A closed form belongs to a rotation

The area a coupler point encloses splits that area into four terms. Three are closed forms: the crank pin’s circle scaled by $1 - u$, the rocker pin’s circle scaled by $u$, and the coupler’s whole turns scaled by $u - u^2 - v^2$. Each of these is exactly nought unless that member goes all the way round. The fourth is an integral of the rocker pin against the crank pin’s motion, and it has no closed form. Where three machines keep one area set a crank-rocker beside the two other four-bars that Roberts’s construction gives for its curve. All three enclosed the same area, but each kept the closed-form part in a different term.

The kind is decided before the lengths are ran the comparison on one triple of each kind. A double crank’s three machines each carried all three closed forms, and they were the same three numbers, 17.6934, 16.3426 and 0.1018, permuted between the columns. A double rocker’s two cognates kept theirs in the same column. The triple rockers kept no closed form at all. The question left was whether the double crank’s permutation was special to it, or whether it was the general fact hidden under the other sets’ zeroes.

It is the general fact, and the reason takes one substitution.

Three machines, three rotations #

Look first at the colours. Each machine has three moving members — crank, coupler and rocker — and each is drawn in one of three colours. Every machine has one member of each colour, and never the same member twice. The original’s crank is colour 1, but in the first cognate colour 1 is the coupler and in the second it is the rocker. Colour 2 is the original’s coupler and both cognates’ cranks. Colour 3 is the original’s rocker, the first cognate’s rocker and the second cognate’s coupler.

Now look at the numbers under the names. The original carries 17.693 in its crank column, 0.102 in its coupler column and 16.343 in its rocker column. The first cognate carries 0.102 as a crank term, 17.693 as a coupler term and 16.343 as a rocker term. The second has them in yet another arrangement. Each number keeps its colour: 17.693 is always colour 1, whichever member of whichever machine carries it.

So the thing that carries a closed form is not a column of the formula. It is a colour — one of three rotations in the figure Roberts’s construction draws — and each machine gives that rotation to a different member.

The substitution #

Roberts’s construction is specific about what it hands back. For an original with ground g, crank a, coupler b, rocker c and tracing point λ = u + iv along its coupler, the first cognate has cranks |λ|b and |λ|c, coupler |λ|a and tracing point 1/λ. The second has cranks |1 − λ|b and |1 − λ|a, coupler |1 − λ|c and tracing point 1/(1 − λ). Three machines, one curve drew the figure that makes these lengths. Here they only need to be put into the area formula.

Take the first cognate’s crank term. Its crank is |λ|b and its tracing point 1/λ has real part $u/|\lambda|^2$, so the term is $(1 - u/|\lambda|^2)\,|\lambda|^2 b^2 = (|\lambda|^2 - u)\,b^2$. That is exactly the original’s coupler term with its sign changed. The first cognate’s coupler term is $(u/|\lambda|^2 - 1/|\lambda|^2)\,|\lambda|^2 a^2 = -(1 - u)\,a^2$, the original’s crank term with its sign changed. Its rocker term is $u/|\lambda|^2 \cdot |\lambda|^2 c^2 = u\,c^2$, the original’s rocker term unchanged. The second cognate works the same way through $|1 - \lambda|^2 = 1 - 2u + |\lambda|^2$.

Every term of every cognate is a term of the original, up to sign, in a different column. The lengths are scaled by |λ| or |1 − λ| and the tracing point is inverted, and the two changes cancel exactly in each closed form. Checked by substitution alone, on two thousand random chains and tracing points with nothing traced, the magnitudes agree to 2.6 × 10⁻¹⁵ of the largest.

The substitution settles which value can appear where. It does not settle which values actually appear, since a closed form is nought unless its member turns. For a machine to carry the original’s crank term in its coupler column, its coupler must go round exactly when the original’s crank does. That is the colouring, and it needs checking on the machines themselves.

A census #

The census draws four-bars at random over lengths from 0.2 to 5, keeps the ones that assemble and sit clear of a region wall, and draws a random tracing point with each. It builds the two cognates, traces every circuit of all three machines, and computes the four area terms from the traced pins. Nothing in it uses the substitution.

The census covers 480 chains and all eight regions, and the table says three things.

Sorted, the three machines’ closed forms are the same three numbers, to at most 1.3 × 10⁻¹⁵ of the largest, in every region. That is the multiset agreement the question asked about, and it holds everywhere.

Paired by rotation, they are equal term by term, to the same precision. Each rotation is switched on in all three machines or in none, 2,268 times out of 2,268 — three rotations on each of 756 ovals. Nothing in the census assumed which member of a cognate carries which rotation: that was read off the substitution, and the traced pins agree with it.

Paired by column, they disagree completely wherever anything turns. In a crank-rocker’s triple the one nonzero value sits in the crank column of the original, the coupler column of the first cognate and the rocker column of the second. Compared column by column, that is a disagreement of the whole value.

It is worth being exact about what those two small numbers measure, because they are smaller than a traced area could ever be. A closed form is not integrated from the trace. Once the traced pin is seen to go round, the term is written from the lengths, as a circle’s area times a factor. So the agreement to 10⁻¹⁵ is the precision of the formula, and it would be there whether or not the substitution were right. What the census measures independently of the algebra is which members go round. That is decided from each pin’s own traced path, with no tolerance to hide behind: a pin either retraces an arc or encloses its whole circle. There are 2,268 such decisions — three rotations on each of 756 ovals — and every one agrees across the three machines. The substitution says the values match if the right members turn. The traces say the right members turn.

The column pattern in the table is therefore just the rotations read through each machine’s labels. A crank-rocker turns rotation 1, which the three machines call crank, coupler and rocker. A double rocker turns rotation 2 — its own coupler — which both cognates call crank, and that is why its two cognates kept their area in the same column. A rocker-crank turns rotation 3, which two machines call rocker and one calls coupler. A double crank turns all three. A triple rocker turns none.

The members of one colour turn together #

The colouring says more than that the terms match. It says that in the machines, the members of one colour rotate together: when the original’s crank goes round, so does the first cognate’s coupler and the second cognate’s rocker. That can be watched directly.

The three lines of each colour lie on top of one another, to within 0.07°. That is the resolution of matching sampled points between machines traced separately. Over the loop, rotation 1 gains exactly one turn and rotations 2 and 3 come back to where they started. In a crank-rocker’s triple only one rotation goes round, and every machine carries its closed form, each in the member it gives to that rotation.

The reason is the figure Roberts’s construction draws. The three machines are joined by parallelograms: the first cognate’s crank pin sits where the original’s crank, coupler and tracing point make a parallelogram with the ground pivot. So the first cognate’s coupler has a side that stays parallel to the original’s crank for the whole motion. A parallel side turns when the side it copies turns, and a closed form counts nothing but whole turns. The same parallelograms account for the other two colours. The first cognate’s crank runs from the original’s ground pivot to a point that makes a parallelogram with the original’s crank and the stretch of coupler from its crank pin to the tracing point. So that crank stays parallel to part of the original’s coupler, and it turns exactly as the coupler does: rotation 2. The second cognate’s crank is built the same way on the other side, from the stretch of coupler between the tracing point and the rocker pin, and it is rotation 2 too. The parallelogram on the rocker’s side also puts a side of the second cognate’s coupler parallel to the original’s rocker: rotation 3. The last two copies are the two cognates’ rockers, which meet at the third pivot. Each keeps a fixed angle to one of the original’s members, as the construction’s similar triangles require — the first cognate’s rocker to the original’s rocker, the second cognate’s to the original’s crank — and the plot above confirms it.

So there are exactly three orientations in Roberts’s figure that can turn independently. The nine moving members of the three machines are three copies of each, and a whole turn of any one of them is a whole turn of all its copies.

One triple of each kind, read by rotation #

The permutation is easiest to see in the double crank, and it is the same in every triple. The crank-rocker’s triple shows it with one nonzero value.

Rotation 1 is the original’s crank, the double rocker’s coupler and the rocker-crank’s output. Each machine carries the same closed form, 1.728, in the one member of that colour. The earlier essay read this as three different decompositions of one area, with the crank pin in one machine, the coupler’s turn in the next and the output pin in the third. Read by rotation it is one decomposition, labelled three ways.

This also gives a short reason for the result of the census of cognate kinds, that the kind of a machine decides the kinds of its cognates. A four-bar’s kind, as far as these regions go, is which of its members go round. Roberts’s construction hands each rotation to a fixed member of each cognate, so which members of the cognates go round is fixed by which members of the original do. The lengths within a region cannot change that, and the tracing point cannot either, since it only scales and inverts. That essay’s census found no exception in twenty-four thousand chains. The rotation picture says there could not be one.

Which cognate a motor can drive #

The rotation picture also answers a practical question about the three machines. A motor drives a member that turns about a fixed pivot: a crank or a rocker, never a coupler. A cognate can therefore be motor-driven through a full turn only if the rotation that goes round is carried, in that machine, by a grounded member.

Read by rotation, this is a lookup. In a crank-rocker’s triple, rotation 1 goes round. The original carries it on its crank and the second cognate on its rocker, so both can be driven, while the first cognate carries it on its coupler and cannot. In a double rocker’s triple, rotation 2 goes round. The original carries it on its coupler and cannot be driven, and both cognates carry it on their cranks and can. In a rocker-crank’s triple, rotation 3 goes round, on the original’s rocker, the first cognate’s rocker and the second cognate’s coupler, so two of the three can be driven. In a double crank’s triple every rotation goes round, so every machine has a grounded member that turns and all three can be driven. In a triple rocker’s triple nothing goes round and no machine can be driven through a turn.

That is the census of cognate kinds’s finding about which machines a motor can use, restated. It found the map from a machine to its cognates’ kinds by census and traced the one place it failed to close to the motor. The rotations say which member the motor would have to hold, and so which machines it cannot.

It also turns a design choice into bookkeeping. A designer who has a curve and wants it drawn by a machine driven from a fixed shaft looks at which rotation goes round and picks a cognate that puts it on a grounded member. Every triple except the triple rockers has at least one, and a double rocker — the one machine of its own triple that cannot be driven — always has two cognates that can.

What this adds to the area #

The area formula has four terms and the census compared three of them. The fourth, the mixed integral, was already known to be one number for all three machines — the earlier essay measured it — and the census confirms it in every region, to between 10⁻¹⁵ and 10⁻⁷ depending on how much of the area it carries. Put together, the whole decomposition is invariant, not only its total. Three closed forms, one per rotation, carried by different members, and one integral carried the same way by all three.

That is the stronger of the two outcomes the question anticipated. Roberts’s construction does not rearrange the area among different terms. It relabels the terms. A designer choosing between three cognates, for size, for which pivots are convenient or for which of them can be driven by a motor, is choosing which member gets which rotation, and the area bookkeeping travels with the rotation.

The triple rockers are the degenerate case of the same statement. No rotation turns, so every closed form is nought in every machine, and the whole area is the integral. They agree as multisets because every entry is nought, and the rotation picture explains that too.

What this does not settle #

The mixed integral. It is the same number in all three machines and it has no closed form. Nothing here decomposes it further, and whether it has one along special families of chains — the ones with a symmetric curve, say — is not examined.

Signs. A closed form carries the sign of its member’s direction of turning and of the circuit’s orientation. The census compares signed values, and they agree, but the signs are measured, not derived. The substitution gives magnitudes, and the parallelograms say the members turn the same way. A derivation of the signs from the construction would finish the argument.

Circuits that do not correspond. The census matches each machine’s circuit to an oval of the original’s curve by position. In every chain drawn, every oval found a circuit in all three machines. Whether a chain exists whose cognates split an oval between two circuits, or cover it twice, is not ruled out by anything here.

Still open: whether a slide keeps the rotations #

A slider-crank is the limit of a four-bar whose rocker has grown without bound. Its coupler curve has a cognate of its own, but in the limit one of Roberts’s three machines has gone to infinity and only two remain. The area formula also changes: the rocker pin’s circle becomes a line, which encloses nothing, and a slide puts part of the curve’s degree somewhere a pin cannot.

The distinct argument there would be the same question with a slide in it. Does the slider-crank and its one cognate still carry each closed form in one member of each, with members joined by a parallelogram turning together? Or does the rotation that belonged to the rocker disappear with the rocker? If it disappears, what carries its share of the area? The measurement would be the same census on offset and centred slider-cranks, with the slide’s own term in the formula.