A growing four-bar reaches none of its departing double points

Three discs decide the real double points found the curve that bounds the region where a growing four-bar’s three departing double points are all real. The four-bar there has its crank and rocker growing together, at a ratio $\lambda$ of rocker to crank, until it becomes the elliptic trammel. On its way it loses the three finite double points every four-bar’s coupler curve owes. Whether one or three of them are real is decided by the sign of one polynomial, quadratic in $\lambda^2$, and its envelope is the circle with the rod as its diameter.

A real double point can be either of two things. At a crossing two real configurations of the machine meet, and the coupler point passes through it twice in a cycle. At an isolated point the two configurations that meet are a complex-conjugate pair, so the curve’s equation has a double point there but the machine never arrives. A point the machine never reaches found one on the standard crank-rocker. The essay on the discs ended by asking where, inside the three-real region, a real departing point stops being one kind and becomes the other, expecting a second boundary.

There is no second boundary in the limit. The machine reaches none of its departing double points: all of them, wherever they are real, end isolated. What does have a boundary is the size at which each one stops being reachable, and that has a law of its own.

Why the machine cannot reach a point a crank’s length away #

The machine is the one a four-bar becomes a trammel along a family of laws built. The crank, of length $t$, is pivoted at $(-t, 0)$. The rocker, of length $\lambda t$, is pivoted at $(0, -\lambda t)$, so both pivot circles pass through the origin, where the two slides of the trammel cross in the limit. A rod of length $b = 3$ joins the pins, and the coupler point sits at $w$ in the rod’s own units. That essay found the three departing double points at distances of order $t$ from the origin, at positions $tZ$ given by the roots of one quartic.

The two pivot circles meet twice: at the origin, and at a second point, $Z_s = -2i\lambda/(1+i\lambda)$ in units of $t$. That is $(-1, -1)$ when the two links are equal. A real configuration has both pins on their circles and within a rod’s length of each other. When the circles are $t$ across and the rod is 3, that is possible only where the circles come within 3 of each other, which is near the two places they meet. So the machine’s real curve sits in two small patches, one about the origin and one about the second meeting, each about a rod’s length across. In units of $t$ the patches shrink to points as the machine grows. The opening figure shows them closing up on the dotted circles’ two meetings while the double points stay where the law put them.

A departing double point is a distance of order $t$ from both patches, so no real configuration reaches it. That is the whole of the limiting argument, and it settles the question asked: in the limit every real departing double point is isolated, inside the three-real region and out.

The same essay explains how the points are reached at all. A configuration that puts the coupler point a crank’s length from the machine’s pivots must have its rod lying along an isotropic direction, $x + iy = 0$ or $x - iy = 0$. Those are the lines through the circular points, and a real rod never lies along them. The configurations that reach a departing double point are complex, and at a real double point they come as a conjugate pair.

How fast each point goes complex #

The limit settles what happens eventually. The machine on a page has a definite size, so the useful question is how fast each point leaves.

At a double point of a four-bar’s coupler curve, the two equations a configuration must satisfy collapse into one. Written in the rod’s orientation, $(\cos\varphi, \sin\varphi)$, it is a line in the plane of orientations. The configurations reaching the point are where that line cuts the unit circle. If the line passes within one of the circle’s centre it cuts it twice, at two real orientations: a crossing. If it passes further away it misses, the two orientations are complex: an isolated point. Exactly at one it touches, the two orientations coincide, and the double point is a cusp. Call that distance $d$.

For a departing point $d$ grows in proportion to the machine’s size. The two pins share their isotropic coordinate $Z$ to leading order, because both are within a rod’s length of the coupler point. Their conjugate coordinates $Z'$, though, differ by an amount of order one in units of $t$, so the rod’s orientation $e^{i\varphi}$ is of order $1/t$. Working that difference out from the two pivot circles gives

$$Z'_B - Z'_A = \frac{Z\,(1+i\lambda)(Z - Z_s)}{(Z+1)(Z+i\lambda)},$$

and so

$$\frac{d}{t} \;\to\; D = \frac{\sqrt{1+\lambda^2}\,|Z|\,|Z - Z_s|}{2b\,|Z+1|\,|Z+i\lambda|}.$$

The step from the orientation to $d$ is short. Write the orientation as $z = e^{i\varphi}$, so that $\cos\varphi = (z + 1/z)/2$ and $\sin\varphi = (z - 1/z)/2i$. A line at distance $d$ from the centre then meets the unit circle where $|z| = d \pm \sqrt{d^2 - 1}$. While $d < 1$ that is a pair of numbers of modulus one, two real angles. Once $d > 1$ it is a number and its reciprocal, neither of modulus one, and for large $d$ the smaller is about $1/2d$. So $d$ is, to leading order, half the reciprocal of how far the reaching configuration’s rod has been turned off the real circle of directions. A rod whose orientation has modulus $b/(t\,|Z'_B - Z'_A|)$ puts the line at $t\,|Z'_B - Z'_A|/2b$.

$D$ is a ratio of four distances. On top are the departing point’s distances from the origin and from the pivot circles’ second meeting; underneath are its distances from the crank’s pivot and the rocker’s. It vanishes only when a departing point sits on the second meeting, which is where one of the machine’s real patches lives.

Each of the four distances has a reading. A departing point near the origin or near the second meeting leaves slowly, because the machine’s real patches sit at those two places and a configuration there is nearly real. A departing point near either pivot leaves fast, since those two distances stand underneath and the rate grows without bound as either shrinks; the reason is not obvious from the machine, and the formula is the only account of it here. The factor $\sqrt{1+\lambda^2}$ is the distance between the pivots, and the rod’s length divides it all: a longer rod reaches further, so every point leaves its reach more slowly.

For the coupler point the opening figure uses, the three departing points leave at $D = 0.477$, $0.278$ and $0.158$. The measured distances fall onto those lines and stay there. At a crank of 100 the law is within 0.82% of the measured $d/t$, at 1,000 within $8\times10^{-5}$, and at $10^4$ within $10^{-6}$: the error shrinks as one over the size squared, the rate at which the departing points themselves close on their law. That holds at every real departing point of seven coupler points at ratios of 0.3, 1 and 3, 39 points in all.

Below the line $d = 1$ the figure shows the crossings. All three points of this machine are crossings at a crank of 2. Two become isolated by a crank of 3, through a cusp each. The slowest, the one with $D = 0.158$, holds on until a crank of $7.36$, against the law’s one-over-$D$ of $6.34$. From then on the machine passes through none of its double points.

Three routes to the same classification #

Two of the classifications above come from the same construction, so a third that shares nothing with them is worth having.

The first route is the line of orientations. The second is the sign of the coupler curve’s Hessian at the point: negative at a crossing, where the curve’s two branches are real, and positive at an isolated point, where they are not. The equation a four-bar satisfies fitted that sextic, and the Hessian is read off the same polynomial, independently of the line. It agrees on all 208 real points. The third route asks the machine directly. The rod is turned through 7,200 angles and refined by golden section. At each angle the crank pin is placed so that the coupler point sits on the double point, and the residuals of the two pivot circles are measured. At every crossing a real configuration closes both circles to within $2\times10^{-14}$. At every isolated point the best a real rod can do misses by at least $0.026$, and by at least $0.18$ at a crank of 100 and $0.23$ at 1,000. Seventy-three of the 208 points are crossings, all at cranks of 10 or less. None is a crossing at 100 or at 1,000.

The last crossing, across the coupler’s plane #

Every coupler point has a size at which its machine loses its last crossing. It was found here by scanning the crank’s length from half a rod to 2,000 in logarithmic steps and bisecting the last step that had a crossing.

No cell keeps a crossing for ever, and none goes without one. At equal crank and rocker the median size is $7.0$ and nine cells in ten are done by $9.9$, so a machine with a crank three times its rod is, for most coupler points, past every crossing it will ever have. The slow cells gather on the circle with the rod as its diameter, darkest at its top and bottom. The slowest of the 252, at $(0.40, -0.50)$, holds a crossing until a crank of $199$: its departing point sits at $(-1.012, -1.010)$, under two hundredths of a crank from the pivot circles’ second meeting, where the rate law’s numerator is nearly nought.

Across ratios the picture keeps its shape and changes its scale. On a coarser 12 by 10 grid the median last crossing is $16.2$ cranks when the rocker is 0.3 of the crank, $7.1$ when they are equal and $4.9$ when it is 3, with the slowest cells at $89.8$, $36.6$ and $27.1$. A short rocker makes the machine slower to let go.

A cusp leaves when the law says, and a pair does not #

There are two ways the last crossing can go, and they turn out to answer to different laws.

A crossing can turn isolated by passing through a cusp. The point stays real, $d$ passes through one, and the rate law applies: the measured size over its prediction runs from $0.89$ to $1.24$ when the rocker is 0.3 of the crank, $0.82$ to $1.19$ at equal lengths, and $0.89$ to $1.24$ at 3. Wherever the exit comes past a crank of 15 it is within a tenth. The law describes $d$ only when the machine is large, $d = Dt + O(1/t)$, and the remainder is what the early exits feel. The slow ones, which matter most, it gets almost exactly: $199$ measured against $200$ for the slowest cell above.

Two crossings can instead meet, collide, and leave together as a complex pair, so that the number of real double points falls from three to one. Then the rate law says nothing, because it describes points that stay real. This is the commoner exit: 192 of the 252 cells at equal lengths, and between 90 and 92 of 120 on each of the coarse grids. Those cells’ exits come later than the rate would suggest, by up to $5.5$ times.

The two exits divide the plane along the boundary the three discs found. A pair exit happens only where two of the three departing points are complex in the limit, because only there can two real points become complex together. In every cell inside the three-real region, 20 of the 252 at equal lengths, the last crossing leaves through a cusp. Outside the region either exit can occur: the surviving real point can go by a cusp, or the doomed pair can go by colliding.

For anyone drawing these machines the practical consequence is that a traced curve and a contour plot of the curve’s equation disagree, and the disagreement grows with the machine. A drawing made by turning the crank shows the two real patches and nothing else. A contour plot of the sextic shows the patches and, in principle, three more points far from them, where the equation is singular and no configuration ever goes. A point the machine never reaches found that a contour plot cannot find even one such point, because the sextic does not change sign there. Here every departing point is of that kind once the crank is a few rods long, so the equation’s picture of the machine and the machine’s own picture diverge in exactly the region where the double points live.

What the departing points were, all along #

The double point a slide cannot keep found that the double point a four-bar loses as its rocker grows into a slide is always real and always isolated. The trammel takes both double points found the slider-crank’s pair real and isolated inside the one-rod disc. Both observations now have one explanation. A point that leaves for infinity in the machine’s own units is reached only by configurations with their rod along an isotropic line, and those are never real, whatever limit carried the point away.

The answer to the question the three discs left is therefore short. Of the three real points inside the region, the machine reaches none, eventually. At every finite size it may reach some. What decides how long it does is a ratio of four distances, and the one place where that ratio is nought is the second place the pivot circles meet.

The two kinds of double point are not two regions of the coupler’s plane in this family. They are two stages of the same point’s history. Branches were components all along treated a four-bar’s configuration space as the thing whose pieces the words circuit and branch name, and a crossing is where two arcs of that space’s real part are mapped to the same place. A growing machine’s real configuration space shrinks onto two patches while its double points stay where the algebra puts them, so every crossing is eventually left behind.

What the reach measurement does not settle #

The cusp’s own law. The size at which a crossing becomes a cusp is found here by bisection, and it is predicted from the large-size slope to within a quarter. An exact expression would need the next term of $d$, the next order of the expansion that gave $D$. That term was not derived.

The pair exit’s size. Where two crossings collide, their collision is a finite-size version of the discriminant the three discs found in the limit. That finite-size discriminant, which would predict the pair exits as the rate law predicts the cusps, was not computed. The figure shows its exits arranged in strands, which suggests it has structure worth finding.

A finite grid. The map’s slowest cell is a cell, not the slowest point. Near the coupler points whose departing point lands exactly on the second meeting the last crossing grows without bound, and no grid can find the largest. Those coupler points are a curve, the preimage of $Z_s$ under the departure law, and it was not drawn.

The rod fixed at 3. Everything scales with the rod: the law has $2b$ in its denominator, so a rod twice as long halves every rate and doubles every exit size. That was checked only through the formula.

Still open: the coupler points that never let go #

The rate law vanishes exactly where a departing double point lands on the pivot circles’ second meeting, the one place a real patch of the machine’s curve survives at every size. For a coupler point whose departing point lands there, the crossing never needs to leave: the machine’s own real patch travels with it.

The distinct argument would be that curve of coupler points found and its crossings followed. The law gives each departing point’s position as a root of one quartic, so asking that a root equal $Z_s$ is one more polynomial condition on $w$. That traces a curve in the coupler’s plane for each ratio. The measurement would trace it, check that its points keep a crossing at every size tried, and ask what the machine does there in the limit: whether the curve’s real patch about the second meeting carries a true crossing that survives into the trammel, or a tacnode where two branches only touch.