Where the machine stops being the function

Every arc in this field’s cost table is a fraction of a turn. The line manages $2.0$ radians, the lemniscate $1.3$, the quintic $0.17$ — about ten degrees.

That is not a limit on how far the sweep was asked to go. It is where the machine stops drawing its curve, and this rung is about what happens there.

A rhombus with nothing left to be #

The reflector has three configurations at which it stops being a function of its inputs, and all three come from the same quadratic.

Its far vertex must lie on the mirror at distance $d$ from the pivot and at distance $s$ from the input’s end, and squaring that gives $d^2 - 2ds\cos(\mu - \theta) = 0$: two roots, $d = 0$ and $d = 2s\cos(\mu - \theta)$.

When the mirror is perpendicular to the input, $\cos(\mu - \theta)$ is zero, the two roots coincide at the pivot, and the rhombus has collapsed. Two such angles in a turn.

When the mirror lies along the input, $d = 2s$: the input, the vertex and the output are all on one line, and the rhombus is flat. The two placements of the output merge there, because the two circles that place it are tangent.

That third one is the one a compiled machine hits, because the doubler reflects the frame direction in $\alpha$, so its singularity is at $\alpha = 0$ — and any working arc worth having passes through the driving angle being zero.

What it does, driven through #

Two lines and they do different things.

The closure residual is flat across the whole scan at $8.3\times10^{-14}$. Every bar is the length it was made, before the crossing and after it. There is no instant at which the mechanism fails to be a mechanism.

The departure — the output angle minus the angle the closed form asks for — is at $10^{-16}$ on one side and rises to order one immediately past $\theta = 0.800$, which is exactly where the mirror was put.

So the gadget goes on being a perfectly good linkage and stops being the function it was built to be, at a configuration that can be predicted in closed form and that nothing about the mechanism’s behaviour marks as special.

Why a continuation is entitled to do this #

It is worth being clear that nothing here is a numerical failure to be tuned away.

At the singular configuration the two solutions of the placement quadratic are equal. Just past it they are distinct again, and a path that arrives at a double root has two ways to leave. Which one a continuation takes is decided by the arithmetic of the step it happens to make — and both are genuine configurations of a genuine mechanism, so there is no sense in which one of them is the numerically correct answer and the other an artefact.

A tighter tolerance does not help. A smaller step does not help. The mechanism has two branches meeting at a point and a physical machine driven through that point would also have to choose, which is the whole content of a singularity and is why this site’s themes include singularities are the point.

The linkages field has the same object in its own terms: at a toggle position a four-bar’s output can go either way, and which way it goes is decided by whatever tips it. The difference is only what the branches mean. There, both branches are motions of the same mechanism doing the same job. Here, one of them computes the polynomial and the other does not.

What the working arc is, and how it is found #

The march that draws every trace in this field carries the departure as its stopping rule. At each driving angle it solves, then compares every gadget’s actual output against the angle its specification asks for, given the machine’s own two arm angles. When the worst departure passes $10^{-7}$, the march stops and records the angle.

The arc reported for each curve is what is left: the interval of driving angle over which every gadget in the machine is still computing what it was built to compute.

That is a stricter test than the curve residual and it is the right one. The curve residual is a single number for the whole machine and it can be small for the wrong reason — two gadgets off in compensating directions would leave the tracing point on the curve while the machine was no longer the machine. The departure is per-part, so it catches the first failure rather than the first failure that happens to matter.

It is also what names the culprit. A machine of four hundred bars produces a departure reading like the third translator in the ripple for term seven, 0.6 radians off, at driving angle 1.04, which is a statement about one bar in an object nobody can see.

The singularities are known in advance #

Unlike almost everything else that ends a mechanism’s motion on this site, these singularities can be written down before the machine is built.

A reflector’s are at $\mu = \theta$ and $\mu = \theta \pm \pi/2$, where $\mu$ and $\theta$ are its two inputs. Both inputs are whole-number combinations of the arm angles, known in closed form. So the driving angles at which each of a machine’s gadgets goes singular are the solutions of a handful of trigonometric equations, and there is no need to drive anything to find them.

The linkages field’s discipline is the relevant precedent: Grashof’s condition predicts from the four lengths which links rotate fully, and the site checks it by sweeping three hundred and sixty positions and seeing. Predicted and then swept — two routes, and the agreement is the evidence.

The same could be done here and was not. The arcs in the cost table are measured by marching until a departure appears, not predicted from the term angles and then confirmed. That is a real gap: a predicted arc would be a second route to the same number and would catch a whole class of error in the march that nothing currently would. It is recorded as one, and the reason it is still open is that the prediction has to account for every gadget in a machine of four hundred, which is a piece of bookkeeping rather than an idea.

Why bracing does not help #

The previous rung removed the parallelograms’ second assemblies with a bar between two midpoints, and it is fair to ask why the same trick does not remove this.

Because the two failures are different objects.

A branch is a discrete alternative: two configurations satisfying the same constraints, separated by a finite distance, one of which is intended. A constraint that is true in one and false in the other removes it, and the midpoint identity is such a constraint.

A singularity is a place where two configurations coincide. There is no quantity that is true at one and false at the other, because at that instant they are the same configuration. Any constraint that excluded the merged placement would exclude the working one too.

So the braced machine’s arc is the unbraced machine’s arc. Measured on the hyperbola: $1.43$ radians either way.

The arc does not follow the size #

The natural guess is that a bigger machine has a shorter arc, and it is nearly right and worth stating precisely because the exceptions are informative.

More gadgets means more chances to hit a singularity, so the trend is downwards. But where each gadget’s singularity sits depends on that gadget’s own mirror, which comes from the term angles, which come from the curve. The folium’s machine has a hundred and sixty bars and works over $2.0$ radians; the lemniscate’s has fifty and works over $1.3$.

Size and arc are different costs. A compiled quintic is four hundred and thirteen bars that draw ten degrees of a curve, and neither number predicts the other.

There is a mild consolation in that, and it is the honest one: nothing forbids a curve whose term angles happen to put every singularity far from the working region, and the folium is close to that. What is not available is a way to arrange it, because the term angles are decided by the polynomial.

A machine’s arc is not its curve’s extent #

One confusion worth heading off, because the two quantities sound alike.

The working arc is an interval of the driving angle — how far the crank turns before a gadget gives way. The portion of the curve drawn is what the tracing point covers in the plane, and the relation between them is the arm’s geometry rather than anything about the compilation.

Those come apart badly near the arm’s own dead positions. When the two arm links are nearly aligned the tip moves quickly for a small change of $\alpha$; when they are nearly folded it barely moves at all. So an arc of a tenth of a radian can cover a respectable piece of a curve or almost none of it, depending on where the machine started.

This field quotes arcs rather than curve lengths because the arc is the property of the machine and the covered length is a property of the machine and its starting point together. Both are in the figures — the traces show what was covered — and only one is in the table.

What lies past the arc #

The machine does not stop at the end of its arc. It goes on turning, and it goes on drawing.

Past the crossing the tracing point is on some other curve — a perfectly definite one, since the machine past the crossing is a perfectly definite mechanism computing a perfectly definite different function. Which curve is not asked here, and it would be a reasonable question: the machine past a flipped doubler is computing $p$ with one term’s frequency changed, which is another polynomial, and the traced set is where that vanishes.

Nothing here pursues it, and it is recorded as a gap rather than implied away. What is established is the negative half: the arc is where the machine is the machine, and outside it every number the field quotes is about something else.

The same failure, three fields apart #

It is worth putting this beside the two other places on this site where a mechanism passes through a configuration and comes out changed, because the three are the same event with three different consequences.

A four-bar at a toggle. Locked and still moving: the output link’s direction is momentarily undetermined and the mechanism can continue either way. The consequence is a machine that sometimes goes backwards, and the repair is a flywheel or a second input.

A parallel platform at a singular pose. One command, six answers: the platform gains a freedom its legs cannot control. The consequence is a machine that is unsafe there, and the repair is to keep out of the region.

A compiled machine at a flat rhombus. The consequence is a machine that goes on running smoothly and computes a different function, and there is no repair.

The third is the odd one out, and the reason is what the machine’s output is. A four-bar’s output is a motion, and a motion that went the wrong way is visibly the wrong way. A platform’s output is a pose, and a pose that is uncontrolled is felt immediately. A compiled machine’s output is a number that nothing measures — the value of a polynomial at a point — so there is nothing to notice.

A singularity is dangerous in proportion to how little its consequence is observed. That is the general lesson, and it belongs to the constraint field as much as to this one.

What this costs the theorem #

Universality says every algebraic curve is traced by some linkage. This rung says the tracing happens on an arc, and it is worth being exact about what that does to the claim.

It does not falsify it. The standard statements of the result are careful about this: the linkage traces the curve in a neighbourhood, and the neighbourhood is where the construction’s gadgets are non-degenerate. What is measured here is the size of that neighbourhood on nine specific machines, which is a thing the theorem does not say and which turns out to be small.

A theorem about what exists says nothing about how much of it is available, and that is the shape of most of what this field has to report. The linkage exists, is enormous, and draws ten degrees of the curve. All three are true and only the first is in the literature.

The gate, and the failure it requires #

The check this field runs on its own instruments is two-sided, and this is where the second side lives.

It requires a gadget driven through its singularity to leave its closed form — a scan in which nothing departed would mean the scan had not reached the singularity, or that the gadget had no singularity, or that the departure measurement was not working — and it requires the closure residual to stay below $10^{-9}$ throughout.

That second requirement is the interesting one. It is an assertion that the closure fails to notice, at a stated level, and it breaks if somebody later changes the solver in a way that makes the two distinguishable. An assertion that a measurement works is worth something; an assertion that it fails in exactly the stated way is worth more.

Whether a longer arc could be arranged #

Given how short the arcs are, it is fair to ask whether anything could be done about them, and the answer separates into two parts.

The starting configuration is chosen and could be chosen better. The compiler already scores candidate starts by how far the machine’s parallelograms are from flat, which is a proxy for staying away from the translators’ degeneracies. It does not score for distance from the reflectors’ singularities, and it could: those are known in closed form too. A start chosen to maximise the distance to the nearest singularity rather than to avoid the flattest parallelogram would give a longer arc for free.

The singularities themselves cannot be moved. They sit where the term angles put them, and the term angles come from the expansion. There is one degree of freedom available — the whole machine can be rotated, which shifts every angle by a constant — and rotating it moves all the singularities together rather than spreading them out.

So the honest position is that the arcs reported here are not the best available and are not far from it either. Better starting configurations would buy some; nothing buys a full turn, because a machine with two dozen reflectors has two dozen sets of three singularities each and they cannot all be avoided at once.

What is left #

Two rungs of this ladder remain, and between them they are the field’s accounting.

One puts the compiled machine beside the mechanisms that have actually been used to draw straight lines, measured the same way, and asks what exactness is worth against what it costs.

The other asks what universality is worth given everything this half of the ladder has measured: a construction that works, produces machines too large to build, over arcs too short to use, and is nevertheless a genuinely different kind of answer from anything the synthesis field can give.

The working arc being bounded by singularities that are known in advance is the practical content, and it turns a limitation into a specification. A compiled machine does not compute its polynomial everywhere; it computes it over an arc, and the arc’s ends are configurations where some gadget flattens. Both facts are available before the machine is built — the gadgets’ singular angles are closed-form, and the machine’s input angle at which each is reached follows from the compilation. So the honest description of a compiled machine has three parts: the polynomial, the bar count, and the arc over which it holds. Quoting the first two without the third describes a machine that works somewhere unspecified. That is the same shape as an approximate linkage quoted without its stroke and as a transmission angle quoted without its interval — a mechanism’s property offered without the range it holds over — and it is the third time this site has had to add the same missing clause.

The arc, predicted rather than walked into #

The arc above is found: the machine is marched until a gadget leaves its specification and the marching stops. That is a measurement of the arc with nothing to check it against, and the arc is computable in advance.

Every gadget in the set is a rhombus and they fail the same way. A reflector’s far vertex sits at 2s·cos(μ − θ) along its mirror; a mean’s is n₁ + n₂, of length 2s·cos(Δ/2). Both pass through nought — the vertex crossing the pivot — and at that configuration the vertex has no direction, so the gadget stops computing the angle it was built for. Every input direction is a whole-number combination aα + bβ + φ known at compile time, and β follows from α through the chain identity: one scalar equation in one unknown, solved by Newton, with no mechanism anywhere in it.

March that, and every departure the mechanism reports lands on a predicted crossing — within 5.6 × 10⁻³ of a crank angle, against a marching step of 8 × 10⁻³, on four machines.

But it is an inclusion and not an equality, and the difference is the interesting part. The circle machine predicts four crossings and departs at one; across the four machines, nineteen crossings pass off with nothing happening at all. A vertex is allowed to cross the pivot and come out on the opposite ray with the gadget working perfectly — the departure test is careful to forgive exactly that. What the crossing marks is where the branch becomes ambiguous, not where it necessarily goes wrong.

And there is a distinction the arc itself hides. A working arc has two kinds of end. One is a departure: a gadget stops computing its angle and the machine goes on assembling perfectly. The other is a refusal: the mechanism cannot be assembled past that crank angle at all. Three of these four machines have one of each, and arc is a pair of numbers with nothing in it to say which is which.

What it draws instead, which is not an altered polynomial #

The obvious account of the off-arc machine — and the one recorded here as the thing to check — is that it computes the polynomial with one term’s frequency altered, so the traced set is where that polynomial vanishes. It is a tidy story and it is wrong.

Off-arc the machine is an ordinary mechanism. Every bar is satisfied at 10⁻¹⁴ and the summing chain still holds its last vertex on the line, so the identity the whole compilation rests on,

Σ ampₜ cos θₜ  +  constant  =  0

holds exactly — measured at 2.2 × 10⁻¹⁶ over two hundred and forty-four off-arc steps. What stops holding is the angle identity: θₜ is no longer mα + nβ + φ.

And the discrepancy is not a half turn. It drifts — 0.10π, 0.23π, 0.35π, 0.48π as the crank turns, over 1.57 radians in all. A sign flip is constant and a frequency change is an integer; neither is a continuously varying angle. So there is no relabelling of terms that reproduces the off-arc locus, and it is not the zero set of any expansion in whole-number multiples of α and β.

The structure underneath the drift is exact and is the part worth keeping. One mean gadget stops taking a mean, by δ. The reflector downstream of it departs by exactly 2δ — verified to 3.8 × 10⁻¹² over two hundred and thirty-nine steps — and the output link inherits that. A reflector doubles its mirror’s error, which is the same amplification the length-sensitivity measurement finds from the other end, arriving here as an angular identity rather than as a fitted gain.