The curve nobody eliminates

A degree counted on a line measured a four-bar’s coupler curve by cutting it with random lines and counting the points, and it chose the four-bar deliberately. That curve already had a fit and an elimination, so a third route could be checked against two others before anybody relied on it.

This essay relies on it. The curve here belongs to a six-bar, and for it the other two routes are not available: nobody has written down its elimination, and the fit, given every advantage, cannot decide its degree at all. The slice counts it anyway, and finds eighteen.

The arm of a dwell linkage #

The machine was already built, for another reason. A dwell made from a curve took a four-bar with ground 3.2, crank 1, coupler 3 and rocker 2.6, found a stretch of its coupler curve that is very nearly a circular arc, and hung a link of that arc’s radius, 3.006, from the coupler point. The far end of that link is carried by an output link 5.170 long from a third ground pivot. While the coupler point runs along the nearly circular stretch, the far end hardly moves, and the output dwells.

The link between the coupler point and the output is the arm. As a chain the machine is a Stephenson six-bar: its two links with three joints each, the coupler and the ground, are not joined to each other directly, which is what the essay on six-bars uses to tell the two six-bar chains apart.

A point on the arm, halfway along it and 0.4 of its length to one side, traces a closed curve as the crank turns. That curve is the object here. It is the path of a point on a link whose ends are both moving, one of them along a sextic and the other along a circle, and it has none of the structure that made the four-bar’s curve tractable. The four-bar’s coupler orientation entered its two circle conditions linearly, and that is why its elimination took three lines. The arm’s position depends on the four-bar’s assembly, on the dyad’s closure and on where the coupler point is along its own sextic, and no comparable shortcut is known to exist.

Thirty-two paths, eighteen arrivals #

The slice needs no shortcut. It needs the machine’s constraints as polynomials and one line.

The unknowns are the three moving pins: the crank pin, the rocker pin and the arm’s far pin, six coordinates in all. Five conditions say the machine is assembled, each a fixed distance between two points, so each is a quadratic. A sixth says the arm’s point is on the line, and because a point on a body is a fixed combination of that body’s two pins, it is linear. Bézout’s number is 2⁵ × 1 = 32, and that is how many paths the continuation tracks.

Eighteen of the thirty-two paths arrive at finite solutions and fourteen leave for infinity. The eighteen are eighteen configurations of the six-bar with the arm’s point on the line, and they put that point at eighteen distinct places. So the curve meets the line in eighteen points.

On a real line drawn across the traced curve the difference between the drawing and the count is stark. The line visibly crosses the curve four times. It meets it eighteen times, and fourteen of those meetings are complex. A reader counting crossings on the page would put the degree at four or more, and would be off by fourteen.

Five bodies, five different arithmetics #

The number of configurations the tracker returns is not automatically the degree, and the six-bar makes that impossible to overlook. The same machine, cut by the same random line, gives a different count for every body a point is placed on, although every system has the same Bézout number of thirty-two.

The table at the head of this essay has one row per body, and each row is a small argument of its own.

The crank: 8 configurations, 2 distinct points, each drawn 4 times, so degree 2. The crank’s point traces a circle, which meets a line twice. Each crank position can be completed in four ways, since the four-bar closes on either side and so does the dyad, and all four completions put the crank’s point in the same place.

The rocker: the same, 8 configurations drawing 2 points 4 times each, for the same reason with the roles of crank and rocker exchanged.

The ternary coupler: 12 configurations, 6 points, each twice, so degree 6. The coupler belongs to the four-bar, so its point draws the four-bar’s coupler sextic, and each position of the four-bar can have its dyad closed two ways without moving the coupler at all.

The output: 12 configurations and only 2 points, each drawn six times, so degree 2. The output link swings about a fixed pivot, so its point is on a circle, which accounts for the two. The six is worth checking because nothing obvious predicts it. With the output held, the arm’s far pin is fixed, so the coupler point must lie on a circle of the arm’s radius about it, and must also lie on the four-bar’s coupler sextic. A circle and a sextic meet in twelve points; the sextic is tricircular, passing three times through each circular point where every circle passes once, so six of the twelve meetings are at infinity and six are finite. Six configurations per output position, as the tracker found without being told any of that.

The arm: 18 configurations, 18 distinct points, each drawn once, so degree 18.

The rows differ in the number of finite solutions, 8, 12 or 18, even though every system has the same thirty-two paths. The rest go to infinity, and how many do depends on which body’s point is asked to lie on the line. The tracker counts configurations; the curve has points, and the degree is the number of points. Only on the arm are the two the same.

A Watt six-bar draws a sextic twice #

Six bars do not make a curve complicated by themselves, and the site’s other six-bar shows it.

The Watt six-bar is built on the standard crank-rocker, ground 4, crank 1, coupler 3.5 and rocker 3. Its rocker is a ternary link carrying a third pin 0.62 of the way along it, and a second dyad, links of 2.4 and 1.8, hangs from that pin to a third ground pivot. The second dyad’s moving link is a second coupler, and a point on it traces a closed curve too.

The second coupler has 12 configurations on the line and 6 distinct points, each drawn twice. Its curve is a sextic, and every point of it is drawn by two configurations of the machine.

The reason is the chain’s shape. The second loop of a Watt six-bar, the ternary rocker, the second dyad and the ground, is itself a four-bar, and the only thing it takes from the first loop is the rocker’s angle. That angle is reached by two assemblies of the first loop, and the second loop cannot tell them apart. So its coupler draws an ordinary four-bar coupler sextic, and draws every point of it once for each assembly of the first loop.

A Watt six-bar’s second coupler is therefore a four-bar’s curve dressed in two more bars, and the Stephenson six-bar’s arm is not. The difference is not the number of links, which is the same, but whether the added dyad hangs from a link that turns about a fixed pivot or from one whose every point is already on a sextic. Counted by configurations alone, 12 and 18 look like a modest difference. Counted by points, it is the difference between degree six and degree eighteen.

The same count on every line #

A count on one line is one measurement, and the claim is that it belongs to the curve rather than to the line.

Each of the three curves was cut by four different random lines, each system solved from scratch with its own start system. The four-bar’s coupler gave 6 configurations and 6 points on every line, from 8 paths. The Watt second coupler gave 12 configurations and 6 points on every line, from 32. The Stephenson arm gave 18 configurations and 18 points on every line, from 32.

A special line, tangent to the curve or passing through one of its double points, would show up as a count that dropped on that line alone, because two meetings would land on one point. None of the twelve systems did, which is what a random choice of line is for.

Every point, and one curve #

Two further questions have to be answered before eighteen is a degree. Are the eighteen points all of the meetings, with none missed by the tracker? And are they on one curve, or might the arm’s path be two or more curves whose degrees add to eighteen?

The first is answered by the trace test, exactly as on the four-bar. The line is slid along a pencil of parallel complex lines, and the eighteen points are carried along with it. Along the pencil the meetings are the roots of one polynomial whose top two coefficients do not depend on the line’s position, so their sum moves as an affine function of that position, provided every root is included.

With all eighteen the sum is affine to 1.5 × 10⁻¹⁵ of its own change across the pencil, which is rounding. Leaving out any one of the eighteen bends it by between 2.1 × 10⁻⁵ and 4.0 × 10⁻³, ten to twelve decades above rounding. So the eighteen are every meeting of this line with the curve they lie on.

The second question is harder on this curve than on the four-bar, and the way it is harder is instructive. Monodromy walks the line’s constant round loops in the complex numbers and records where each witness point comes back; points carried onto one another are on the same irreducible curve. On the four-bar’s coupler six points joined into one orbit within sixteen loops. On the arm, after forty loops, seventeen of the eighteen had joined and one had never moved, from every seed and loop radius tried.

Monodromy can only join points. An orbit is certainly inside one irreducible component, but a point that no loop has happened to move looks exactly like a component of its own. The loops cannot say which it is.

The trace test can. The witness points of any whole component, taken on their own, have an affine trace, so if the lone point were a curve of degree one by itself, its trace, and the trace of the other seventeen, would each be affine separately.

Neither is. No union of orbits short of all eighteen points has an affine trace; the smallest bend among them is 6.9 × 10⁻⁵. So no proper subset of the witness points is a curve, and the arm’s path is a single irreducible curve of degree eighteen, one that the loops tried had simply not finished joining. The Watt second coupler’s twelve witness configurations and the four-bar’s six joined into one orbit each within sixteen loops.

This is a place where two instruments are needed, and each is used for the half it can do. Monodromy proves joins and cannot prove separations. The trace test proves that a set is complete and cannot say how to split it. Together they settle irreducibility, and neither would have alone.

What a fit makes of the same curve #

Against that, the fit.

It is given every advantage the four-bar’s fit did not have. The arm curve is traced at 1,600 points, twice the four-bar’s 800. The columns are products of Chebyshev polynomials on the curve’s bounding box rather than bare powers, which keeps high-degree columns from differing wildly in size. And the singular values are those of the design matrix itself, from a QR factorisation, rather than square roots of the normal matrix’s eigenvalues, which would square the conditioning and floor the answer at about 10⁻⁸. If a fit can decide this degree at all, this is the fit that can.

On the four-bar it decides emphatically. At degree five the ratio of the second-smallest singular value to the smallest is 1.49; at degree six it is 4.3 × 10¹⁰, with the smallest singular value at 4.6 × 10⁻¹⁶ of the largest; at degree seven it is back to 1.12, because a sextic times anything also vanishes. That is the site’s standing discipline, a rank decided by a gap rather than by a residual, working exactly as intended.

On the arm it decides nothing. The ratio is 7.45 at degree six, 12.37 at ten, 14.07 at fourteen, 1.44 at seventeen and 1.36 at eighteen, the degree the curve actually has.

The smallest singular values are more alarming than the ratios. At degree fourteen the smallest is already 1.1 × 10⁻¹⁵ of the largest, which is rounding. A reader going by the residual, as the field’s first fit was built not to, would conclude that a polynomial of degree fourteen or less vanishes on the arm curve, and would be wrong: it has degree eighteen and nothing of lower degree vanishes on it. The fit’s matrix has become numerically singular for a reason unrelated to the curve. A hundred and twenty columns of degree-fourteen products, sampled on one closed curve, are close enough to dependent that double precision cannot separate “this combination vanishes on the curve” from “this combination is small on the curve”. By degree eighteen, with a hundred and ninety columns, every direction is in that condition and none stands out.

That is not a failure that more points would repair. The fit’s difficulty is that it asks for the coefficients, and at degree eighteen there are a hundred and ninety of them, most of which are individually tiny on the region the curve occupies. The slice never asks for a coefficient. It asks for eighteen points, each of which is a well-conditioned solution of a polynomial system with six unknowns, and the curve’s degree enters it nowhere except as the number of answers.

What the degree costs each route #

The essay on the four-bar’s slice predicted this in terms of cost, and the six-bar bears it out.

A fit’s cost is set by the curve: (d + 1)(d + 2)/2 coefficients, twenty-eight at six and a hundred and ninety at eighteen, each a column the traced points must pin down while the columns grow more nearly dependent.

An elimination’s cost is set by whether a short combination of the equations exists. For the four-bar one does. For the arm nobody has published one, and none is attempted here.

The slice’s cost is set by the machine. Five quadratic closure conditions and one linear line give thirty-two paths, whichever body is sliced. The crank’s circle and the arm’s curve of degree eighteen cost exactly the same, because the tracker’s work depends on the machine’s equations and not on what any one of its points draws.

That is why the slice is the route that generalises. A mechanism of pins and bars is always a system of distance conditions, so its Bézout number is always a power of two, and the degree of any point’s curve is one tracking run away. The equation of that curve may be out of reach for ever.

What this essay does not establish #

Eighteen is measured, not derived. It rests on four random lines agreeing, a trace test showing the eighteen points are complete, and monodromy together with the trace test showing they are one curve. That is as strong as a numerical statement about a degree can be made, and it is still probability one rather than proof. The literature’s figure for this class of Stephenson curve is also eighteen, which is agreement with a quoted result rather than a second route run here.

The curve’s equation is not produced. Nothing above yields its coefficients, its symmetries, its behaviour at infinity or its double points. The slice says where the curve meets lines and nothing about its shape between them.

One geometry. The count is made on the site’s dwell six-bar with the arm’s point at one place on the arm. A different point on the arm, or a Stephenson chain with different lengths, might have a lower degree through some special coincidence; the generic figure is not surveyed.

The real picture is open. A real line crosses the drawn curve four times in the figure. How many real meetings any real line can have is not settled by anything here.

What comes next #

How the degree splits between finite and infinite. Thirty-two paths, eighteen finite. The other fourteen leave for infinity, and on the four-bar the two that left went to the circular points. Where the arm’s fourteen go, and with what multiplicity, is the six-bar’s analogue of tricircularity. It is readable from the same tracked paths, by the directions in which they leave, and it would say how many times the arm curve passes through each circular point.

The arm curve’s singular points. A curve of degree eighteen has room for up to a hundred and thirty-six double points, and its genus is set by how many it actually has. The essay on circuits measured the six-bar’s configuration curve at genus five or seven from its branch points. Counting the arm curve’s double points directly, as the curves field did for the four-bar’s three, would connect the two and say how far the curve in the plane is from the configuration curve it is drawn from.

The dwell, read as a degree. The dwell six-bar was designed around a stretch of the coupler curve that is nearly a circle. Its output’s curve has degree two and is drawn six times per point, which is the algebraic shadow of the dwell: six configurations of the machine per output position. Whether the length of the dwell can be read off how close two of those six come to coinciding is a measurable question, and it would tie a design figure of merit to a count.