The mesh inside keeps the half

Every rational gear ratio has a degree built a geared five-bar’s curve as a Laurent polynomial in one variable. Write the ratio as $p : q$ in lowest terms, the input angle as $q\varphi$ and the output as $\beta_0 - p\varphi$, and put $w = e^{i\varphi}$. The two crank pins’ isotropic coordinates are then Laurent monomials in $w$; eliminating the pin against a line leaves one polynomial $F(w)$; and the degree of the curve is the span of $F$’s exponents, which is $4\max(p, q) + 2\min(p, q)$.

It ended with a list, and the first item on it was a different machine:

With an internal mesh the second crank’s angle is the offset plus $p\varphi$, so its pin’s $Z$ carries $w^{p}$ and its partner $w^{-p}$, and both crank pins’ $Z$-coordinates would grow together as $w$ does, as a pin machine’s do. The circularity instrument would say whether a co-rotating gear keeps the half that a counter-rotating one loses.

It does, and the answer is cleaner than the question expected: half, exactly, at every ratio.

One sign #

The change to the machine is one gear pair. Two gears side by side turn against each other; a pinion inside a ring turns with it, and so does a crank driven by a belt rather than by a mesh. Either arrangement gears the two cranks of a five-bar together, and the second gives the output angle $\beta_0 + p\varphi$ where the first gives $\beta_0 - p\varphi$.

The change to the algebra is the same sign in one exponent. Where the external mesh has

$$B = d + r_2 e^{i\beta_0} w^{-p}, \qquad \bar B = d + r_2 e^{-i\beta_0} w^{p},$$

the internal mesh has the two exponents swapped, so that both crank pins’ $Z$-coordinates carry positive powers of $w$ and both conjugates carry negative ones. Nothing else in the elimination changes: the same two circle conditions, the same subtraction, the same Cramer’s rule, the same single polynomial at the end.

The mechanism’s own position formula is written with the ratio in it as $\beta_0 - k\alpha$, so the internal mesh is the same machine with a negative $k$ — which is convenient for computing and is also the honest statement of how small the difference is.

It is worth being clear what is and is not a new mechanism here. The geared five-bar is a five-bar chain with two freedoms and a gear taking one away, and everything in that description survives the change of mesh: the count and the rank still agree that one freedom is left, the motion still closes exactly when the ratio is rational, and the machine still comes back to where it started after $q$ turns of its input. What changes is the shape of the closed path, and the shape is what an algebraic curve is.

The same degree #

The first question is whether the degree law survives, and it does, without alteration.

At 1 : 1 both are 6; at 2 : 1 both are 10; at 3 : 2 both are 16; at 8 : 5 both are 42. Every one is $4\max + 2\min$, and the count is the number of genuine roots of $F$ — roots that give a pin satisfying both circles and the line, not merely a root of the polynomial.

That is not obvious in advance. The span argument that produces the law reads the exponent ranges of $F$’s two factors, and swapping a sign moves both ranges. It happens that they move so as to leave the total span alone, which is worth stating as a fact rather than as a consequence: two machines that are not the same machine, whose polynomials are not the same polynomial, draw curves of the same degree at every rational ratio.

And a different curve #

The degree is what a general line sees. A line through a circular point sees something else, and that is where the two machines part.

The circular points are the two points at infinity that every circle passes through, and how many times a curve goes through them is its circularity — the quantity the four-bar’s tricircular sextic is named for, and the one measured there by counting a line’s finite meetings and subtracting from the degree.

The external mesh’s circularity is twice the larger of the two tooth counts: 2, 4, 6, 6, 10, 10, 16 at the seven ratios. As a share of the degree that is $2\max/(4\max + 2\min)$, which depends on the two counts separately and wanders — 33.3% at 1 : 1, 42.9% at 3 : 1, 37.5% at 3 : 2, 38.1% at 8 : 5.

The internal mesh’s circularity is 3, 5, 7, 8, 12, 13, 21. Every one is exactly half the degree.

A curve of even degree $n$ cannot pass through a point more than $n/2$ times and still meet a general line in the remaining points, so half is the ceiling, and the internally geared machine is at it at every ratio while the externally geared one never is. The flat line in that figure is not flat to a tolerance: the share is the ratio of two integers and it is $1/2$ in every row.

The two circular points give the same count in every case, which they must for a curve with real coefficients and is checked rather than assumed. It is worth checking because the instrument can fail in a way that looks like a result: a line through a circular point makes $F$ carry a factor whose roots put the two crank pins on top of each other, and those roots satisfy the polynomial without giving a pin at all. They are counted as spurious and excluded, and the count of them is itself reported, so a circularity that came from the wrong roots would be visible rather than plausible.

Two things follow that are worth separating, because the essays below measured one of them and not the other. The degree is a count of a line’s meetings with a curve and it is blind to where those meetings are; the circularity is a count of the meetings that have gone to infinity in a particular way. So a machine can be changed in a way that leaves the first alone and moves the second, which is what has happened, and it means the degree is a coarser description of a coupler curve than it looks.

It also means a fit cannot tell the two apart. A fit finds a rational degree by watching the singular values of a Vandermonde-like matrix drop at the right count of coefficients, and both machines have the same count. Whatever a fit reports for one it reports for the other, so the sign of the mesh is invisible to every instrument used here except the one aimed at the circular points.

What the curve pays for it #

Circularity is not free. A point a curve passes through $c$ times accounts for $c(c-1)/2$ of its double points, and a curve of degree $n$ has at most $(n-1)(n-2)/2$ of them altogether. So the two circular points take $c(c-1)$ between them, and what is left is the most the drawn curve can have in the finite plane.

With $c = n/2$ the arithmetic collapses:

$$\frac{(n-1)(n-2)}{2} - \frac{n}{2}\left(\frac{n}{2} - 1\right) = \left(\frac{n}{2} - 1\right)^2 .$$

So the internal mesh’s curves are left with a perfect square every time — 4, 16, 36, 49, 121, 144, 400 — and the external mesh’s with 8, 24, 48, 75, 163, 210, 580, which follow no such pattern because their circularity is not a function of the degree alone.

That is a real difference in what the two curves can do. A double point is a place the tracing pin reaches from two different configurations, so the budget is a ceiling on how often each curve can cross itself, and the internally geared machine spends a much larger share of its allowance getting to infinity. At 8 : 5 the external mesh keeps 580 of its 820 and the internal mesh keeps 400: the same degree, and nearly a third less room for the curve to be interesting in the part of the plane anybody can look at.

Two routes on a real line #

Every count above comes from the roots of one polynomial, and a polynomial is a claim about a curve nobody has drawn. The check is to draw it.

A root $w$ of $F$ on the unit circle is a configuration the machine can actually be in, so it is a real meeting of the line with the drawn curve. Counted on the polynomial and counted on the polyline — both assembly branches, 7,200 samples a turn — the two agree in all eight rows, for both meshes.

Counting on a polyline is not free of judgement, and the judgement is the sampling. A crossing is a sign change between consecutive samples, so two crossings closer together than one sample would be missed as a pair; at 7,200 samples a turn on a curve of this size that would need two real meetings within a thousandth of a turn of each other, which the root list would have shown as two nearly equal roots and does not.

The internal mesh’s real counts are lower at the small ratios, 2 against 4 at 1 : 1 and 4 against 6 at 2 : 1, and equal at 3 : 2 and 5 : 3. That is a fact about this line and these lengths rather than about the machines: the degree is the count of complex meetings and how many are real is a separate and less stable question, which the real-count map is about.

A curve that is a shape and a curve that is a size #

There is one more reading, and it is about which of a coupler curve’s numbers a designer can choose.

A geared five-bar has a great many lengths to pick — two cranks, two couplers, a pivot spacing, a gear phase — and a ratio to pick alongside them. The degree is decided by the ratio alone: $4\max + 2\min$, with no length in it. The circularity, it now turns out, is decided by the ratio and the sign of the mesh, again with no length in it. Everything a designer actually adjusts to make the curve pass where it should — the lengths — changes nothing about either count.

That is the same division a lift is a size and a law is a shape draws in the cam field, arriving in a field where it has no obvious right to hold. A curve’s degree and circularity are topological facts about the machine’s wiring, and a machine’s lengths move the curve about inside the class its wiring put it in. The mesh’s sign is part of the wiring, which is why it moves the class and why it is the only thing on this page that does.

Why the half is the half #

The half has a reason, and it is visible in the exponents rather than in the geometry.

A curve’s circularity is how many of $F$’s roots fail to give a finite pin when the line is taken through a circular point — equivalently, how much of $F$’s span belongs to the factor that vanishes there. With the external mesh, one crank pin’s $Z$ carries $w^{+q}$ and the other’s carries $w^{-p}$, so the two pins’ positive and negative powers pull against each other and only the larger of the two survives into that factor: hence $2\max$. With the internal mesh both pins’ $Z$ carry positive powers and both conjugates carry negative ones, so the factor collects the whole of one side and none of the other, which is exactly half the span.

That is also why the external mesh’s share rises towards a half as the ratio becomes lopsided: $2\max/(4\max + 2\min)$ tends to $1/2$ as $\min/\max \to 0$, and at $\min = \max$ it is $1/3$. A one-to-one external mesh is the furthest a geared five-bar gets from being circular, and a one-to-one internal mesh is at the ceiling with a degree-six curve through each circular point three times — which is a tricircular sextic, the same description a four-bar’s coupler curve answers to.

The four-bar, recovered #

The one-to-one internal mesh deserves a paragraph of its own, because it lands on something already built.

Its curve has degree 6 and passes three times through each circular point: a tricircular sextic. That is the exact description of a four-bar’s coupler curve, which this site fitted and checked years of essays ago, and it is the classical characterisation of the class.

The two machines are not the same — a five-bar with two cranks geared one to one and turning together is five bars and a gear pair, and a four-bar is four bars and no gear — but their curves belong to the same class, and the double-point budget agrees too: 10 in total, 6 at infinity, 4 left, which is the four-bar’s own allowance with its three finite double points sitting inside it.

So the sign of the mesh is the difference between a curve that is a four-bar’s kind of curve and one that is not, at the same degree and from the same five bars. That is a sharper way to say what circularity measures than any amount of talk about points at infinity.

What this does not settle #

The same lengths throughout. Both machines are drawn on one set of bar lengths, pivot spacing and gear phase. The degree and the circularity are counts and should not depend on those, and the essays below establish that for the external mesh; nothing here re-establishes it for the internal one.

The genus is not computed. The double-point budget is a ceiling and the actual number of double points is a measurement nobody has made here. Counting the arm curve’s own double points is a polynomial system in two copies of the machine, and the same system would settle whether the internal mesh’s curves use their square or fall short of it.

Real is not counted systematically. Eight rows on one line say the two instruments agree; they do not say how many real double points or real branches each curve has, which is the question a designer choosing between the two meshes would actually ask.

Only the convergents and a few neighbours. Seven ratios are measured — the golden ratio’s first few convergents plus 3 : 1 and 5 : 2 — and the claims are about all rational ratios. The span argument and the exponent argument are general; the measurements are not, and a ratio with a large $\min$ and a small $\max$ is exactly the region the external mesh’s share is least well sampled in.

No mechanism has been compared. An internal mesh puts one gear inside the other, which constrains the centre distance to be the difference of the pitch radii rather than their sum, so the two machines cannot be built to the same proportions at the same ratio. The curves compared here share a set of bar lengths, and the gears that produce them are different sizes.

Still open: whether the half survives a third crank #

The half came from both crank pins carrying powers of $w$ of the same sign, and that is a property of the mesh rather than of the count of cranks. A geared six-bar — three cranks, two meshes — has three pins, and each mesh can be inside or outside, so there are four machines and their pins’ exponents can be made to agree in sign or not.

Its distinct argument would be that table: for each of the four sign patterns, the degree of the curve its pin draws by the same span argument, its circularity by the same instrument, and whether “every pin carrying the same sign” is again exactly the condition for the circularity to be half the degree. Two outcomes are worth the work. If it is, then maximal circularity is a statement about the senses of a geared chain and nothing else, which would be a rule a designer could apply without computing anything. If it is not — if a third crank can be at the ceiling with mixed senses, or off it with matched ones — then the half belongs to the two-crank case and the reason given above is a coincidence of two terms rather than an argument.