Exact costs more than close

The straight line is the one demand this subject has been asked for by people who needed it, and it is the only place where an exact mechanism, an approximate one and a compiled one can be put on the same axis.

The five, measured the same way #

Each mechanism is swept over its own working arc — walked out to its dead centres and back a tenth — and the departure of its traced path from the straight line through its endpoints is measured as a fraction of the stroke. Comparing four straight-line mechanisms over one fixed range of crank angle would be comparing three of them over strokes they do not have.

Watt, four bars, 9.0 per cent. Two rockers and a coupler; the coupler’s midpoint traces a figure of eight whose middle is nearly straight. It is in every beam engine ever built.

Chebyshev, four bars, 12.4 per cent. A different four-bar with a different point, optimised by a different criterion, and slightly worse on this measure over its own arc.

Peaucellier, seven bars, $4\times10^{-16}$. The inversor: a rhombus and two long arms turn a circle through the pivot into a straight line, exactly, because the product of two distances is constant.

Compiled with a slide, five bars, $4.8\times10^{-14}$.

Compiled all-revolute, twelve bars, $2.7\times10^{-14}$.

The gap is the finding #

Between twelve per cent and $10^{-14}$ there is nothing.

That is not a statement about these five mechanisms; it is a statement about what exactness is. An approximate straight-line linkage is one whose coupler curve has a high-order contact with a line — the curvature field computes exactly how high — and the residual error is governed by the first derivative that does not vanish. Improving it means making that derivative smaller, and there is no arrangement of four bars for which it is zero.

An exact one is a linkage in which an algebraic identity holds. Peaucellier’s cell is exact because $|OP|\cdot|OQ|$ is constant, which follows from the rhombus and the two arms, and it is either true or it is not. There is no partial version of an identity.

So the axis has two populations and nothing between them, and the empty region is not empty for want of trying. Nineteenth-century mechanism design produced dozens of approximate straight-line linkages and exactly one family of exact ones, and the gap between the best approximation and the worst exact mechanism was, as it remains, about fourteen decades.

Where each of the five gets its answer from #

The five mechanisms are exact or approximate for five different reasons, and setting them out is the quickest way to see why the axis has a hole in it.

Watt. The coupler point’s curve has a point of inflection with contact of order three against the tangent line. Three derivatives vanish and the fourth does not, so the error grows as the fourth power of the distance from the design point. That is a curvature statement and the curvature field computes exactly how many derivatives can be made to vanish with a given number of free lengths.

Chebyshev. The same object with a different criterion: instead of maximising the order of contact at one point, the error is equalised across the stroke, which is the minimax answer. It is worse at the centre and better at the ends, and over a whole arc it comes out slightly larger on this measure. Two reasonable definitions of best giving two different linkages is the ordinary situation in approximate synthesis.

Peaucellier. An inversion: the rhombus and the two arms make $|OP| \cdot |OQ|$ constant, and an inversion sends a circle through the centre to a straight line. The crank puts $Q$ on such a circle. Nothing is fitted and no derivative is being made to vanish.

Compiled with a slide. A prismatic pair holds a point on a line, and the linkage in front of it holds the relation between that point and the tracing point. The line is assumed rather than produced.

Compiled all-revolute. The same, with Peaucellier’s cell doing the holding — so its exactness is Peaucellier’s, wrapped in five bars of general-purpose apparatus.

Read down that list and the boundary is visible: the two approximations are fitting, the three exact ones are satisfying an identity, and there is no mechanism in the subject that does a bit of each.

What the strokes look like #

Watt’s figure of eight and Chebyshev’s symmetric bow are the classic pictures, and they are the shapes an approximation makes: the error is not noise, it is a definite curve with a definite order of contact, and it goes to zero at the points where the design was fitted.

The exact traces are flat, at this scale and at every scale. Zooming in on them shows the solver’s residual and nothing else.

What the compiled machine is doing there #

Five bars, exact, and fewer bars than Peaucellier. That looks like a win and it is not one, for a reason worth stating carefully.

The compiled machine closes its summing chain with a slide — one prismatic pair, holding a point on a vertical line. Kempe’s whole interest was in avoiding exactly that: a slider is a straight-line device, and using one to build a straight-line mechanism assumes what was to be produced.

Closed the honest way, with a Peaucellier cell doing the holding, the compiled machine costs twelve bars. That is Peaucellier’s seven plus five bars of overhead, and it is exactly the right answer for a general procedure applied to a case with a famous special solution: the compiler rediscovers Peaucellier and charges for the discovery.

The five-bar figure is quoted because it is the count of the compilation, and the twelve-bar figure beside it is the count of the mechanism.

Four bars is not the boundary between the populations #

A reader might guess that the split in the figure is a split by size — four bars approximate, seven and up exact — and it is worth showing that it is not, because the guess is natural and it would make the empty axis look like an artefact of the sample.

There are four-bar mechanisms with exact properties. A parallelogram holds two links exactly parallel, at every configuration, for ever; that is an exact statement made by four bars. An antiparallelogram enforces an exact relation between two angles — a constant product of half-angle tangents — at every configuration. A Cardan arrangement makes a point travel on an exact straight line by rolling one circle inside another of twice the radius, and that is exact and is not seven bars.

So four bars can be exact about something, and the reason no four-bar is exact about a general straight line is not a count. It is that the property wanted — a coupler point on a line — is not one of the identities four bars can express. Adding a fifth and a sixth bar does not gradually make it expressible.

The population a mechanism belongs to is decided by whether the demanded relation is among the ones its topology can hold, and that is a yes-or-no question with no size in it. Peaucellier’s seven bars are seven because that is how many the inversion identity needs, not because seven is the number at which accuracy arrives.

The exactness is checked two-sidedly #

There is a way to make this measurement say nothing, and this site has recorded falling into it once.

If the sampling is too narrow, an approximation looks exact. Watt’s linkage over a hundredth of a radian of crank is straight to the solver’s floor, because its error is fourth-order in the departure from its design point. A measurement that reported all five mechanisms at $10^{-16}$ would be a measurement of the sampling rather than of the mechanisms.

So the gate asserts both directions. The approximations must be approximate — a relative error above $10^{-6}$ — and the exact ones must be exact, below $10^{-9}$. The first half is the one that matters, and it is why the arcs are found by walking to the dead centres rather than chosen: a hand-picked range is a range that can be picked too small.

That check has been on this site since its foundation, in the linkages field, and this rung reuses it rather than writing a second one.

The arcs are part of the answer #

The residual is one number and the stroke it was measured over is another, and quoting the first without the second is what makes accuracy claims about linkages unreliable.

Peaucellier’s cell in this figure works over $3.6$ radians of its crank — the longest arc of the five — and Chebyshev’s over $1.1$. So the two are not only fourteen decades apart in error, they are three times apart in how much motion they deliver, and the exact one wins both.

The compiled machines work over $2.0$ radians, which is respectable and is the best arc in this field’s whole catalogue. That is not a coincidence: a line’s expansion has two terms, so its machine has one reflector-free path from the arm to the chain and almost nothing that can go singular. The curve with the fewest terms has the longest arc, and the relation holds loosely across the catalogue and fails on the folium.

What made the difference historically, and it was not this axis #

Watt’s linkage is in every beam engine ever built. Peaucellier’s cell is in museums and in the occasional demonstration model. On the axis above that is the wrong way round, and the reason is instructive because it is the same reason the compiled machines in this field will never be built either.

Watt’s linkage is four bars and three pins. It fits in the space available, it can be made from forged parts by a smith, and nine per cent of the stroke was — for a beam engine’s parallel motion — a great deal better than nothing and entirely good enough. Watt himself described it as the invention he was proudest of.

Peaucellier’s cell is seven bars, eight pins, and a rhombus that has to be made accurately. Every one of those pins has clearance, and clearance is a link: the assembled mechanism’s exactness is the exactness of the identity, degraded by the play in eight joints. The practice field measures what that does, and the honest summary is that an exact linkage made to ordinary tolerances is not more accurate than a good approximation made to the same ones.

So the axis in this rung is the axis of the ideal mechanism, and it is the right axis for the question this field asks — what can a linkage express exactly — and the wrong axis for the question an engineer asks.

What happens to the gap when the parts are real #

The axis above is drawn for ideal mechanisms and the whole of the practice field exists because ideal mechanisms are not the ones anybody has. It is worth asking what the fourteen decades become when the bars have tolerances.

Give every length in Peaucellier’s cell an error of a thousandth and the identity no longer holds exactly; the traced path is straight to something like a thousandth rather than to $10^{-16}$. Give Watt’s linkage the same errors and its nine per cent becomes nine per cent, because a thousandth is lost in it entirely.

So the gap closes from fourteen decades to about two, and the exact mechanism is still better — but it is now better by a factor of a hundred rather than by a factor beyond counting, and it costs nearly twice the parts. That is a trade a designer might genuinely refuse, and it is why the history came out the way it did.

For a compiled machine the arithmetic is worse still, because the amplification of a length error grows with the machine — reaching ninety on the largest machine measured. An exact construction whose errors are amplified ninety times is not, in any practical sense, more exact than an approximation whose errors are not amplified at all.

Exactness is a property of the design and accuracy is a property of the artefact, and this axis measures the first. Every number on it is true and none of it is a prediction about a machine.

The place the trade actually bites #

If exactness is not what decides, it is fair to ask what this field’s construction is for, and the answer is not accuracy at all.

It is generality. Peaucellier’s cell draws a straight line. It does not draw a cubic, and no amount of adjusting it will make it draw one. The compiled machine draws whatever polynomial it was compiled from, by a procedure with no insight in it, and the price of that generality is the fourth-power growth in the bar count and an arc measured in degrees.

The comparison this rung actually supports is therefore narrower than it looks. It is not the compiled machine is a good straight-line mechanism. It is: on the one demand where all three kinds of answer exist, the exact ones are separated from the approximate ones by fourteen decades, and the compiled one is on the exact side. Everything else the compiled machine can do, neither of the others can do at all.

What a designer takes from the empty axis #

Two things, and the second is the one that generalises.

A linkage does not get more accurate by getting bigger. There is no sequence of five-, six- and seven-bar mechanisms creeping down from nine per cent towards zero. The question is binary: does an identity hold. If one does, the mechanism is exact at any size; if none does, adding parts moves the error around and does not remove it. That is why approximate synthesis is an optimisation problem and exact synthesis is an algebra problem, and why the two fields have never merged.

A measurement over the wrong interval turns the first fact into its opposite. Sampled narrowly enough every mechanism is exact, and sampled over its own arc the populations separate by fourteen decades. Any claim about a mechanism’s accuracy that does not say over what range it was measured is not a claim about the mechanism.

There is a third thing, smaller, that this rung is the only place to record. Four bars have been enough for a great many things and there is a reason beyond convenience: Grashof’s condition tells a designer in one line whether a set of four lengths gives a link that turns fully, and no comparable statement exists for anything larger. A designer choosing between a four-bar and a fifty-bar compiled machine is choosing between an object whose behaviour is characterised by an inequality and an object whose behaviour has to be found by driving it and watching. That asymmetry has kept the subject small for two centuries and it is not going to be changed by a construction with four hundred bars in it.

The second of the two lessons is the more useful, and it is not about linkages. It is the reason every arc in this field’s cost table is quoted beside its residual, and the reason the singularity rung spends its length on where a machine stops working rather than on how well it works while it does.

The empty axis is the finding and it has a design reading that is worth separating from the historical one. A designer does not choose a point on a scale of accuracy; they choose which population to be in. The approximate population is four bars, cheap, compact, and wrong by a per cent or several; the exact population is seven bars or a compiled machine, larger, with more joints and more clearance, and right to the arithmetic. There is no intermediate design that is wrong by a tenth of a per cent with five bars, and adding a bar to Watt’s linkage does not produce one. So the decision is discrete, it is made once, and it is made on whether the application can tolerate a per cent — not on how much accuracy the budget will buy. That is an unusual shape for an engineering trade and it is the reason the two populations coexisted for a century: they are not competing points on one curve, and the argument between them was never really about accuracy at all.