The curve the other assembly draws

The curves field has an equation for the site’s standard crank-rocker. It was fitted to six hundred and two traced positions and then derived exactly, and both essays explained why the tracer solves the linkage on both of its assemblies rather than one: a coupler curve is one closed curve and a mechanism covers it in two pieces, one per branch, with the two meeting at the limit positions.

For the linkage those essays were written about, that sentence is false. The way it fails is more useful than the sentence was, because it says something about the equation that nothing else about coupler curves here has said: the curve an equation describes can be larger than anything one machine draws.

Each assembly closes on an oval of its own #

Split the traced positions by the assembly each was solved on, and each half is a closed curve by itself.

The measurement is simple. The linkage has ground 4, crank 1, coupler 3.5 and rocker 3, with the coupler point at u = 0.45 along the coupler and v = 0.5 off it. It is solved at every half degree of crank angle, 720 angles, once from a guess on each side of the line joining the crank pin to the rocker pivot. Both assemblies assemble at all 720. So on each assembly the coupler point is a continuous function of a crank angle that goes all the way round, and a continuous function of a full turn has to come back to where it started. There is no half-curve here and no join. There are two complete ovals.

They are not close to touching. At their closest the two ovals are 1.712 apart, on a linkage whose ground is 4. All 1,440 solved points satisfy the eliminated sextic to 6.6 × 10⁻¹⁶ of the polynomial’s largest term, so both ovals are on the one curve. Neither is an approximation to it or a neighbour of it.

That makes the real part of this sextic two ovals, and a built machine draws exactly one of them. A drawing made by driving a real crank-rocker round and marking the coupler point is a drawing of half the equation’s curve, and it gives no hint that the other half exists.

The six hundred and two points the fit was given are 301 on each oval, which is why the fit worked. What was wrong was the description of what they were.

The same four bars, the same crank angle, a different machine #

An assembly is decided when the last pin goes in. With the crank held, the rocker pin must be a coupler’s length from the crank pin and a rocker’s length from its pivot, and two circles meet at two points, mirror images of each other across the line joining their centres. The essay on the four-bar introduced the pair, and every sweep here carries the previous solution forward so that a drawing never jumps between them.

At each crank angle, then, the coupler curve has two points: one on each oval. As the crank turns, both points move and neither ever crosses to the other oval. A sweep of one assembly through a full turn finds a whole oval and nothing of the other, and a sweep of the other assembly finds the other oval and nothing of the first.

The metrology field met the same pair as a trap. One set of lengths, two machines recovers a four-bar’s lengths exactly from a calibration run and then finds the other assembly out by 268°, because nothing in the lengths says which assembly was built. Here the ambiguity takes a different form. The lengths determine the curve, and the curve contains both machines’ paths.

That changes what “the coupler curve of a four-bar” should mean. If it means the path a built machine’s coupler point follows, it is one oval and depends on how the pin was put in. If it means the zero set of the equation the lengths determine, it is both ovals and does not. The essays on coupler curves have used the second meaning for their algebra and the first for their drawings, and the sentence that has now been corrected was the place where the two were assumed to agree.

Nothing the crank does brings one oval to the other #

For a machine to pass from one oval to the other without being taken apart, its two assemblies would have to coincide somewhere, at a configuration belonging to both, so that the motion could leave by the other side. Two assemblies coincide exactly when the two circles the rocker pin must lie on are tangent, which means the distance from the crank pin to the rocker pivot is either the sum of the coupler and rocker or their difference.

For the site’s crank-rocker that distance runs between 3 and 5 as the crank turns, the ground less the crank and the ground plus it. The sum of coupler and rocker is 6.5 and their difference is 0.5. The distance never reaches either, so the circles are never tangent, the assemblies never coincide, and the two ovals are separated by the geometry rather than by a margin somebody could shave.

The figure measures this directly rather than arguing it. Both assemblies are solved at every whole degree and the distance between their rocker pins is recorded. For the crank-rocker it never drops below 4.132 and rises to 5.686. Nothing about the turn of the crank brings the two machines anywhere near each other.

The triple rocker drawn beside it (ground 4, crank 3.2, coupler 3.4, rocker 3.3) is the contrast. Its crank pin can get as far as 7.2 from the rocker pivot, and the coupler and rocker together reach only 6.7, so there is a range of crank angles where the linkage does not assemble at all, and at each edge of that range the two circles are tangent. It assembles at 274 of 361 whole degrees. At the last whole degree before an edge its two rocker pins are 0.485 apart, and at the edge itself they are the same pin.

That edge is a dead centre, and it is where the old sentence’s two pieces genuinely meet.

Where two assemblies make one curve #

On the triple rocker the joined configurations form a single circuit of 1,094 solved positions, and it uses both assemblies. The coupler point runs out along one assembly as the crank swings towards a dead centre, arrives at the configuration where the two assemblies are one, and comes back along the other assembly as the crank swings away again. The two halves of the traced curve are the two assemblies, and they meet exactly twice.

So the corrected sentence describes this linkage accurately, with one word changed: the pieces meet at the dead centres, not at the limit positions. The difference in wording is probably where the error came from. A crank-rocker has limit positions too, the two places its rocker reverses, and they are real, measurable and important for a quick-return mechanism. But a limit position of the rocker is a reversal of one link inside one assembly. It is not a place where two assemblies coincide, and nothing crosses from one to the other there.

A triple rocker’s dead centres are places the crank cannot go past. A crank-rocker’s limit positions are places the rocker turns back. Calling both “limits” allowed a description of the first to be written beside a drawing of the second.

The practical content is the same in both cases, and worth stating plainly. A triple rocker driven slowly from one dead centre to the other and back really does draw its whole coupler curve, provided whatever drives it can carry it through the dead centre onto the other assembly. That takes inertia, a second input or a guide, as the change-point discussion in the essay on Grashof’s condition notes for a parallelogram. A crank-rocker can be driven round forever and will draw half its curve.

Grashof’s condition decides how many pieces #

Two linkages are two examples, and the claim is about a class, so it is checked over a census. One hundred and sixty four-bars are drawn with the ground fixed at 4, the other three lengths between 0.6 and 4, a coupler point anywhere near the coupler, and nothing within 0.05 of a change point. Each is solved on both assemblies at 240 crank angles, and its solved configurations are joined into circuits by the same rule as above: neighbouring angles on one assembly are joined, and where an assembly stops existing it is joined to the other assembly at the same angle.

All 66 crank-rockers came back with two circuits, all 22 double rockers with two, and all 72 triple rockers with one. No exceptions, and none near one: the number of ovals a coupler curve has is the number of circuits its linkage has, and the classification Grashof published in 1883 predicts it from four lengths.

The double rockers show that “one oval per assembly” is itself too simple. A double rocker’s input cannot turn fully, so each of its circuits uses both assemblies, joined at dead centres as in a triple rocker, and yet it has two circuits. Measured on one of them, the input swings between 28.5° and 55.5° on one circuit and between 304.5° and 331.5° on the other, two mirror-image arcs that share no angle. Its two ovals are separated by where the input is, not by how the pin was put in. Assembly and circuit coincide only for a crank-rocker.

The reason the count is one or two is not a fact about sampling. Branches were components all along showed that a four-bar’s configurations form connected loops and that Grashof’s condition is the test of how many. The coupler point carries each loop to a closed curve in the plane, so the curve has as many ovals as the configuration space has loops. And no two circuits of one linkage drew the same oval: in every two-circuit linkage of the census, some point of the second circuit is at least 0.670 from the whole of the first. That is what makes the count of ovals and the count of circuits equal rather than merely related.

The census had no double cranks in it. With the ground fixed at 4 and nothing longer than 4, the ground is almost never the shortest link. That class is predicted, not measured.

The equation knows the oval the machine never drew #

Two ovals of one equation should be recoverable from each other. If the sextic is determined by the lengths, then a sextic read off one oval should vanish on the other, which is a sharper test than the elimination because the fit knows nothing about lengths, assemblies or the second oval.

Each oval is taken alone, 300 solved points, and fitted for the degree-six polynomial that vanishes on it, on one normalisation common to both. Each fit is decided by a gap: the smallest singular value sits below the next by a factor of 1,964 on the first oval and 2,504 on the second. On its own oval each fitted polynomial vanishes to 3.5 × 10⁻¹⁵ relative to its largest term. On the other oval, which it has never seen, the first leaves 3.5 × 10⁻⁴ and the second 5.1 × 10⁻⁵. For comparison, the sextic of a coupler point moved 0.05 further along the coupler, a genuinely different curve that passes near both, leaves 0.225 on the same points.

So one oval predicts the other to within about four decades of the difference between two neighbouring curves, and each fitted polynomial agrees with the exactly eliminated one coefficient by coefficient to 8.3 × 10⁻⁶ and 4.1 × 10⁻⁶. The residual on the unseen oval is not at the rounding floor, and it should not be. A fit pins its coefficients best where its points are, and the second oval is 1.7 away from every one of them; evaluating there is extrapolation, and it degrades the way extrapolation does.

The fit had to be done with some care, and the care is part of the finding. The site’s usual route to the smallest singular direction goes through the matrix multiplied by its own transpose, which squares every singular value. With two ovals spread across the plane that is harmless. With one oval alone, on a normalisation shared with the other, the next-smallest singular value is only about 4 × 10⁻¹⁴ of the largest. Squared, it sits a dozen decades below anything double precision can resolve, level with the true zero, and the first attempt returned a polynomial that vanished on its own oval and missed the other by a relative 2.9. Decomposing the matrix itself keeps small singular values to relative accuracy, and the gap of about two thousand came back. The earlier essays’ claim that half an algebraic curve determines all of it is true; it needed the right instrument to be measured on the half that is actually half.

Two ovals that cross #

Separate ovals need not be far apart, and they need not avoid each other. The census found the two ovals crossing in 11 of its 66 crank-rockers and 2 of its 22 double rockers.

The linkage drawn has ground 4, crank 1.4, coupler 2.5, rocker 3, and its coupler point at u = 0.3, v = −0.6. Its two ovals cross twice. At the crossing near (0.368, −1.759) the machine is solved twice: on one assembly at a crank angle of −16.4°, on the other at −140.0°. Both put the coupler point on the crossing to 1.5 × 10⁻¹⁵. Two different machines, at two different crank angles, share a point.

A point of a curve that two configurations share is a double point, and that is the next object to take up. Two things about it are worth noticing here, while the ovals are still the subject.

The first is that a crossing of ovals is not a crossing of paths in any sense a built machine experiences. The machine on one assembly passes through that point once per turn and never meets the other machine there, because the other machine does not exist unless somebody builds it. The crossing is a fact about the equation. It is not an event in the motion.

The second is the ring. This linkage’s curve has a third double point, at (3.771, 0.555), and neither oval passes through it. It is a point of the sextic in the plane, real, and unvisited by any configuration of either assembly. What such a point is, and why every coupler curve has at least one real double point whether or not its ovals cross, is a point the machine never reaches.

The correction, and what survives it #

The sentence that has been wrong appeared in two essays and in the reasoning of the tracer. It is now corrected in the equation a four-bar satisfies and in three linkages, one equation. Three things those essays said about the traced curve deserve to be sorted, because they fail differently.

Solving both assemblies was right, and for a better reason than the one given. The two assemblies do not draw two halves of one curve. On a crank-rocker they draw the two ovals of one equation, and a fit given only one would still have found the sextic but would have described a curve half of which was never shown to it.

“One closed curve covered in two pieces meeting at the limit positions” was false for the crank-rocker and true, with “dead centres” substituted, for a triple rocker. The cognate the tracer was actually written for, a double rocker, is neither: its two circuits each use both assemblies, and its curve is two ovals separated by input angle.

“Half of an algebraic curve determines the whole of it” was true, and is now measured rather than assumed: one oval of 300 points recovers the eliminated sextic to 8.3 × 10⁻⁶ and predicts the other oval to 3.5 × 10⁻⁴.

None of the numbers in either earlier essay changes. The 602 positions are 301 per oval, the agreement of the three cognates is still 5 × 10⁻⁷, and the degree is still six. What changed is what the drawing was of.

What this essay does not establish #

The census is a sample. One hundred and sixty linkages agreeing with Grashof’s classification is evidence that the joining of configurations into circuits works. It is not a proof that circuits and ovals correspond for every four-bar; that rests on the classification, which the essay on configuration space describes.

The circuits are assembled from solved positions at a fixed spacing. Two circuits closer together than one step, as they are near a change point, could be joined by mistake, and that is why the census stays 0.05 away from every change point. Linkages on a change point are excluded here and are the subject of a sextic that comes apart, where the two circuits are two factors of the polynomial and touch.

No double crank was sampled. That no two circuits of one linkage draw the same oval was observed in every linkage examined and is not proved here.

What comes next #

The ovals open three questions, and each needs its own measurement.

The points two ovals share, and the point neither reaches. A crossing of ovals is a double point, and a coupler curve’s finite double points are fixed in number. A point the machine never reaches counts them by construction rather than by search, and finds that the only real double point of the site’s standard crank-rocker is one no assembly visits.

The linkages whose two circuits are two polynomials. On a parallelogram chain the two circuits meet at change points, and the sextic is no longer one irreducible curve with two ovals but a circle times a quartic. A sextic that comes apart divides it exactly.

A synthesis defect as a question about ovals. The synthesis field rejected 1,065 of 1,176 syntheses because the positions they were built to reach could not all be reached without taking the linkage apart. For a crank-rocker that should be the same statement as prescribed coupler points falling on both ovals of one sextic, and for a double rocker it should not, since its two circuits each use both assemblies. That can be tested on the same syntheses, oval by oval, and would turn a check the field runs on every candidate into a statement about which of the equation’s curves each prescribed point lies on. It has not been run, and the expectation stated here is a hypothesis.