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):  where  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 (C_A) and (C_B) 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 (C_A) (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 (C_A) 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.

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: php.math.unifi.it/archimede/archimede/curve/geomeccan0.php?id=8

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