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THREEBAR LINKAGE CURVE
The curves traced here are halfcurves of the threebar 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 threebar 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:

A curve of the threebar 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 A_{1} is the lefthand intersection point between (C_{A}) and (AB), and A_{2} the righthand one, then the point P passes by A_{1} iff and the point P passes by A_{2} 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 ; 
a = 4 ; b =2 ; c = 7 ; 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 rhomboid (a = c et b = d) 
The curve is generated by two other ways thanks to two other threebar 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.
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: quasitriangle 
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 threebar 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 nonparallel 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 ellipsetracer of Archimedes.
When one of the curves is a line and the other is a circle, we get the curves of the slidercrank 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 Lsystem with three bars and the fourbar linkage curve.


The points on the cyclist's leg describe (in the frame linked to the bike), curves of the threebar linkage. The points on their thigh describe mere arcs of circles!  The feet of this athlete describe curves of the threebar linkage. The machine is called elliptic bike, but where are the ellipses? 
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© Robert FERRÉOL, Alain ESCULIER 2017