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The curves traced here are half-curves of the three-bar linkage

Curve studied by Chebyshev in 1868, Cayley and Roberts in 1875, Darboux in 1879 and Koenigs in 1897.
KOENIGS, leçons de cinematiques, Hermann, 1897, p. 246 aa 299 ; BROCARD LEMOYNE T 2 p 116 ;
BRICARD, cinematique et mecanismes, A. Colin , 1947 ; H. Lebesgue, Leçons sur les constructions geometriques, 1950 ;
ROSENAUER, WILLIS, kinematics of mechanisms, Dover, New York, 1967 ; HUNT, Kinematics, geometry of mechanisms, Oxford University Press, 1978 ; 
F. RIDEAU, les systemes articules, Pour La science n° 136, fevrier 1989 ; Au dela du compas expos. du Palais de la Decouverte p. 54 aa 57.
F. PECAUT : the articulated quadrilaterals
See also the website of Alain Esculier for the explanations on the animations on this page.

A three-bar linkage curve is the locus of a fixed point M on the plane linked to the rod [PQ] of an articulated quadrilateral (APQB), A and B being fixed. The three "bars" are [AP] and [BQ] (called the cranks or the rocker arms) - and [PQ] (called the connecting rod, or the couple).
We choose A(0, 0), B(0, a), AP = b, BQ = c, PQ = d.

Writing with small letter the affixes of the points, we have:
, where t and u are linked by the relation .
Cartesian equation (Roberts)
Tricircularsextic, that is elliptic except when , in which case the curve is rational (and decomposed?).
There are three double points (real or not) on the circle LP = MN.

A curve of the three-bar linkage is therefore the locus of a point linked to a segment line of constant length joining two circles (the circles (CA) and (CB) with centres A and B and radii b and c).

When the cranks have the same length and M is the middle of [PQ], we get a Watt curve.

The curve is not empty if and only if there exists at least a quadrilateral the lengths of the sides of which are a, b, c and d, i.e. iff each of these four numbers is less than or equal to the sum of the others, or also iff

The point P is said to have complete revolution if it can describe the whole circle (CA) (same definition for Q); in Mechanics, the arm [AP] is called a crank when there is complete revolution, and a rocker arm otherwise; it can be proved that if A1 is the left-hand intersection point between (CA) and (AB), and A2 the right-hand one, then the point P passes by A1 iff  and the point P passes by A2 iff. The point P has complete revolution iff these two conditions are met, and for the point Q, there are two similar inequalities obtained by swapping b and c.

The curve is composed of only one component if and only if, for P and Q, one of the two above inequalities is satisfied(??).

a = 4 ; b  =1,5 ; c = 2  ; d = a + c - b ;
(Limit) case where P has complete revolution; Q crosses the axis on the right but does not have complete revolution.

a = 4 ; b  =2 ; c = 7 ; d = a  ;
No point with complete revolution, but P and Q cross the axis on the left.
a = 4 ; b = 1,5 ; c = 2 : d = a 
P has complete revolution, but Q does not cross the axis: the curve is composed of two components (that overlap: it is therefore connected!).

Case of the parallelogram and the antiparallelogram (a = d et b = c): only case where P and Q have complete revolution; the curve is composed of a circle and a rational bicircular quartic, which is a the pedal of an ellipse when a < b  or of an hyperbola when a > b (the connecting rod is the focal axis of a centred conic rolling on an equal conic).

Case of the rhomboid (a = c et b = d)
According to the principle of crank-rod swap (see below), the curve obtained is similar to the previous one.

The curve is generated by two other ways thanks to two other three-bar linkages [BRSC] and [CTUA] (Roberts theorem of triple generation) :
The three white quadrilaterals are parallelograms, and the three red triangles are similar; therefore:
r = b + m q ; u = a + m p ;
s = r + (m q)(m r)/(p q) ;
t = m + (m q)(m u)/(q p) ;
c = t + s m.
(b', d', c') is proportional to (d, c, b) and
(b" ,d", c") to (c, b, d).
The triangle (ABC) remains fixed.

Superb animation of the triple generation of Roberts due to Alain Esculier.

We derive from this the crank-rod swap principle, stating that if the connecting rod and the crank are swapped, then the curves described by this new mechanism are similar to the initial ones.

In the case b = c = d and equilateral triangle (case studied by François Rideau): the triple generation mechanism has an order 3 symmetry, thus so does the curve; with k = 3b / a we get the following shapes:
k < 3/2 : rounded triangle

k = 3/2: quasi-triangle

 3/2 < k < 3

k > 3

The case a = 2d, b = c =QM = MP, corresponds to the Roberts mechanism that provides an almost linear curve (see also Watt curve).

the curve passes by A, B and the middle of [AB]

the complete curve (in fact, to be completed with its symmetric image with respect to [AB]) does not have, theoretically, any linear portion.

Envelope of the rod?
The curves of the three-bar linkage are special cases of glissettes, when the fixed and sliding curves are circles.

These curves can be generalised to the case of a segment line with fixed length (still called connecting rod) the ends of which are constrained to move on two given curves, which also provides a special case of glissette.

When these two curves are non-parallel lines, the points linked to the rod describe ellipses, (La Hire theorem), while the rod envelopes a tetracuspid (?); this mechanism is at the base of the ellipse-tracer of Archimedes.

When one of the curves is a line and the other is a circle, we get the curves of the slider-crank mechanism and, more generally, when one is a line and the other is a conic, we get the polyzomal curves.

Website to visit:

See also the L-system with three bars and the four-bar linkage curve.
The points on the cyclist's leg describe (in the frame linked to the bike), curves of the three-bar linkage. The points on their thigh describe mere arcs of circles! The feet of this athlete describe curves of the three-bar linkage. The machine is called elliptic bike, but where are the ellipses?

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© Robert FERRÉOL, Alain ESCULIER 2017