What a coupler point draws
The coupler of a four-bar is the link attached to nothing fixed. It rotates and translates at once, and a point rigidly attached to it goes somewhere more interesting than a circle.
That somewhere is a coupler curve, and it is an algebraic curve of degree six. Every four-bar has a two-parameter family of them, one for each place a point could be attached, and they are startlingly varied: ovals, figure-eights, kidney shapes, curves with cusps, curves with a nearly straight segment.
Attaching a point properly
A tracing point cannot be given coordinates, because coordinates would make it a free joint and the mechanism would gain a degree of freedom it does not have.
It is specified instead in the coupler’s own frame: a fraction u along the line from A to B, and a fraction v perpendicular to it, both scaled by the coupler’s length. As the coupler moves, the point moves with it, and the two constraint equations that say so go into the same solve as everything else.
That is more than an implementation detail. It means the traced curve is the output of the same machinery as every other position on this site: 240 solved configurations, each converged, each asserted. A traced curve that jumped assembly branch part-way round would be caught, because the trace is checked for a step larger than its own sampling interval.
Why degree six
The result is Roberts’s, and the outline is worth having even without the algebra.
The coupler point’s coordinates satisfy a polynomial relation obtained by eliminating the two link angles from the loop-closure equations. Each elimination step raises the degree, and the answer comes out sextic — degree six in x and y together.
Two consequences follow that matter more than the number. A sextic can have up to three separate branches, which is why some coupler curves come in disconnected pieces that a single linkage cannot trace continuously. And a sextic has enough freedom to approximate a great many shapes, which is why four-bars were the universal solution to motion generation for two centuries.
Roberts’s other result
There is a second theorem, from the same person, that is more surprising than the first.
Every coupler curve can be generated by three different four-bar linkages. Given one linkage and its coupler point, two others — the cognates — trace exactly the same curve with entirely different link lengths.
That is practically useful rather than merely elegant. If a linkage traces the wanted curve but will not fit in the space available, or its transmission angle is poor, one of its cognates may be better proportioned and will trace the identical path.
Why synthesis is hard
The forward problem is easy: give the lengths and the attachment, get the curve. The inverse problem — give the curve, get the lengths — has no simple solution.
There are exact methods for a handful of prescribed positions. Burmester’s theory finds the linkage passing through three or four specified positions of the coupler, and the construction is classical. But asking for a whole curve rather than a few points is a different problem, and it is one where the sensitivity above becomes an obstacle rather than a feature: small changes in the attachment produce large changes in the curve, so the inverse map is badly conditioned.
The practical answer for most of the twentieth century was the atlas — Hrones and Nelson’s 1951 volume printed some seven thousand coupler curves, and a designer found the one closest to what was wanted and read off the proportions. Numerical optimisation has replaced that, and the underlying difficulty is unchanged.
The curve people wanted
Of all the shapes a coupler can trace, one was worth more than the rest: a straight line.
Before 1800 a long, true flat surface was among the hardest things to manufacture, so a slide was expensive and a pin joint was cheap. Guiding a piston rod straight without a slide was therefore a real industrial problem, and Watt solved it approximately in 1784 with a coupler point on exactly this kind of linkage.
Sensitivity, seen directly
The five curves in the opening figure come from attachment points that differ by fractions of a coupler length, and they are barely recognisable as relatives.
That is the practical character of this subject. A four-bar has five effective parameters — three length ratios and two attachment coordinates — and the map from those five numbers to the traced shape is smooth but violently nonlinear. It is why the atlas approach worked at all: nothing short of looking at thousands of examples gives an intuition for it.
Why the tracing point is attached rather than positioned
A detail of the machinery that turns out to be the whole reason these curves can be trusted.
The coupler point is specified as a fraction along the coupler and a fraction perpendicular to it — coordinates in the link’s frame, not the world’s. Two constraint equations say that the point is at those coordinates relative to the link’s two pins, and they go into the same solve as the bars.
The alternative would be to compute the point’s position afterwards, from the solved pin positions, by applying the same transformation by hand. That would give identical numbers and would put the transformation in the drawing code rather than in the constraint set — which means it would not be checked. Anything that is not in the solve is not asserted, and anything not asserted is a place a figure can go wrong quietly.
Putting it in the solve also means the tracing point participates in the mobility count, which is a useful cross-check: attaching a point must not change the mechanism’s degrees of freedom, and if it did, the attachment would be wrong.
The atlas, and what replaced it
Hrones and Nelson published Analysis of the Four Bar Linkage in 1951: about seven thousand coupler curves, printed, with the proportions that produced each one. A designer with a wanted path found the nearest printed curve and read off the linkage.
That sounds primitive and was a rational response to the problem. The inverse map — from a wanted curve to link lengths — has no closed form, the forward map is smooth but violently nonlinear, and before computers there was no way to search it except by having already computed a great many examples.
Numerical optimisation has replaced the book and has not changed the underlying difficulty. A modern synthesis tool minimises a distance between the traced curve and the target over the five parameters, and it has the problems that optimisation over a badly conditioned map always has: many local minima, sensitivity to the starting point, and solutions that fit the curve while being mechanically unusable — poor transmission angle, or requiring a linkage larger than the space available.
That last point is where Roberts’s cognate theorem earns its place: given one linkage that traces the curve, there are two others that trace it exactly, and one of them may fit.
The instant centre, and the shape of the curve
A coupler curve’s local behaviour is completely determined by where the coupler’s instantaneous centre is at that moment, and that observation explains most of the features people notice in the curves without any algebra.
At each instant the coupler is rotating about a point found by extending the crank and the rocker until they meet. Every point on the coupler is therefore moving perpendicular to its own line to that centre, at a speed proportional to its distance from it. A coupler point close to the instantaneous centre is barely moving; one far from it is moving fast.
Three consequences show up directly in the traced curves.
Cusps. A coupler curve has a cusp exactly where the tracing point coincides with the instantaneous centre — the point’s velocity is zero, and the curve comes to a sharp stop and reverses direction. Points that pass near the centre produce tight loops rather than cusps.
Near-straight segments. Where the instantaneous centre is momentarily far away and moving slowly, the tracing point travels on a large circular arc, which over a limited range is indistinguishable from a line. That is the mechanism behind the approximate straight-line linkages, stated in the language of curves rather than of error functions.
Dwells. Where the tracing point’s distance from a fixed reference is momentarily stationary to several orders, the curve has a segment of nearly constant radius, and a link pinned to it will hold nearly still while the crank keeps turning.
The instantaneous centre also explains why the curves are so sensitive. Its position is the intersection of two lines, and the intersection of two nearly parallel lines moves violently when either line rotates slightly. Near the dead centres the crank and rocker approach parallel, the centre runs off toward infinity, and small changes in proportion move the whole curve.
What the curve is used for now
Coupler curves are usually presented historically, and they are a current design tool in at least three places worth naming.
Suspension kinematics. The wheel of a double-wishbone suspension is carried by the coupler of a four-bar, and its camber, track and roll-centre behaviour through the travel are readings off the coupler curve. The proportions are chosen to shape that curve, and the design method is exactly optimisation over the four lengths and the point.
Walking mechanisms. A leg that must move forward through the air and backward in contact with the ground, at constant speed, is a coupler point with a specified curve — the Jansen linkage and the Klann linkage are both this, and both are proportions found by search rather than by construction.
Film and packaging transports. Intermittent film advance, carton folding and bottle handling all want a point that follows a specified path and dwells at one end of it. A cam does it precisely and expensively; a four-bar does it approximately and cheaply, and where the approximation is good enough the four-bar wins.
In each case the curve is the specification and the linkage is the answer, which is the inverse of how this essay presents it — and the inverse direction is the hard one, which is why it took an atlas and now takes an optimiser.
What the sensitivity means for building one
The measured sensitivity — a small proportional change in one link producing a large change in the traced curve — is usually presented as an obstacle to synthesis. It is at least as much an obstacle to manufacture, and that is the part that decides whether a coupler-curve mechanism is worth designing.
A linkage whose curve shifts visibly for a 1% change in a link length will shift visibly for a manufacturing tolerance of 1%. Machining the links to 0.1% is routine; the pin clearances that let the mechanism turn at all are frequently worse than that, and they vary with wear.
So the honest specification of a coupler-curve mechanism includes the tolerance, and the tolerance comes from the same sensitivity calculation that makes synthesis hard. The derivative of the curve with respect to each link length is the quantity that answers both questions: how hard the optimiser’s job is, and how tightly the parts must be made.
Where the sensitivity is highest — near the dead centres, where the instantaneous centre runs off to infinity — is also where the transmission angle is worst, which is not a coincidence. Both are readings of the same near-degeneracy in the geometry, one as a velocity effect and one as a force effect, and a design that avoids one avoids the other.
The general lesson is the one this site keeps arriving at from different directions. The behaviour of a mechanism is a function of its configuration and its proportions, and summarising it with a single number — a ratio, an angle, a curve name — throws away the variation that decides whether the machine works. The variation is what the sweep measures, and the sweep is what a drawn figure cannot do.
Six degrees, and what that number rules out
The coupler curve of a four-bar is an algebraic curve of degree six, and the degree is not a technicality — it settles several questions at once.
A curve of degree six meets a straight line in at most six points. It follows immediately that a coupler curve cannot contain a straight segment, because a segment meets a line in infinitely many points. That is the formal reason the straight-line problem had no four-bar solution, and why the answer required more links.
The degree also bounds the number of features. A sextic can have at most a fixed number of double points, which limits how many crossings, cusps and loops a coupler curve can display, and explains why the curves in an atlas fall into a manageable number of visual families rather than being arbitrarily complicated.
And it explains the tricircular property: the curve passes through the circular points at infinity three times each, which is the algebraic statement of the fact that a coupler curve is built from two circular motions combined. That property is what Roberts’s cognate theorem turns on — three different four-bars can produce the same sextic because the curve determines the mechanisms only up to a threefold ambiguity.
None of this is used to draw the figures, which come from solved configurations like everything else here. It is used as a check on what the figures could possibly show: a traced curve that appeared to contain a straight segment would be evidence of a bug rather than a discovery, and knowing the degree is what makes that judgement available in advance.
That is the useful role for theory on a site that measures. It does not produce the pictures, and it says which pictures are impossible, which is exactly what is needed to tell a surprising result from a broken one.