the roll-sliding number of associated curves
TRANSCRIPT
Mechanism and Machine Theory, 1976, Vol. 11, pp. 419-423. Pergamon Press. Printed in Great Britain
The Roll-Sliding Number of Associated Curves
Jtirgen T61ket
Received 6 October 1975
Abstract Curves ~' which possess an envelope ~ ' in spatial kinematics are called associated curves. We show that the roll-sliding number of such curves X, ~ ' is a projective invariant and give a geometrical interpretation as a cross ratio.
Introduction IN PLANE kinematics a curve ~-- f ixed in the moving plane--normally has an envelope ~ ' in the fixed plane. During the motion ~( is rolling and sliding on ~ ' . This process is described by the roll-sliding number. Due to Miiller there exists a famous interpretation for this roll-sliding number as a cross ratio[I]. Bottema gave two contributions to questions on this matter[2, 3].
If we go to spatial kinematics only special curves can have an envelope. Koenigs[4] has studied such pairs of curves which are known as associated curves. But a geometrical meaning of their roll-sliding number was still a problem to be solved. We close this gap and show. The roll-sliding number is a projective invariant and has an interpretation as a cross ratio.¢
1, Notation Let E3 and E~ be two coincident 3-dimensional Euclidean spaces. We attach to E3 and E~
cartesian frames of reference {O; el, e2, e3} and {O'; e',, e;, e;} respectively, both having a right handed orientation. For a point X holds the relation
O ' X = O X - O 0 ' or O X = O ' X - O ' O . (1)
Using the Einstein convention we write
x': O t X ! t i t = =X et, O X : = x = x i e l
d O' : = c' = c"e~, Od' : = c = c'e, (2)
and denote by x', x, c', c the column vectors
c 2 .
L r'J L 'J Lcrt.!
A motion (J of E3 with respect E~ is then described by the matrix equation
x ' = C ' ( x - e), (3)
where C' is a direct orthogonal (3,3)-matrix the elements of which, like those of the (3,1)-matrix c are supposed to be real functions of a real parameter t.
With every motion/~ there is a uniquely determined motion/3', the inverse motion of E~ with
tProfessor of Mathematics, Gesamthochschule Siegen, 59 Siegen 21, Hrlderlinstr. 3, BRD. ¢See Appendix.
419
420
respect to E3. This motion is evidently described by
x = C ( x ' - c ' ) ,
where by definition
C'C = CC' = I (I is the (3,3)-unit matrix).
(4)
(5)
2. V e l o c i t y
We suppose the motion fl(t) (and hence/3'(t)) to be of the class Ck(k >t 2). Differentiation with respect to the parameter t will be denoted by placing a dot over the function symbol. Now let us differentiate the eqn (1.3), which yields to
x ' = C ' { B ( x - c ) - ~ + i } with B:=C(Y, (1)
when we allow x to be a function of t. By virtue of (1.5) B is a (3,3)-skew symmetric matrix, the so called infinitesimal transformation matrix, which determines the quadratic complex of the trajectory tangents. So we can write for the matrix B
B:= _ 3 0 % and q:= q2 . [. q2 _ql q
(2)
We want to give (1) a more elegant form. To do this let p be a solution of the equation
(/:, q) , B(x - c) = c - ~ q (3)
where (,) is the symbol for the inner product. Now we form
4: = Bp + kq, (4)
where the parameter invariant function
k: -- (c, q) _ (t~, q) (q, q) - (q, q) (5)
is the pitch (Steigungsparameter) of the motion ~(t). Evidently ~ does not depend on a special solution p of (3). These solutions
a : y = p + / z q (6)
form the instantanous screw axis. With (4) we get instead of (1) [5], p. 150.
~,' = C ' { B x - ~ + x}. (1')
Especially we get for the translation velocity v, of the motion
v~ = -kq. (7)
For a later use we need the analogues equations for the inverse motion fl'(t). It was for this reason, that we did not normalize p as done in [6, (5.10)]. For the inverse motion, we have instead of (1)
x = C { B ' ( x ' - c ' ) - c ' + x ' } with B':=C'(~. (8)
In virtue of (1.5) we have
C'B + B'C' = 0,
and so, if we put
0 q,3 B': = _q,3 0
q,2 _q , ,
the eqn (9) is equivalent to
421
(9)
-q' l Fq"l and q':=/q'2/
La' j (2')
q = -Cq ' , (9')
because every element of a direct orthogonal matrix is equal to its subdeterminant. By (3) and (4) we have
c ' = - C ' c or c = - C c ' . (10)
If we put further
p ' : = C ' ( p - c ) and ¢1' = - C ' ~ (11)
an easy computation leads to
i, = C{B'x' - ¢ + ~,'}. (8')
3. Assoc ia t ed Curves
In plane kinematics a curve fixed in the moving plane normally possesses an envelope in the fixed plane. Not so in space! Koenigs [4] in 1910 determined all curves ~ which have an envelope. These pairs of curves are known as associated curves (courbes associ~es). As Koenigs has shown, the problem of the determination for the curves ~ can be solved by two quadratures. They are the integral curves of a Monge-differential equation (compare also [7], p. 170f.).
Let ~ be a curve, which has the curve ~ ' as an envelope. We assume that ~' does not lie on the moving polode, so that ~o rolls and slides on ~ ' during the motion/3(t).t In the given position let X be the characteristic point, which means in virtue of (2.1') ~ is a solution of the differential equation system of the form
x = A ( B x - ~ + t ) (1)
and ~ ' a solution of
Ax' = B ' x ' - ~ ' +x' . (2)
We call A = A (t) the roll-sliding number (Rollgleitzahl) of the associated curves ~, ~ ' . For the effective determination of A for a known curve ~ see [7], p. 172. In the following we exclude the cases A = 0 and A = oo. Indeed, this is valid if ~ ' is an orbit of the fixed point x = ~ or ~ is the orbit of the fixed point x' = ~ ' under the inverse motion respectively.
4. The Geometrical Meaning of , t(t) In plane kinematics there is a famous interpretation for the roll-sliding number A (t) due to
Miiller [1]. If we designate as po the pole of the plane motion and the fixed curve ~ contacts its
t l / ~ lies on the moving polode the motion/~ is a pure rolling[7], p. 151. The curves ~, ~ are then easy to determine [7], p. 184. Because ~ does not lie on the moving polode, we have in (1) A # 1.
422
envelope ~ ' in X for the given position, there holds
CR (po, x; x~,, x~) = a (t),
where CR is the cross ratio and, x~, and x~e are the centers of curvature of the curve ~ ' and respectively. So A is a projective invariant. Bottema has made two contributions on this matter [2, 3].
In space, so far nothing is known about a geometrical meaning of the roll-sliding number of associated curves. We will close this gap.
If you try, as a first attempt, to prove a relation like (1), you have first to modify it. A natural attempt will be to cut the three lines: the instantaneous screw axis and the axes of curvature of ~ ' and ~ with the perpendicular through X onto the instantaneous screw axis. We thus obtain the points ¢o, x~e,, x~e. It is easy to show, that the modified eqn (1) is equivalent to k = 0, which characterises pure rolling motions.
Now let us consider the plane containing the characteristic point X of ~, ~ ' and the instantaneous screw axist
{ z ' - x ' , p ' - x ' , q ' ) = 0 or ( z - x , p - x , q ) = 0 . (2)
We determine the characteristic of (2) under the motion/3' and/3 respectively. Besides (2) there holds for the characteristic lines in question
(z ' - x', 1~'- ~k', q')+ ( z ' - x', p ' - x', q ' ) - (x ' , p ' - x', q'> = 0 (3)
and
( z - x, l i - / i , q) + ( z - x, p - x, q ) - 0k, p - x, q) =0 (4)
respectively. Because of (2.9') and (2.11) we have in virtue of (2.2) and (2.3)
p' = C ' (p - k~) and q' = -C 'q (5)
and hence instead of (3)
( z - x, l i - i - vl, q) + ( z - x, p - x , / I ) - <x + vl, p - x, q) = 0 (3')
with the abreviation
vr: = B x - ~ . (6 )
The eqns (2), (3') and (2), (4) respectively lead to the required representation of the characteristic lines
and
z~.,(p)- x - { p - x (q, p - x, 1~> } <p- x, x+ v~,q> (q ,p_x, / i ) q ( l + p ) + p [q ,p-x , ( l ) q (7)
(q, p - x, li) } <p- x ,L q) z~e(~) -x= p - X - < q , p _ x , q ) q ( l+ /~)+ /Z(q ,p_x , / l ) q. (8)
Hence cutting these characteristic lines with the line through X parallel to the instantaneous screw axis, we get in virtue of (3.1) for the division ratio
DR(x; z,~(-l), z~,(-1)) = ,~. (9)
tWe designate (a, b, ¢) as the determinant of the column vectors a, b, c.
423 Theorem. The roll-sliding number A of two associated curves ~, ~ ' is a projective invariant. A
is equal to the cross ratio
CR (~ , / ]x ; g~t', g~t) = A,
where a is the ins tan taneous screw axis, g~r, g~ are the characterist ic lines of the plane containing X and a under the mot ion // ' and /3 respectively, and fix is the line connect ing X with intersect ion point of g=r and g~.
A p p e n d i x
Four concurrent straight lines 1, of a plane have a projective invadant, the cross ratio
CR(I,12; ld,).
This value is equal to the cross ratio CR(L~, L2; L3, L4) of the intersection points of any straight line (not through the center c) with the straight lines l,
LI
L 3 ~ L 4
C ~ -- 14 ~" L~L3 L,L, OR(l,12; I,I,) = CR(L,, L,; L3, L,,) = ~ :~,L~
and remains invariant under any projectivity.
R e f e r e n c e s
l. H. R. Miiller, Zur Kinematik des Rollgleitens. Arch. Math. 4, 239-246 (1953). 2. O. Bottema, Zur Kinematik des Rollgleitens. Arch. Math. 6, 25-28 (1955). 3. O. Bottema, The A-pairs or curves for a cycloidal motion. J. Mechanism Machine Theory 10, 189-195 (1975). 4. G. Koenigs, M6moire sur ies courbures ¢onjugu6es dam le mouvement relatif le plus g~n6ral de deux corps solides (1910),
published in Mtmoires pr~sentts par divers savants ?l l'Acadlmie des Sciences 35, 2 s~r., Paris (1914). 5. H. R. M011er, Kinematik. Berlin (1963). 6. G. R. Veldkamp, Canonical systems and instantaneous invariants in spatial kinematics. J. Mech. 3, 329-388 (1967). 7. R. Gamier, Cour de Cin6matique, Tome II. Paris (1956).
ZUM ROLLGLEITEN ASSOZIIERTER KURVEN
J. TSlke
Kurzfassun~ - Kurven ~jdie in der r~umlichen Kinematik eine H~llkurve x besitzen;helssen
nach G. Koenigs aasoziierte Kurven. Wit zeigen, dass die Rollgleitzahl solcher Kurven x,~'
eine proJektlve Invariante ist, ttnd geben eine geometrische Deutung an.