The equation a four-bar satisfies

This field has been asking, for six phases, how many configurations a mechanism has: two for a four-bar, forty for a Gough platform, and the difference between the real count and the complex one. The unknowns have always been joint coordinates and the polynomials have been the mechanism’s own constraints.

This rung turns the object round. The polynomial here is not the mechanism’s constraints; it is the equation of the curve its coupler point draws, in the plane’s own coordinates, and the question is what degree it has.

What is being asked #

A four-bar’s coupler point traces a closed curve. Classically that curve is a sextic: a polynomial of degree six vanishes on it, and no polynomial of lower degree does.

That statement can be derived. Write the loop closure, treat the coupler point’s coordinates as knowns and the crank angle as an unknown, and eliminate — the resultant of the resulting system is a polynomial in $x$ and $y$ alone, and counting its degree gives six.

It can also be measured, which is what happens here, and the two are different activities with different failure modes.

Fitting an equation to a set of points #

Trace the coupler point. At each of six hundred and two solved positions, record where it is. Now ask which polynomials vanish on those points.

A polynomial of degree at most $d$ in two variables has $\binom{d+2}{2}$ coefficients — twenty-eight at degree six. Build a matrix with one row per point and one column per monomial; a polynomial vanishing on every point is a vector in that matrix’s null space.

There will not be an exact null space, because the points carry the solver’s residual. So the question becomes: is there a direction the matrix nearly annihilates, and is it clearly separated from the others.

That second half is the whole measurement. A least-squares fit always returns an answer — it returns the smallest singular direction whether or not one exists in any meaningful sense — so a residual on its own establishes nothing. What establishes something is the gap.

Degrees four to seven #

Degree four, fifteen monomials: nothing near zero.

Degree five, twenty-one monomials: worst residual $2.7\times10^{-4}$, smallest singular value $1.0\times10^{-3}$ against a next-smallest of $1.9\times10^{-3}$. Less than a factor of two between them, which is no gap at all — the fit is returning the least-bad quintic and there is nothing distinguished about it.

Degree six, twenty-eight monomials: worst residual $7.0\times10^{-12}$, smallest singular value $3.3\times10^{-8}$ against a next of $9.3\times10^{-5}$. A factor of $2.8\times10^{3}$. One direction, clearly separated, annihilated by every point.

Degree seven, thirty-six monomials: also finds a vanishing direction, and it has to — any multiple of the sextic by a linear factor vanishes on the same points. Nothing is learned, and nothing should be.

The answer is six, read off the gaps, and it is the textbook answer arrived at by measurement.

Degree seven, and the trap it would have been #

The degree-seven row is the one a careless version of this measurement would report as a discovery, so it is worth saying exactly what it contains.

At degree seven the fit finds a vanishing direction with a residual as small as the sextic’s, and it must: multiply the sextic by any linear polynomial and the product vanishes on every point of the curve. So the null space at degree seven is at least three-dimensional — the sextic times $1$, times $x$, times $y$ — and the fit returns whichever member of it the eigenvector routine happens to hand back.

A measurement that read degree seven fits as well as degree six as evidence of anything would be reading an inevitability. What the row is for is the opposite: to show that the machinery does not keep improving when given more freedom, because there is nothing left to improve on.

The right quantity at degree seven would be the dimension of the near-null space rather than one residual, and it would come back as three. That is a stronger check than the one run here and it is not run; the four rows as reported establish the degree by the gap at six and the absence of one at five, which is enough for the claim being made.

Why the gap and not the residual #

A point worth dwelling on, because it is the same discipline this site applies to numerical rank everywhere and the reason is the same.

A rank is a decision about which quantities are zero, and a decision needs a separation. The pairs field’s surface census reports the ratio of the smallest singular value kept to the largest discarded on every row, and a row reading $10^{15}$ is not near being reclassified by anybody’s tolerance. A row reading $10^{2}$ would be — and that is exactly how a bug was found there: a cone came back with two freedoms instead of one, with a clean integer and a plausible classification, and the only thing wrong with it was that its rank gap was a hundred instead of $10^{15}$.

A numerical rank without its own error bar is an assertion. Here the gap at degree six is $2.8\times10^{3}$ and at degree five it is $1.9$, and those two numbers are the finding.

The same instrument, on a curve that is not a sextic #

A determination is worth more when the instrument has been shown to say something else on a different input, so it is worth running the fit on a curve whose degree is known for another reason.

The computing field’s machines trace curves whose polynomials are given rather than discovered — a lemniscate is degree four by construction, a folium degree three. Fitting a traced arc of one of those returns its own degree with its own gap, and returns nothing at degrees below it.

That is the control the coupler measurement needs. Without it, an instrument that always reported six would report six here and be right by accident; with it, the instrument is known to report whatever degree it is shown, and its answer on a curve nobody had an equation for is evidence.

A measurement that has only ever been run on cases with known answers is a calibration, not an instrument. The coupler curve is this one’s first real use, and the compiled curves are its calibration.

Two routes, and what each is good for #

The classical derivation and this measurement answer the same question and neither makes the other unnecessary.

Elimination gives the degree exactly, as an integer, for every four-bar at once, and gives it as a theorem. What it does not give is the coefficients for a particular linkage without doing a great deal of symbolic work, and it says nothing about a curve that arrives as a set of points from somewhere else.

The fit gives the coefficients of one particular curve directly, from points, with a confidence attached. It does not prove anything about four-bars in general and it works on any sampled curve at all, including ones no mechanism produced.

The site’s standing habit is to have both wherever both exist, and this is a case where they are complementary rather than redundant: one is general and abstract, the other is specific and numerical, and each catches errors the other cannot.

What a curve’s equation is good for #

Having the polynomial rather than the points changes what can be asked, and three of the changes are worth naming because they are why this rung exists at all.

A part determines the whole. Fit on the arc a double-rocker cognate covers — fifty degrees of input, a hundred and sixteen points — and the polynomial that comes back is the same one the full trace gives, to seven digits. So the fit reconstructs branches the mechanism cannot reach without disassembly, which no amount of tracing can.

Two curves can be compared exactly. Comparing sampled traces is comparing where two mechanisms went, and two different curves that run close over the sampled arcs pass that test. Comparing coefficient vectors is comparing curves, and that is what makes the cognate check beside this one a test of Roberts’s theorem rather than of a sampling.

A curve becomes an input. Once a curve is a polynomial it can be handed to something that takes polynomials, and the computing field takes polynomials and returns linkages. Without the fit there is no way to feed a mechanism’s own curve back into a construction, and the round trip at the end of this essay would not exist.

What the fit needs to be given #

Three choices, each of which changes the answer if made badly.

Normalisation. The points are centred and scaled to unit radius before the matrix is built. Without that, a sixth power of a coordinate of size three is a column two thousand times the size of the constant column, the smallest singular direction is decided by the units, and the fit reports something about metres.

Both branches. On this crank-rocker the two assembly branches draw two separate ovals of one sextic, and a built machine draws only one of them. Handing the fit one branch gives it one oval; it would still find the sextic, because one oval of an algebraic curve determines all of it, and the conditioning would be worse for no reason.

Enough points. Twenty-eight coefficients need more than twenty-eight points, and rather more than that for the high-degree columns to be well determined. Six hundred is comfortable; the shortest cognate trace in the neighbouring rung is a hundred and sixteen and is what limits that comparison’s agreement.

The elimination this rung does not do #

It is worth being precise about the route not taken, since the essay claims two routes are complementary and only walks one of them.

The four-bar’s loop closure is two equations in the coupler point’s coordinates and the two link angles. Substituting half-angle tangents turns them into polynomials; eliminating the two angles by resultants leaves one polynomial in $x$ and $y$, and its degree is six. That is a symbolic computation of a size the algebra field’s existing machinery could carry — the field already computes characteristic polynomials in exact integers and takes resultants of the Freudenstein equation.

It was not done here, and the honest reason is that it is a different piece of work rather than a harder one: an exact elimination produces a polynomial whose coefficients are expressions in the four lengths, and comparing those with the twenty-eight numbers this fit returns needs a normalisation reconciling a symbolic result with a numerical one on scaled coordinates.

A second route quoted but not run is not a second route. The claim in this rung rests on the fit alone, and the derivation is cited as the classical result rather than reproduced as a check. Closing that is the obvious next rung on this ladder and it is recorded as open.

What the coefficients look like #

A last concrete note, because twenty-eight coefficients stays abstract otherwise.

The fit returns them as a unit vector, so the polynomial is determined up to scale — which is right, since $p$ and $2p$ vanish on the same set. The sign is fixed by making the largest coefficient positive, a convention without which two runs of the same fit would agree up to an overall sign and a comparison would report a difference of two on a third of them.

On the site’s standard example the coefficients are of very unequal size: the sextic terms carry most of the weight and several of the low-degree ones sit near $10^{-3}$. That spread is not noise. It says the curve is close to being describable by its top-degree part alone over most of its extent, which is what a curve with a distant centre looks like after centring, and it is why the normalisation matters as much as it does.

A coefficient near zero in a fitted polynomial is not evidence that the monomial is absent. Establishing absence would need its own error bar, taken from the same singular-value structure that decides the degree, and this rung does not compute one.

What this buys the algebra field #

The field’s existing machinery is about counting solutions: how many configurations satisfy a mechanism’s constraints, by Bézout, by monodromy, by continuation. Every one of those treats the mechanism as the polynomial system.

This is the first machinery in the field that treats a curve as the polynomial. It is a small addition and it opens the direction the computing field runs the other way: there a polynomial is the input and a mechanism is compiled from it, and the two together make one round trip.

The round trip is worth stating because it is the sharpest thing in either field. A four-bar traces a sextic; that sextic, handed to the compiler, produces a machine of hundreds of bars. Four bars and hundreds of bars draw the same curve exactly, and the difference between them is that one was found and the other was constructed.

Why a curve arriving as points is the normal case #

It would be reasonable to think that fitting an equation to a mechanism’s own curve is a roundabout way to get something the mechanism’s equations already contain, and for a four-bar that is nearly fair. The reason to build the machinery anyway is that the four-bar is the easiest case in the subject and almost nothing else is like it.

A six-bar’s coupler curve has a much higher degree and an elimination that nobody does by hand. A cam profile arrives as an envelope, computed point by point, with no closed form at all. A conjugate tooth flank is generated from its mate by a construction that produces coordinates and not an equation. In each of those the curve is a set of points because that is how it was made, and asking what polynomial it satisfies is a question with no symbolic route to it.

So the fit is general-purpose machinery arriving in the field where it can be checked against a known answer. That it agrees with the classical sextic on a four-bar is what licenses using it on the cases where nothing is known — and this rung is deliberately the easy one, for that reason.

The bit that is not established here #

The classical description is that a coupler curve is a tricircular sextic — passing through the circular points at infinity with multiplicity three, which is what distinguishes coupler sextics from general ones and is why a general sextic is not a four-bar’s curve.

Nothing in this rung tests that. The degree is measured and the tricircularity is not, because it is a statement about the curve’s behaviour at infinity and this measurement is made entirely in the plane the mechanism moves in. A projective version of the fit would answer it, and would be a different measurement with a different normalisation.

It is worth adding what the untested half would look like as a measurement, so the gap is a stated piece of work rather than a shrug. Homogenise the fitted polynomial to three variables, restrict to the line at infinity, and check that the resulting binary form has the circular points as roots of multiplicity three. That is a root-multiplicity question on a degree-six form in one variable, which is a computation this field’s machinery already does; what it needs is a fit whose high-degree coefficients are determined well enough for a multiplicity-three root to be distinguishable from three nearby simple ones, and whether the $10^{-8}$ singular value here is good enough for that is not obvious either way.

It is left recorded as a gap. The consequence of the untested half is the interesting one: most sextics are not coupler curves, which is why the synthesis field’s problem is hard and why a compiler that draws any sextic at all is doing something a four-bar cannot.

The elimination, which is shorter than its reputation #

Everything above establishes the degree by fitting: polynomials of degrees four to seven are least-squared against a traced curve, and the answer is read off the gaps. That is a measurement, and it can only ever report a degree it was asked about. The classical elimination was cited beside it and not reproduced, which is a second route quoted rather than run.

Run, it takes three lines. Put O₂ at the origin and O₄ at (g, 0), let the coupler’s direction be e = b(C, S) with C² + S² = 1, and put the coupler point at (u, v) in the coupler’s own frame:

A = P − u e − v e⊥
B = P + (1 − u) e − v e⊥

The machine is exactly |A|² = a² and |B − O₄|² = c². Both look quadratic in the orientation and neither is: their quadratic parts collect into b²(u² + v²) and b²((1−u)² + v²), which are constants the moment C² + S² = 1 is used. So the two circle conditions are linear in the coupler’s orientation, eliminating it is Cramer’s rule, and imposing C² + S² = 1 on the result gives

(β₁γ₂ − β₂γ₁)² + (α₂γ₁ − α₁γ₂)² − (α₁β₂ − α₂β₁)² = 0

with α and β linear in x and y and γ quadratic. Degree three squared is six; the determinant’s square only reaches four. The sextic falls out of the degree count as an integer, with no fit, no traced points and nothing that could have been a degree somebody happened to ask about.

What the fit could and could not have told anybody #

Having both routes makes a question about the first one answerable, and it had been an open one: the fit returns twenty-eight numbers with a singular value at 10⁻⁸ under them, and nothing in that says whether the high-degree coefficients are pinned or are noise it was free to put anywhere.

They are pinned. Put both on one normalisation and every coefficient agrees to about three parts in ten million, with the degree-six terms among the better ones rather than the worse. The fit is excellent and the doubt about it was unfounded.

It still could not have established what the next essay establishes. The exact leading form’s odd coefficients are identically nought; the fitted ones are 1.3 × 10⁻⁷. A curve whose leading form were a perfect cube plus a genuine perturbation of that size would be a different curve of a kind no four-bar draws, and no fit to six hundred traced points tells the two apart. A statement about a multiplicity is a statement about an exact zero, and a residual is the wrong instrument for one — however small it is.