Three linkages, one equation

Roberts’s theorem is one of the pleasant surprises of the subject: given a four-bar and a coupler point, there are two other four-bars, with different lengths and different fixed pivots, whose coupler points trace the same curve.

The site has drawn that. What it has not done is test it in a way that could fail for the right reason, and this rung does.

What the existing check establishes, and what it does not #

The cognates essay traces all three linkages and compares the traced points, measuring the distance from each point of one trace to the polyline of another. On the site’s standard example that comes back at the solver’s floor, and it is a real check: it caught a comparison that had been made at the nearest sampled crank angle and was wrong by $10^{12}$ on two correct cognates.

But it is a comparison of samples. Two mechanisms sampled at three hundred positions each, and the claim is that every point of one is near the other’s polyline. That is a strong statement about where the two mechanisms went and a weaker one about the curves they are on: two different curves that happen to run close together over the sampled arcs would pass it, and a mechanism covering only part of its curve makes the arcs the thing being compared.

The stronger statement is that one polynomial vanishes on all three.

Fitting an equation to a trace #

A coupler curve is an algebraic curve of degree six — a fact the neighbouring rung establishes by measurement rather than by quotation. So there is a polynomial of degree six, with twenty-eight coefficients, that vanishes on it.

Finding it is least squares with no right-hand side. Build a matrix with one row per traced point and one column per monomial of degree at most six; the coefficient vector wanted is the one that makes every row’s dot product zero, which is the eigenvector of the normal matrix with the smallest eigenvalue.

Two details make it a measurement rather than a fitting exercise.

The points are centred and scaled first. A sixth power of a coordinate of size three is a column two thousand times the size of the constant column, and without normalisation the fit would be about the units rather than about the curve.

The decision is made by a gap, not a residual. A curve that genuinely satisfies a polynomial of this degree leaves one singular direction near zero and the next one far from it. Here that gap is a factor of $2.8\times10^{3}$, which is what makes the answer a determination rather than a best effort.

Where the cognates come from, restated #

The construction is worth having in front of the measurement, because the measurement’s independence from it is the whole argument.

Take a four-bar with ground $g$, crank $a$, coupler $b$ and rocker $c$, and a coupler point at a complex position $\lambda$ along the coupler. Roberts’s construction says: put a third fixed pivot at $\lambda$ of the way along the line between the two existing ones, and build two more four-bars on the two segments that creates, with every length scaled by $|\lambda|$ and $|1-\lambda|$ respectively and the tracing point placed at the reciprocal position along each new coupler.

That is a formula in complex arithmetic, four lines long, and it produces the lengths. Nothing in it mentions curves, points traced, or polynomials.

The three linkages it produces are then handed, separately, to a solver that positions them from their own bar lengths. Each one’s coupler point is recorded. The cognate arithmetic is upstream of everything and appears nowhere downstream of it, which is what makes the agreement of three independently fitted polynomials evidence about the construction rather than an echo of it.

The common normalisation, which is load-bearing #

Here is the part that has to be got right and would be easy to get wrong.

Each cognate covers a different portion of the shared curve — the original is a crank-rocker and sweeps a great deal of it, and the first cognate is a double-rocker whose input link swings through about fifty degrees. So the three traces have different centroids and different extents.

Normalising each trace by its own centroid and scale would put the three polynomials in three different coordinate systems. They would then disagree, in coefficients, while describing the same curve — and the disagreement would be about the normalisation and nothing else.

So the centre and scale are computed once, over all three traces together, and every fit uses them. What is then compared is three polynomials in one coordinate system, and a difference between them is a difference between curves.

Which points the fit is given #

One more choice, and it is the one that decides how much the measurement is worth.

A four-bar’s coupler point is traced by driving the crank and solving at each position, and this site’s tracer does that on both assembly branches. On the original, a crank-rocker, the two branches do not draw two halves of one closed curve: each draws a whole oval of its own, the two ovals never meet, and both are the zero set of the same sextic.

Handing the fit only one branch would give it one oval of the curve. It would still find the sextic, because one oval of an algebraic curve determines the whole of it, and the fit’s conditioning would be worse for no reason. So both branches go in, and the trace for the original is six hundred and two points, three hundred and one on each oval.

The cognates get the same treatment, and it is why the first cognate contributes a hundred and sixteen points rather than the two that a naive full-turn sweep of a double-rocker returns. A trace is what the mechanism can reach, not what one sweep of one branch happened to find, and the difference between those two was worth a correction when it was found.

The result #

Three linkages. Three independent traces, of six hundred, a hundred and sixteen and a hundred and fourteen solved positions. Three separate fits, each finding its own smallest singular direction. Twenty-eight coefficients agreeing to $5\times10^{-7}$.

That is not the solver’s floor and it should not be: a least-squares fit to sampled data on an arc has a conditioning of its own, and the cognate covering the smallest arc has the least to constrain its high-degree coefficients. The agreement is limited by the shortest trace, not by the arithmetic.

Two mechanisms drawing near curves would pass the old test #

To see why the equation is a stronger statement, it is worth constructing the case the trace comparison would let through.

Take two mechanisms whose curves agree over the arc both of them cover and diverge elsewhere. Every point of one trace is near the other’s polyline, so a distance comparison over the sampled arcs reports agreement at whatever the solver’s floor is. The curves are different objects and nothing about the comparison could say so.

That is not a hypothetical worry in this subject. A coupler curve has branches the mechanism cannot reach without disassembly, and two four-bars agreeing on one branch and differing on another is exactly the shape of a circuit defect — which the synthesis field spends a rung on, and which disqualified 1,065 of 1,176 otherwise-correct syntheses.

The equation does not have that blind spot. A polynomial fitted on an arc describes the whole curve including the parts nothing traced, so two mechanisms whose polynomials agree agree everywhere, on branches neither of them can reach.

What the twenty-eight coefficients are #

It is worth being concrete about the object being compared, because twenty-eight coefficients is easy to read as an abstraction.

A polynomial of degree at most six in two variables has one constant term, two linear, three quadratic, four cubic, five quartic, six quintic and seven sextic — twenty-eight in all. The fit returns them as a unit vector, so the object is a point on a twenty-seven-dimensional sphere, and two curves are the same exactly when their points coincide up to sign.

The sign is fixed by convention: the largest coefficient is made positive. Without that the three fits would agree up to an overall sign and the comparison would report a difference of two on a third of the runs, for no reason but the arbitrary orientation of an eigenvector.

Small conventions like that one are where a comparison of high-dimensional objects goes wrong, and they are worth stating because there is no way to notice from the outside that they were handled. A measurement reporting agreement to $10^{-7}$ has either fixed the sign or been lucky.

Why this is a test the theorem could have failed #

The chain from Roberts’s construction to these numbers has no shortcuts in it, and that is the point.

The construction gives three sets of lengths, computed from the original’s lengths and the coupler point by a complex-arithmetic formula. Each set is handed to the solver, which positions the linkage by Newton–Raphson on its own loop-closure equations and knows nothing about cognates. The traced points go into a least-squares fit that knows nothing about linkages. What comes out is a list of coefficients.

Nothing in the fit could produce agreement that the mechanisms did not have. If the three linkages traced three nearby but different curves, the three coefficient vectors would differ by an amount reflecting the difference, and on this normalisation that amount would not be $10^{-7}$.

That is the site’s standing habit — two routes that share no inputs — applied to a theorem rather than to a quantity.

The cognate that nearly broke the comparison #

There is a hazard in this measurement that the earlier essay records and that bites here in a different way.

The original four-bar in these figures is a crank-rocker, so its input link goes right round and it covers the whole curve in one sweep. Its cognates need not be, and the first one is a double-rocker whose input swings through about fifty degrees. Driving it from zero to three hundred and sixty degrees finds two positions out of three hundred and sixty-one.

For the trace comparison that meant a cognate that appeared to draw nothing. For the fit it means something subtler: a trace covering a short arc of a sextic constrains the fit much less well than one covering the whole of it, and the conditioning shows up in the answer. The $5\times10^{-7}$ is dominated by the shortest arc, and a version of this measurement quoting only the two long traces would report a smaller number and would be reporting less.

Quoting the worst pair rather than the best is the whole discipline here, and it is why the number is not at the floor.

Degree five is not enough, and that is checked too #

A fit at degree six that comes back small is only evidence if a fit at degree five comes back large, and it does.

Fitting the same trace with the twenty-one monomials of degree at most five gives a worst residual of $2.7\times10^{-4}$ and a smallest singular value that is not separated from the next one. There is no quintic vanishing on a coupler curve, and the fitting machinery says so rather than quietly returning the best quintic available.

That two-sidedness is the difference between a determination and a fit. The algebra field’s rung beside this one runs the same comparison at degrees four through seven and reads the answer off the gaps: nothing at four or five, a drop of ten decades at six, and nothing further bought at seven.

An assertion that a measurement finds the right answer is worth something; an assertion that it refuses the wrong ones is worth more, and it is the reason this site’s gate for the fit checks both.

What the equation says that the picture does not #

Three linkages drawing one curve is a striking picture and the equation adds two things to it.

The curve is an object with an identity independent of any mechanism. A sextic in twenty-eight coefficients exists whether or not anything traces it, and saying that three linkages satisfy the same one is a statement about them rather than about a drawing.

A part of a curve determines the whole of it. The first cognate covers a fifty-degree arc and its fit produces the same polynomial the full trace does, to seven digits — so the short arc contains enough information to reconstruct the entire sextic, including the branches the mechanism never reaches.

The second is the more interesting, and it is the beginning of an idea the compiled field is built on: a curve given as an equation is a completely different object from a curve given as a set of points. Everything the synthesis field does works with points, and gets approximation. Everything the equation supports is exact, and the passage between the two is a fit like this one.

A second use for the same machinery #

The fit built for this rung is not specific to coupler curves, and it is used twice more in this collection for a different purpose.

Handed any sampled curve, it answers what is the lowest-degree polynomial that vanishes on this, with the gap between the smallest singular value and the next as its confidence. That is a general question about a set of points, and the only kinematics in it is where the points came from.

The compiled field uses it the other way round: there the polynomial is the input and the curve is the output, and the fit is what checks that a compiled machine’s trace is on the curve it was compiled from — though for that purpose evaluating the known polynomial directly is cheaper and is what the field actually does.

Where the fit earns its place is at the junction between the two directions. A four-bar is a mechanism whose curve has an equation; a compiled machine is an equation that has been made into a mechanism. Reading one direction against the other is how the smallest linkage that draws a given sextic turns out to be four bars where a compiler would use hundreds, and that comparison needs both halves.

Where this leaves the coupler field #

The four-bar’s coupler curve has been in this collection from the beginning, described as what a point on a coupler draws and studied by tracing.

It now has an equation, measured to be degree six, shared exactly by the three cognates, and reconstructible from a fifty-degree arc of it. That is a different handle on the same object, and it connects a field built on tracing to one built on polynomials.

One thing this rung deliberately does not claim. A coupler curve is classically described as a tricircular sextic — it passes through the circular points at infinity with multiplicity three, which is a statement about its behaviour in the projective plane and is what distinguishes coupler curves from general sextics. Nothing here tests that. The fit establishes the degree and the sharing; the tricircularity is a property of the same polynomial and would need a different measurement, made in projective coordinates rather than in the plane the mechanism moves in.

It is recorded as a gap rather than glossed over, because a reader who knows the classical description will expect the word and should know that this rung has not earned it.

It also settles a question the tracing could not. The Roberts construction produces cognates by an algebraic formula, and whether the formula is right has been checked by drawing. It is now checked by an invariant of the curve, computed three times, from three mechanisms, with no cognate arithmetic anywhere in the computation.

Tricircular, and why that word is doing the work #

Every textbook calls a coupler curve a tricircular sextic, and this site has been quoting the adjective without testing it. The fit that establishes the degree says nothing whatever about behaviour at infinity: it is a statement about six hundred points in a bounded patch of the plane, and tricircularity is a statement about where the curve meets the line at infinity.

The elimination settles it, and settles it exactly rather than to a tolerance. Only the top-degree parts of each factor reach degree six, and they collapse:

top(β₁γ₂ − β₂γ₁) = (x² + y²) · 2b[(xv − yu) − (vx + ky)] = −2b y (x² + y²)
top(α₂γ₁ − α₁γ₂) = (x² + y²) · 2b[(kx − yv) + (xu + yv)] =  2b x (x² + y²)

with k = 1 − u, so that the coupler point’s own coordinates cancel to 1 in both. The determinant’s square reaches only degree four. So the whole degree-six part is

4b²(x² + y²)³

exactly, for every four-bar and every coupler point — a perfect cube of x² + y², measured on three quite different linkages including one whose coupler point is off the far side of the coupler, with a worst coefficient deviation of exactly nought.

That is tricircularity: the curve meets the line at infinity only at the circular points, and three times at each. And it is what makes the synthesis problem hard, which is the reason it is worth more than a word. A general sextic has twenty-seven coefficients to choose; a coupler sextic must have its leading form be that cube, which is four conditions satisfied identically and not by choice. Most sextics are not coupler curves, and the four-bar has no way to draw one that is not.

The amplitude is the only part that varies, and it varies in one number: 4b², the coupler length squared. Two four-bars with the same coupler and different everything else have the same leading form. Nothing else about the machine reaches degree six at all.