, «3) = (P-f-/i)Mi^W2^ + '^sina.«i^«2«3 —2Pcosa.«iM2"3^ +
+ dsiax.U2U3^ + iR-h)u3'^ = 0. (5.5.1) We find the double points by solving the simultaneous equations df/dui = 0,
I = 1,2,3, or
M2[(P+^)"i«2+^sina.«iM3-Pcosa.M3^] = 0, (5.5.2) 2(P-l-//)«i^«2-l-rfsina.Wi^M3-2Pcosa.MiM3^-l-É?sina.«3^ = 0, (5.5.3)
£?sina.«i^M2 —4Pcosa.MiM2M3-t-3</sina.M2W3^-f-4(P-/!)«3^ = 0. (5.5.4) Substituting «3 = 0 we find the double points Pi and B2, each counted once, and no others. For the third double point we may assume «3 = 1. (5.5.2) is satisfied by «2 = 0, but (5.5.4) is not. Therefore we may as well suppose that «2 7^ 0, and the system to solve is
(P-f/?)MiM2-l-^sina.Mi —Pcosa = 0,
2(P-l-/!)«i^«2 + ^sina.Mi^ —2Pcosa.«i+Ê?sina = 0, fi?sina.«i^M2 —4Pcosa.Mi«2 + 3rfsina.M2-f4(P —A) = 0. We denote
p = {R-h)^iR+h)-i (5.5.5) and find the following double points.
( 1, -ƒ>) if (5.4.3) is valid, ( 1, ;») if (5.4.4) is valid, ( - 1 , -/)) if (5.4.5) is valid.
5.6 THE CASE dsin a-Rcos <x = {R^-h^)^
The third double point is (1, —p). Hence in the double position (py =in, cos(>)2 = hR-\ sin(p2 = -{R^-h^R-K Fig. 6.
As for the geometrical meaning of this result we refer to section 5.8. We choose an other triangle of reference with the origin at Pj (1, —p). The transformation formulas are
M, = H - i - l - l ,
(5.6.1) "2 = W2-p.
Substituting this in the FR we obtain
(P-HA)Wi^H'2^-F[PCOSa-(P^-/!'')*]H'i^W2 + 2 ( P + /ï)WiM'2^ + -P(P-A)*(P-hA)~*COSa.M'i^-2(P^-/!')*WiH'2H-(P-|-/!)W2^ = 0. (5.6.2) From (5.5.5) it follows R + h = 2R{p^ + \)-\ R-h = 2p^R(p^ + \)-\ {R^-hy = 2pR{p^ + \)-\ Fig. 6
After denoting (1 +p^) cos a = 2A: we can rewrite the FR as
Wi^W2^ + {k—p)Wy^W2 + 2WiW2^ — kpWy^ — 2pWiW2 + W2^ = 0. (5.6.3) The double point tangents at P3 are given by
kpwy^ + 2pWyW2 — W2^ = 0.
The discriminant of their equation is A =p^ + kp.
As A>0 the double point tangents at P3 are real and diff"erent. P3 is a node. The double point tangents at Pi are
H'2 = p, or «2 = 0, and 1^2= —k, or «2 = —k—p. By is a node too.
The double point tangents at P2 coincide: Wy = — 1, or «i = 0. P2 is a cusp. Fig. 7.
Fig. 7
5.7 THE PARAMETER REPRESENTATION OF THE PC The conies of the pencil
W2{Wy + \) = tWy (5.7.1) pass through Pi and P2 and have their tangent at P2 in common with the PC.
Thus we can use the pencil to derive a parameter representation. From (5.6.3) and (5.7.1) we obtain
Wy = {k+p)(-t^ + 2pt + kp),
W2 = [t^ + ik-p)t-kp](-t^ + 2pt+kp), (5.7.2) W3 = ik+p)[t^ + {k-p)t-kp].
The values of t for the double points are for P i : ty =p, t2= —k,
for 82'. t= CO,
for P3: the real and different roots of t^-2pt-kp = 0.
5.8 THE COUPLER CURVE A N D THE RS
In order to obtain the equations ofthe coupler curve and the RS we apply the transformation formulas (5.6.1), which, in homogeneous coordinates, are
«1 = ^ 1 - 1 - ^ 3 , "2 = ^ 2 - ^ * ^ 3 . (5.8.1) « 3 = W3. We denote fy=ik+p)\ ƒ2 = it+k){t-p), (5.8.2) ƒ3 = -it-p)'.
Then we obtain from (5.7.2)
"1 = / i ,
«2 = / 2 / 3 , (5.8.3) «3 = ik+p)f2.
The equations of the coupler curve and the RS can now be written as x = 2cosP[XyRy{k+p)fyf2{f3' + ik+p)'} +
+ X2R2ik+p)fAfi' + {lc+p)'f2'}] +
y = )nRdfi'-(k+P?f2'][f3'+ik+p)']+ -X2R2[fl' + {k+p)'f2'][f3'-{k+p)'] +
+ih(Xy -X2)[fy' + (k+p)f2^][f3^ + (lc+pn (5.8.4) z = 2sin^[^iPi(A:+/7)/i/2{/3'-F(A:-f;7)^} +
-X2R2ik+p)f3{fi' + (k+pff2'}] +
+ cosP(Xydy-X2d2)Ui' + (k+pyf2'][f,' + ik+p)'], y^=ai+^2)lfi' + (k+pff2'][f^ + ik+p)'].
Each ofthe double points P , , P2 and P3 indicates a hnkage position. To Pi corresponds the position for which (py=n, (P2 = 0.
The coordinates of Ay are ( —^sinj?, ^h + R, dcosP) and those of A2: (0, —\h + R, 0). We find for the equations ofthe double generator
x{-dsinP)-^ = iy + ih-R)h-^ = z{dcosP)-K
In the same way we find for the double generator corresponding to P2 x{-dsinp)-'' = iy+ih + R)li-' = z(dcosP)-K
P3 gives
X = z tan P, y = \h.
So Pi and P2 correspond to parallel double positions of the coupler. In the double position indicated by P3 the coupler is perpendicular to P2. This result can be checked in the following way: in this double position we have
(py = in, cos(p2 = hR'^, sin(p2 = -{R^-h^)^R~^.
That means that A2 lies on SDy (fig. 6). Hence in its double position the coupler is found in the plane through Mi perpendicular to c. Now it follows from fig. 6 that dsin a — R cos a = (R^~h^)^ = ^ 2 ^ 1 - On the other hand is dsin a —P cos a the projection on SDy of DyMyAy, so ^1^2 is perpendicu-lar to P2.
P i , P2 and P3 also indicate double points of all coupler curves.
5.9 THE CASE rfslna-i?cosa=-(/?2-A^)*
The third double point of the PC is (Up) and we have in the corresponding double position
Fig. 8
The origin 83 of a new triangle of reference is chosen in this double point. We obtain the transformation formulas
«1 = Wy + l,
"2 = W2 +p.
We denote (1 +p^) cos a = 2k. The FR becomes
Wy^W2^ + (k+p)Wy^W2 + 2WyW2^+kpWy^ + 2pWyW2 + W2^ = 0.
(5.9.1)
(5.9.2)
(5.9.2) follows from (5.6.3) by replacingp by -p. We refer to the sections 5.6, 5.7 and 5.8. There is, however, a remarkable diff"erence from the preceding case: the discriminant of the equations of the tangents at P3 is
A =p^-kp= -pd sin <x.{R + h)~K
So P3 is an isolated double point of the PC. Fig. 9. In the corresponding position the coupler is an isolated generator ofthe RS, and any coupler curve has an isolated real double point.
5.10 THE CASE dsin a + R cos x = (R^-h^)^
The third double point of the PC is ( - 1, -p). The FR becomes
Wi^W2^-(k+p)Wi^W2-2wyW2^+kpwy^ + 2pWyW2 + W2^ = 0. (5.10.1) By changing Wy into —Wy and M'2 into —H'2 we find (5.9.2). We refer to section 5.9.
5.11 THE TYPES
From the graphs ofthe PC (fig. 7 and 9) we derive that the mechanism related to (5.4.3) is a crank rocker and that (5.4.4) and (5.4.5) indicate double rockers.
5.12 THE CASE OF INTERSECTING ROTATION AXES
Up to now we have assumed 0<h< R. Now we investigate the four-bars with h = 0 and h = R respectively.
If /j = 0 we have intersecting rotation axes. As d = 0, M2 coincides with Dy. In section 3.3 we proved that in this case the coupler curve has a real double point in the finite. For then the coupler can take up two positions AyA2 and Ay'A2', symmetric with regard to the plane through Mi and M2 perpendicu-lar to c. In these symmetric positions we have (p{ = n—(pi, or «;«/ = 1. And indeed: if/; = 0, (5.1.2) changes into itself if we replace Uy by «i"^ and «2 by «2"^. For the properties ofthe PC, the RS and the coupler curve we refer to the sections 5.5 up to and including 5.11. We indicate some differences, apart from the fact that the coupler curve has an "extra" double point.
If h = 0 (5.4.3) becomes dsin a—P cos a = P. This equation has one useful solution tan ^a = Rd~^ with d^ R. In the double position ofthe coupler all links are in the plane through My and M2 perpendicular to c. It is evident that d = g = b.
Substituting /; = 0 in (5.4.4) we find no useful solution except a = 0. We refer to section 5.15.
From h = 0 and (5.4.5) it follows ö^sin a-l-P cos a = P with the solutions a = 0 and tani<x = dR~\ The third double point ( - 1 , - 1 ) of the PC is isolated. The hnkage cannot move out of its double position unless we discon-nect it.
5.13 THE CASE A = /?
If /i = P we have rf sin a = P cos a. The FR is
W2[2«i^M2 + COSa(«i-1)^] = 0.
The motion is degenerate. We refer to section 4.2. Iftx^in we find </= 0, and we have a degenerate simply skewed four-bar which is dealt with in section 6.30.
5.14 THE CASE OF FOUR EQUAL LINKS
We suppose P'^ —/j^ = d^. As g^ — h^ = d^ andè = g we have four equal links. The conditions for a rational motion are now
rfsina —Pcosa = <y, (5.14.1) fi?sina —Pcosa = —(/, (5.14.2) tfsina + P c o s a = ^. (5.14.3) (5.14.1) has one useful solution a = in.
The PF is (R + h)ui^U2^ + duy^U2+du2 + R-h = 0. As R^-h^ = d^ we obtain
[(R + h)u2 + d][iR + h)Uy^U2+d] = 0. (5.14.4) The PC is degenerate. It consists ofthe straight line (R + h)u2+d= 0 and the
rational cubic (R + lr)ui^U2 + d= 0. The "extra" double points are their inter-sections (1, —p) and ( - 1 , —p). As for the kinematical aspects we refer to the sections 4.2 and 4.3.
(5.14.2) has one useful solution tan i<x = (R — d){R + d)~K We refer to section 5.9.
(5.14.3) allows a discussion similar to that of (5.14.1).
5.15 PLANAR MOTIONS
In the preceding sections we assumed a 7^ 0. If a = 0 one finds dy=d2 = 0. Then we have a simply skewed four-bar which is dealt with in the next chapter.
CHAPTER VI
THE SIMPLY SKEWED FOUR-BAR
6.1 THE FUNDAMENTAL RELATION OF THE SIMPLY SKEWED FOUR-BAR
In the preceding chapters we supposed that dy and d2 are not both equal to zero. Now we take dy = d2 = 0. Then h - g. We noticed that in section 1.3. The mobility of the simply skewed four-bar has been discussed by HARRIS-BERGER [7] and by BOTTEMA [3].
We investigate whether its motion can be rational. From (2.1.5) it follows
A = Ry^ + R2^+g^-b^ + 2gRy+2gR2 + 2RyR2, 8 = 0, C = 0, D= Ry^ + R2^+g''-b^ + 2gRi-2gR2~2RyR2, (6.1.1) £ = Pi^ + P 2 ' + g ^ - f t ^ - 2 g P , + 2 g P 2 - 2 P i P 2 , £ = — SPiPjCosa, G = Pi^ + P 2 ^ + g ^ - è ^ - 2 g P i - 2 g P 2 + 2PiP2. We denote Si = - P , + P2 + è + g , Tl = - P i - f P 2 + è - g , S2= Ry-R2 + b+g, T2= Ri-R2 + b-g, « 3 = Ry+R2-b+g, T3= Ry+R2-b-g, 5 ^ = Ri+R2 + b-g, T^= Ry + R2 + b+g. (6.1.2)
As the length of any Hnk is shorter than the sum of the other link lengths we see that S; > 0 (/ = 1, 2, 3, 4) and T^. > 0.
From (6.1.1) and (6.1.2) it follows A = SjT^, (A > 0) D=-S2Ty, (6.1.3) £ = - S 1 T 2 , G = S4T3. Hence the FR is S3T4Mi^M2^-S2TiMi^-SiT2M2H£«lW2 + S4T3 = 0 . (6.1.4) Of course the PC has double points at P i ( l , 0, 0) and at P2(0, 1, 0). We
investigate if there are any other double points. From (2.3.3) it follows P4 = - 4 ^ £ , P3 = ^1 = 0, (6.1.5) 6P2= --4AG-4DE + F\ Po = -4DG. (2.3.2) becomes
A = P4U2'' + 6P2U2^ + Po.
The equation A =0 has two coinciding roots in the following cases.
1. P o = 0 , (6.1.6) 2. P4 = 0, (6.1.7) 3. 9 P 2 ' - P o P 4 = 0. (6.1.8)
We investigate these three cases in the next sections.
6.2 THE CASE Po = 0
From (6.1.5) we obtain £)G = 0. If £) = Owe find Ti = 0 or Pi-|-g = P2-(-è. From J = 0 and Po = 0 it follows «2 = 0. Substituting in (6.1.4) Ti = 0 and « 2 = 0 one obtains «, = oo.
Hence in the double position we have (py = n, (p2 = 0 and the four links ofthe mechanism are in line with the F-axis.
It is stretchable. The PC is rational. Two of its double points coincide with By (here it has a selfcontact), one of them coincides with P2- ^1 corresponds to a hnkage position, P2 does not.
If G = o we find 7^=0 or Ry + R2 = b + g. Substituting in (6.1.4) T3 = 0 and «2 = 0 we find «i = 0. In the double position we have (py = (p2= 0- The four-bar is stretchable as well. The PC is rational. Its double points coincide with Pi, P2 and O. O corresponds to a hnkage position. Pi and P2 do not.
6.3 THE CASE ^4 = 0
From (6.1.5) it follows AE = 0 or £ = 0. If we interchange Pi and P2' "1 and «2, D changes into £ and conversely.
The FR changes into itself. Hence for the case P4 = 0 we may refer to the preceding section.
6.4 THE CASE 9P2^--Poi'4 = 0
Now the equation zl = 0 has four roots, coinciding two by two. So the PC has two extra double points, and it is degenerate.
We will deal with this case in the sections 6.31 and 6.32. First we shall discuss the stretchable simply skewed four-bar.
6.5 THE EIGHT CASES OF SINGLE STRETCHABILITY The FR of the simply skewed four-bar can be written as
S3r4«i^M2^-S2Ti«i^-SiT2M2^-8PiP2COSa.MiM2 + S4T3 = 0. (6.5.1) The mechanism is stretchable if T = Ti T2T3 = 0. We speak of single stretcha-bihty if only one of the factors T, = 0.
If we interchange Pi and P2, «1 and «2, we see that Ti and Tj, Si and S2 are interchanged and the FR remains the same. We conclude that for single stretchabihty we can restrict ourselves to two cases: Ti = 0 , T2T3 # 0, and T3 = 0, T i r 2 # 0 .
Therefore we distinguish the following possibihties.
I II III IV V VI VII VIII T, 0 0 0 0 > 0 > 0 < 0 < 0 T2 > 0 > 0 < 0 < 0 > 0 < 0 > 0 < 0 T3 > 0 < 0 > 0 < 0 0 0 0 0
If Tl = 0 we find S, = 2g, S2 = 26, S3 = 2P, S4 = 2P, If T3 = 0 we have Si = 2P, S2 = 2P, S3 = 2g, S4 = 2b, We denote p= \S2TyS3- q=\SyT2S3-Ty =0, T2=2{b-g), T3 = 2 ( P - g ) , T^ = 2(P+g). Ti = 2 ( P - g ) , T2 = 2 ( P - g ) , T3=0, T^ = 2{b+g). 'T,-'\, 'T^-\ (6.5.2) (6.5.3) r = 4RyR2S3~^T^~^ cosa, s = \S^T3S3-'T^-'\. (6.5.4)
p, q, r and s are dimensionless parameters.
In the sections 6.6 up to and including 6.25 we discuss the eight cases mentioned.
6.6 FIRST CASE OF SINGLE STRETCHABILITY
If Ty = 0, T2 > 0, Tj > 0 it follows from (6.5.2) and (6.5.4) P = 0, q = g{b-g)R2-\Rr+gy\ r = P i ( P i + g ) " ^ c o s a , s=Ry{R2-g)R2-\R,+gy'. The FR is «l^M2^ —9«2^ —2rMiM2 + J = 0. (6.6.2) Writing (6.6.2) as u^^'iuy^ — q) — 2ruyU2+s = 0 we introduce the discriminant
Dy=Uy^{r^-s)-itqs.
In a similar way we find D2 = U2^iqu2^+r^—s). Uy exists if D2 ^ 0, «2 exists if Dy è 0.
From (6.6.1) we derive
r ^ - 5 = P i ( è g - P i P 2 s i n ^ a ) P 2 - H ^ , + ^ ) " ' . So we have to distinguish three subcases:
1.1 èg-P,P2sin^a > 0 , 1.2 èg-PiP2sin^a = 0,
1.3 Z?g-PiP2sin^a < 0.
6.7 FIRST SUBCASE (I, 1)
i g - P i P j s i n ^ a > 0, or sin^a < bg{RyR2)~^. If bg<RyR2 a can vary from 0 to arc sin(èg)*(PiP2)~*. If è g ^ P1P2 a can vary from 0 to in.
As r ^ - i > 0 it follows from Dy and Z>2 that «j exists for any «2 and con-versely. So the linkage is a double crank. Fig. 10.
We obtain the parameter representation of the PC by intersecting it with the pencil of conies
«1 «2 — r = q^U2 + t, (6.7.1)
each specimen ofthe pencil having at P , and at P2 a tangent in common with the PC. We find in homogeneous coordinates
«1 =2qt[(r + t)^-s],
U2=(r^-s-t^f, (6.7.2) U3 = 2q''t{r^-s-t^).
The parameter values for the double points are for P i : t= ±(r^-sy,
fox 82'. ty = 0, /j = 00.
As r ^ - i ' > 0 we have at Pi two real branches with a selfcontact.
6.8 THE COUPLER CURVE AND THE RS IN THE FIRST SUBCASE We denote
/ i = 2?*r,
f3=r'-s-t\
Now it follows from (6.7.2) «1 = r / i ƒ2,
« 2 = ƒ 3 ^ «3 = / l / 3
-Substituting this in (3.5.1) we find the equations ofthe coupler curve and the RS.
X = 2cOSp[XyRyq^f2f3if^+f3') + l2R2fJz{qf2^+f,')l
y = hR,{f^+fi')iqf2^-fz')-^2R2{fx^-f^)iqf2'+f3')+ + ^g(^l-^2)ifl'+f3')iqf2'+f3%
z = 2sin P[XyRyqifJ3(J y^+f3^)-X2R2fJ^{q f2^-^ f3^)1 W={Xy+X2)ify'+f3^){qf2^+f3').
In the double position we have (py =n, (p2 = 0. Therefore the y-axis is a double generator of the RS, and any coupler curve has a real double point on the 7-axis.
6.9 SECOND SUBCASE (I, 2)
fcg-PiP2sin^a = 0, or sin^a = è g ( P i P 2 ) " ' . This case can only arise if è g ^ ^1^2- (6.6.2) becomes
(UyU2-r + q'^U2)iUyU2-r-q^U2) = 0.
This equation represents two conic sections, with intersections at Pi (three-fold) and at P2. At Pi they have the tangent «2 = 0 in common. Their tangents at P2 are «i = ± q^. Uy exists for any «2 and conversely. So the type is a double crank.
The equations of the conic sections are UyU2-r + q'U2 = 0,
and
UyU2 — r — q^U2 = 0.
We consider the first one. We denote ƒ = r—tq^. Then we can write the para-meter representation of the conic section as
«1 = ƒ» «2 = '^. «3 = r.
The equations of the coupler curve and the RS are x = 2cos^[;.iPi/r(r^ + l) + ;.2P2(/' + / ' ) ] , y = XyRy{p-t^)(t^ + \)-X2R2t{P + t^) +
z = 2 sin/?[/liPi ƒ?(?' + l ) - A 2 P 2 f ( / ^ + ?^)], w = (Ai+A2)(/^ + r^)(/^ + l).
The equations corresponding to the second conic section are found by chang-ing q^ into ~q^.
We conclude that the RS is degenerated into two surfaces ofthe fourth order. The y-axis is a common generator of these surfaces. The coupler curves are degenerated into two space curves of the fourth order. The latter intersect each other at a point of the y-axis. This point is indicated by the coupler po-sition for which (py=n, (p2=0.
6.10 THIRD SUBCASE (1,3)
bg-RyRiSin^a < 0, or sin^a > bg{RyR2) ^ It is only possible if bg <
RyR2-The parameter representation of the first subcase is valid here; that for the PC as well as that for the coupler curve and the RS. There is, however, a difference. As r^ — s<0 the point Pi is an isolated point of the PC, and therefore the y-axis is an isolated double generator of the RS. The coupler curves have an isolated double point on the y-axis. In fig. 11 the PC is mapped. We conclude that we have a double rocker.
Fig. 11
6.11 SECOND CASE OF SINGLE STRETCHABILITY If Ti = 0, Tj > 0, Tj < 0 we have P = 0, q=g{b-g)R2'\Ry+gy\ ^ ^ , , , ^ (6.11.1) r = Pi(Pi-l-g) ^cosa, s = Ry(g-R2)R2-\Ri+gy\ The FR is Uy^U2^-qU2^-2rUyU2-s = 0. (6.11.2) Dy =uy\r^ + s)-qs, D2 = U2^{qu2^->rr^+s).
The type is a crank rocker.
We derive the parameter representation from (6.7.2) by replacing shy —s. Fig. 12.
For the equations of the coupler curve and the RS we refer to section 6.8, wherein s must be replaced by — .y as well.
Fig. 12
6.12 THIRD CASE OF SINGLE STRETCHABILITY If Ti = 0, T2 < 0, 73 > 0 we find P = 0, q = g{g-b)R2-\Ry+gy\ (6.12.1) r = Ri(,Ry+g)-^cosx, s = Ry(R2-g)R2-\R,+gy'.
The FR is
Ui^U2^ + qu2^-2rUyU2+s = 0. (6.12.2) Di = Ui\r^-s)-qs,
D2 = U2^(r^-s-qu2^).
The motion is only possible if r^-s>0, or sin^ a < i g ( P i P 2 ) ~ \ Then «i exists if «2^ g (r^ — 5)9~' and «2 exists if «1 ^ ^ qs{r^ — s)~^.
The type is a double rocker.
The parameter representation of the PC is found by the aid of the pencil of conic sections MI«2 = U2t+y, wherein y is one ofthe real roots of y^ —2ry-f
-1-5 = 0. We find
«1 = 0^ + qWr-y)t^ + qy],
«2 = 4 ? ^ ( r - y ) ^ (6.12.3) «3 = 2r(/^-l-9)(r-y).
The parameter values of the double points are for P i : ?i = 0, ?2 = 00,
forP2: ' = ± ( - ? ) * •
At Pi we have a selfcontact. P2 is an isolated double point.
6.13 THE COUPLER CURVE AND THE RS IN THE THIRD CASE We denote
/ i = t^ + q, f2 = (2r-y)t^ + qy, ƒ3 = 2 / ( r - y ) .
Then it follows from (6.12.3) "l =fif2.
" 2 = ƒ 3 ^ «3 = / l / 3
-The equations of the coupler curve and the RS can be written as X = 2cOSiS[AiPi/2/3(/,'+/3') + A2P2/l/3(/2'+/3')]. y = XyRyify'+f3^)(f2'~f3')-X2R2(f'-f3')if2'+f,') + + i^(A,-A2)(/i^+/3^)(/2^+/3'), Z = 2 s i n ^ [ A i P i / 2 / 3 ( / , ' + / 3 ' ) - A 2 P 2 / l / 3 ( / 2 ' + / 3 ' ) ] , W = ( A i + A 2 ) ( / i ^ + / 3 ' ) ( / 2 ' + / 3 ' ) .
The y-axis is a double generator of the RS. The coupler curves have a double point on the j-axis.
6.14 FOURTH CASE OF SINGLE STRETCHABILITY If Ti = 0, T2 < 0, T3 < 0 we have p = 0, q = gig-b)R2-HRr+gy\ (6.14.1) r = P i ( P i + g ) * cos a, s = Ry{g-R2)R2-\Ri+gy'. The £ P is Uy^U2^ + qu2^-2ruyU2-s = 0. (6.14.2) Dy = uy^{r^+s) + qs, D2 = U2^ir^+s — qu2^).
«1 exists if «2^ g {r^+s)q~^, «2 exists for any «j. We have a crank rocker. We obtain the parameter representation ofthe PC from (6.12.3) by replacing s by —s.
6.15 THE COUPLER CURVE AND THE RS IN THE FOURTH CASE We refer to section 6.13, where we replace sby —s.
6.16 FIFTH CASE OF SINGLE STRETCHABILITY If Ti > 0, T2 > 0, Tj = 0 we obtain p^Ri{R2-g)g-\b+gy\ q = R2iRi-g)g-\b+g)-\ (6.16.1) r = P i P 2 g - H * + 5 ) " cosa, s = 0.
The FR is
Uy^U2^ — puy^ — 2ruyU2 — qu2^ = 0. (6.16.2) So the third double point coincides with the origin.
Dy=uy\puy^+r^-pq), D2=^U2^{qu2^ + r^-pq). From (6.16.1) we derive
r ^ - M = PiP2(6^-PiP2sin2a)g-^(è-Hg)"^ Again we distinguish three subcases:
V, 1 è g - P i P 2 s i n ^ a > 0, V,2 Z)g-PiP2sin^a = 0, V, 3 è g - P i P 2 s i n ^ a < 0.
6.17 FIRST SUBCASE (V, 1)
è g - P i P 2 s i n ^ a > 0, or sin^a < bgiRyR^)"^. If bg<RyR2 a can vary from 0 to arc sin(Z)g)^(PiP2)~*. If bg ^ PiP2 a can vary from 0 to in.
Uy exists for any «2 and «2 does for any «i. The type is a double crank. To obtain the parameter representation ofthe PC we take the pencil of conic sections UyU2~q^U2 + tUy = 0 and we find «1 = 2{r + q't){qt^ + 2rq''t+pq), "2 = {t^-p){qt^ + 2rqit+pq), (6.17.1) « 3 = -2it^~p)(rqi + qt).
The parameter values for the double points are for By-, t = ±pi,
for 82'. ty= —rq~^, ?2 = 00, for 0:t = 9 " * [ - J - + ( r ^ - ; ; 9 ) ^ ] .
Fig. 13
6.18 THE COUPLER CURVE AND THE RS IN THE FIRST SUBCASE We denote
fy=2{r + qit), f2 = qt^ + 2rqi+pq, ƒ3 =
t'-P-Now (6.17.1) can be written as "1 = / l / 2 >
"2 = ƒ2 ƒ 3 . "3 =
-q^fifi-The equations of the coupler curve and the RS are
X = 2cos P[-XyRyqif2f3{q f 1'+f2')-^2R2q^fJ2if2'+ qf3')], y = -XyRy{qfy'+f2'){f2'-qh') + ^2R2iqfi'-f2')if2' + qf3') +
+ igai-^2)iqfl'+f2'){f2'+qf3^),
z = 2sinP[-XyRyq%f3(qfy'+f2') + X2R2q^fj2if2' + qf3')l
W=iXy+X2)iqfy'+f2'){f2'+qf3')-The points Pi and P2 do not correspond to linkage positions, but 0 does. It indicates a double position with <?>i = (P2 = 0- So the y-axis is a double gener-ator of the RS and any coupler curve has a double point on the y-axis.
6.19 SECOND SUBCASE (V, 2)
bg-RyR2 sin^a = 0, or sin^a = bg(RyR2)~K
This case is only possible if i g ^ PiP2- The PC is degenerated into two conic sections
UyU2-p^Uy+q^U2 = 0 and
UyU2+P^Uy—q^U2 = 0.
We consider the first one. Its parameter representation is
«1 = -qh,
U2 = t(t-pi), (6.19.1)
«3 = t-pi.
We denote ƒ = /-/>*. Then (6.19.1) can be written as
«1 = -q^'t,
"2= ft,
« 3 = / .
The equations of the coupler curve and the RS are X =: 2cosP[-XyRyqift{t^+l) + X2R2t{f^ + qt^)], y = XyRy{t'-l)ip + qt')-X2R2(t' + l)ip-qt') +
+igai-^2)it'+m'+qt'),
z = 2sinP[-XyRyqift{t^ + l)-X2R2tif^ + qt^)], w={Xy+X2)it' + l)(f + qt').
We find the parameter representation ofthe second conic section by replacing /»* by -/>* and q^ by — q^.
In the same way we derive the equations ofthe coupler curve and the RS. We conclude that any coupler curve is degenerated into two space curves of the fourth order, which have a point ofthe y-axis in common. The RS is degener-ated into two surfaces of the fourth degree with the y-axis as a common generator.
6.20 THIRD SUBCASE (V, 3)
è g - P i P 2 s i n ^ a < O, or sin^a > bg{RiR2)~^.
This case can only present itself if 6g ^ P1P2. "1 exists if «2^ ^ (pq — r^)q~^, «2 exists if «i'^ ^ (pq — r^)p~^.
The type is a double rocker.
The parameter representation of the PC is that of (6.17.1). We derive the equations of the coupler curve and the RS as we did in the first subcase. There is a geometrical difi"erence: O is an isolated double point ofthe PC. So the coupler curves have an isolated double point on the y-axis, which is an isolated generator of the RS. See fig. 14 for the graph of the PC.
Fig. 14
6.21 SIXTH CASE OF SINGLE STRETCHABILITY If Ty > 0, T2 < 0, 73 = 0 we have
p = Ry{R2-g)g-\b+gy\ q = R2{g-Ry)g-\b+gy\
r = RyR2g-Hb+g)-Uosa, s = 0.
The FR is
Ui^U2^—pUi^-2rUiU2 + qu2^ = 0. (6.21.2) Dy=uy\pui^+r^+pq),
D2 = U2\-qu2^ + r^+pq).
Uy exists if «2^ ^ (r^+pq)q~^, «2 exists for any «j.
The type is a crank rocker.
For the parameter representation ofthe PC we use the pencil of conic sections
UyU2—p^Uy—tU2 = 0
and we obtain
«1 =it^ + q)(pt^~2pht-pq), «2 = 2(p^t-r)(pt^ -2p^rt-pq),
U3 = 2(t^ + q){pt~ph-).
The parameter values of the double points are for P i : ty =p'^r, ?2 = 00,
f o r P 2 : ? = ±{-q)\
for O: t = p~^[r±(r^+pq)%
By and O are real double points with two diff"erent tangents. P2 is an isolated
double point.
6.22 THE COUPLER CURVE AND THE RS IN THE SIXTH CASE We denote fi=t' + q, ƒ2 =pt^-2pirt-pq, f3 = 2{ph-r). N o w we can rewrite (6.21.3) as " l = / l / 2 . "2 = / 2 / 3 . «3 =
