A four-bar becomes a trammel along a family of laws

A four-bar’s coupler curve is a sextic with three finite double points, points the coupler point reaches by two different configurations, and an odd number of them real. The double point a slide cannot keep grew the four-bar’s rocker without bound until its pin’s circle flattened into a slide. One double point had to go, since a slider-crank’s quartic has only two, and it went along a law in the rocker’s units that knew nothing of the crank: the real one, because a complex one cannot leave without its conjugate. The trammel takes both double points then grew the slider-crank’s crank the same way into a second slide. The machine became the elliptic trammel, whose coupler points draw ellipses, and both remaining double points left along a second law in the crank’s units.

Both limits were taken one link at a time. A four-bar can become a trammel directly, its crank and its rocker growing together, their pivots receding at right angles to each other, until near the point where the two pins’ circles cross both have flattened into slides. All three double points must leave, and the trammel essay ended by asking how. Perhaps along a law that is neither of the two; perhaps one first on the four-bar’s law and the pair after on the trammel’s. Growing one link much faster than the other ought to recover the two one-step limits in one order or the other, and the question was whether growing them together is merely the composition of the two steps.

It is not. The direct limit is a family of laws, one for every ratio of the two growth rates, and the two one-step orders are only its two ends.

Two pivots receding at right angles #

The machine is set up so that the trammel appears at a fixed place. The crank, of length $a$, is pivoted a crank’s length to the left of the origin, and the rocker, of length $c$, a rocker’s length below it, so both pins’ circles pass through the origin. The coupler is a rod of 3 between the two pins, and the coupler point is $w = u + iv$ in the rod’s own units, $u$ along the rod from the crank pin and $v$ across it. As $a$ and $c$ grow, the crank pin near the origin runs on a nearly vertical arc and the rocker pin on a nearly horizontal one, and in the limit the two arcs are the trammel’s two slides crossing at right angles at the origin.

The two grow together as $a = t$ and $c = \lambda t$. The ratio λ is the new number. When λ is enormous the rocker becomes a slide long before the crank does, which is the first one-step limit followed by the second; when λ is tiny the crank goes first. Every finite λ is a way of taking both at once.

The double points are found as the earlier essays found them, with no reference to the limit: the three finite double points of a four-bar’s coupler curve, from the construction that locates them on the circle through the three cognate pivots. They were solved for crank lengths from 10 to 10,000 at several ratios and several coupler points.

All three leave together #

At equal rates, λ = 1, the coupler point (0.3, 0.2) has three double points, and divided by the crank they settle at (−0.220, −0.322), (−1.655, −0.352) and (−0.119, −1.384) from the slides’ crossing. All three are real. All three leave at the same rate, in proportion to the machine. None of them goes first.

That already differs from either order of the steps. With the rocker grown first, one point leaves at the rocker’s scale and the other two at the crank’s, so after both have grown at the same rate the far one sits at the four-bar’s law, (−0.328, −1.149) rocker lengths below the crossing, and the pair at the trammel’s law, (−0.405, −0.505) and (−1.595, 0.505) crank lengths from it. The direct limit’s three points are up to 0.86 crank lengths away from those. With the crank grown first it is the same comparison with the machine turned over.

The law is one quartic #

The law comes from the same device the two earlier laws used: asking what a configuration of the machine looks like when its coupler point is as far away as the machine is large. In isotropic coordinates, a point written as the pair $z = x + iy$ and $\zeta = x - iy$, scaled by $t$ to $(Z, Z')$, a real rod of length 3 has components whose product is 9, which is negligible at the machine’s scale, so one of them must be nearly nought. Either the two pins share their $z$-component, or they share their ζ-component.

In the first kind of configuration the crank pin sits on its circle, $(Z_A + 1)(Z'_A + 1) = 1$, the rocker pin on its own, $(Z_B + i\lambda)(Z'_B - i\lambda) = \lambda^2$, and $Z_B = Z_A$. The coupler point is then $Z = Z_A$ with $Z' = Z'_A + \bar w(Z'_B - Z'_A)$, and eliminating the pins leaves one curve,

$$Z' = -(1 - \bar w)\,\frac{Z}{Z + 1} + i\lambda\bar w\,\frac{Z}{Z + i\lambda}.$$

The second kind is its mirror image. A double point is reached twice, so it needs one configuration of each kind at the same point, and substituting one curve into the other gives $Z$ times a quartic. $Z = 0$ is the crossing itself. The quartic’s four roots are points of the plane in units of $t$, and three of them are the double points.

The fourth is the second place the two pivots’ circles meet. Both circles pass through the crossing, so they meet again at its reflection in the line joining the two pivots, at $(-2\lambda^2, -2\lambda)/(1 + \lambda^2)$ in units of $t$. Every configuration with both pins at that point reaches it, which makes it a solution of both conditions without making it a double point of the curve. It plays the part that the crank pin’s own crossing of the slide played in the trammel’s derivation, and it was found to be a root to $10^{-14}$ at every coupler point and ratio tried.

Against the solved double points the law’s three are exact in the limit, with a remainder that falls as one over the machine’s size squared. For four coupler points, two inside the one-rod circle and two outside it, the gap in crank lengths runs from $10^{-2}$ or $10^{-1}$ at a crank of 10 down to between $1.6\times10^{-8}$ and $1.3\times10^{-7}$ at 10,000, and every line has slope −2.000 on log axes. So each double point is at the machine’s size times the law plus a remainder that shrinks like one over the size. That is the trammel law’s rate, one power better than the four-bar’s law.

The two orders are the family’s ends #

The law depends on λ, and the one-step orders sit at its two extremes.

At a ratio of ten thousand one of the three roots is ten thousand times further out than the other two. Divided by the ratio it sits at (−0.3283, −1.1491), against the four-bar’s law, (−0.3283, −1.1491): this is the rocker growing into a slide first. The other two sit at the trammel’s law to $10^{-4}$, at the crank’s scale, symmetric through the crank’s pivot. The hero figure shows the same thing continuously. Divided by $a + c$, which keeps both ends in view, the three points spread across the plane at intermediate ratios, and as the ratio runs to either end one point settles on a cross while the other two run into the crossing.

How unequal the arms must be for the one-step picture to serve is a number the law gives directly. With the rocker ten times the crank, the pair is still 0.12 crank lengths from the trammel’s law and the far point 0.036 rocker lengths from the four-bar’s; at thirty times, 0.041 and 0.012; at a hundred, 0.0125 and 0.0036; at a thousand, 0.0012 and 0.0004. Both gaps fall as one over the ratio, about 1.2/λ crank lengths for the pair and 0.36/λ rocker lengths for the far point. So the one-step picture is good to a hundredth of a crank only when one arm is more than a hundred times the other, and at the proportions a real pair of guide arms would have it is the family, not either end of it, that says where the double points are.

The other end is the same statement read backwards. The law at λ is exactly the law at 1/λ for the coupler point $1 - \bar w$ — the same point described from the rocker pin’s end of the rod — reflected across the diagonal and scaled by λ. Measured, the two agree to $5\times10^{-15}$. Swapping which link grows faster is the same as turning the machine over and reading its coupler from the other end, so the crank-first order is the rocker-first order of the mirror machine.

So the composition question has a clean answer. The direct limit is the composition of the two steps only at its ends, where one link grows infinitely faster than the other. At every finite ratio the three double points leave at one rate and along positions that neither step contains, and they move continuously from one end of the family to the other as the ratio changes. The paths that leave found that how a machine’s algebra goes to infinity depends on which link grows. Here it depends on how fast each grows relative to the other, and the answer is a continuous family, not a choice between two.

Which of the three are real #

The four-bar always has an odd number of real double points: one, or three. At the limit, how many of the three departing points are real depends on the coupler point, and in the direct limit it depends on the ratio as well.

At a ratio of 100, three real double points fill most of the disc one rod from the crank pin. That is what the ends predict: the far point is real by parity, as the double point a slide cannot keep found, and the trammel’s pair is real inside that disc. At 0.01 the shaded region is the disc one rod from the rocker pin, by the symmetry. At equal rates it is neither. It shrinks to an oval about the rod itself, reaching between 0.50 and 0.59 rods from the rod’s midpoint, and covers 7.7% of the window against 23.8% at either end. Nowhere is the count even: every coupler point has one or three, at every ratio.

So three real departing double points are rarest when crank and rocker grow together. At the coupler point (−0.5, 0.8), inside the crank pin’s disc, the trammel’s own order makes all three real, and the direct limit agrees only once the rocker grows a thousand times faster: at equal rates, and even at a ratio of a hundred, only one is real.

The one-rod circle loses its meaning on the way. The trammel essay found the slider-crank’s pair at infinity at every crank length for coupler points one rod from the crank pin, because its law’s denominator vanished there. In the direct limit that circle is ordinary. At ratios of 0.3, 1 and 3 the coupler points (0.6, 0.8), (−0.6, 0.8) and (0.8, −0.6) on it have three finite double points in units of the machine, and the solve meets the law to $10^{-7}$ at a crank of 10,000. The circle is special only at the end of the family where the crank grows infinitely faster than the rocker.

The point that stays at the crossing #

One circle stays special throughout.

The trammel moves every point of the circle with its rod as diameter along a straight line through the slides’ crossing, and the trammel essay found one of the slider-crank’s double points settling on that crossing instead of leaving. The direct limit keeps it. For coupler points on that circle at 50°, 70° and 130° round it, and ratios of 0.3, 1 and 3, the law has a root at the crossing exactly, and the solved double point nearest the crossing approaches it with slopes from −1.001 to −0.994 on log axes: it settles at one over the machine’s size. And the quartic has its spurious root there twice, so a second double point sits exactly on the pivot circles’ second meeting. On this circle only one of the three is free to leave.

A trammel’s straight-line path is, as an algebraic curve, a line counted twice, singular all along its length, and the trammel essay read the staying point as the machine’s curve approaching a curve that is all double points. In the direct limit the same reading holds with a third point added, and that point is pinned to the one other place the two pivots’ circles agree on.

The law knows only the point and the ratio #

The derivation dropped the rod’s length at the step where the rod’s isotropic product became negligible, and the census tests that. Ninety machines, each with a random rod length, ratio and coupler point, solved at a crank of 10,000, meet the law to at most $1.1\times10^{-5}$ crank lengths, with a median of $2\times10^{-7}$. The gaps are largest at the smallest ratios, where the rocker is only a few hundred rod lengths long and the remainder, which scales with one over the shorter link, is largest. Neither the rod nor anything else moves the three points by more than that remainder.

What a designer sees #

No one builds a four-bar with a crank and a rocker each a hundred times its coupler, but every trammel built with two pivoted arms in place of its slides is one, and the arms are rarely equal. A point the machine never reaches is what most of these double points are: isolated points, or complex ones, far from the curve the coupler draws. What the family adds is that where they are, and how many are real, depends on the arms’ ratio as well as on the coupler point, continuously, with the one-step pictures correct only for very unequal arms. And on the circle with the rod as its diameter, where a trammel draws straight lines, a real crossing of the curve sits at the slides’ crossing for every ratio, so a nearly straight path from two long arms of any proportion folds back on itself there.

What the family does not settle #

Other angles between the slides. The pivots recede at right angles, which is what makes the limit the elliptic trammel. Pivots receding at another angle give a trammel with oblique slides, whose coupler points still draw ellipses; the law would change, and the reflection of the crossing in the line of the pivots would no longer be where it is.

Unequal powers. Both links grow linearly in $t$. A rocker growing as $t^2$ against a crank growing as $t$ is a path to the trammel that is neither a finite ratio nor one of the two ends, and whether it simply follows the large-ratio end has not been measured.

The oval. At equal rates the region of three real double points is an oval about the rod. Its boundary was located only by bisection along rays from the rod’s midpoint, and nothing here says what curve it is.

Still open: the curve that bounds the real triples #

The reality maps show the region of three real double points shrinking from the crank pin’s disc to an oval about the rod and growing back out to the rocker pin’s disc as the ratio runs from one end of the family to the other. At its ends the boundary is a circle a rod’s length from a pin. In between it is not a circle at all.

The distinct argument would be that boundary found exactly. Two of the three double points become real together where the quartic has a repeated root, so the boundary is where the quartic’s discriminant vanishes, a curve in the coupler’s plane for each ratio. The measurement would compute that discriminant in closed form, check it against the bisected boundary, and ask whether it passes through fixed points for every ratio — the two pins, the points where the two discs’ circles cross — as the circles at the ends both pass through one pin each. If it does, the family of boundaries would be a pencil of curves with a description as short as the family of laws.