What a synthesis assumes it knows

The synthesis field is handed a demand and returns a linkage. Three positions of a coupler plane, at stated places in the plane, and out comes a four-bar that reaches all three.

Scale the demand and the answer scales exactly with it. That is obvious and it has consequences that are not.

A construction that discards the size it was given #

Burmester’s construction for three positions is ruler and compass: perpendicular bisectors of corresponding point pairs, intersected to find a circle centre. Every step is a line, a circle or an intersection.

A similarity carries lines to lines, circles to circles and intersections to intersections. So scaling the three prescribed positions about any point scales every construction line, every circle centre and every returned linkage by the same factor.

The site’s implementation is not ruler and compass — it solves for the dyads algebraically — and the property survives, because the algebra is homogeneous. Every equation in it relates coordinates to coordinates, with no absolute constant anywhere, so multiplying all the inputs multiplies all the outputs.

That is a checkable statement and it is the kind of thing a stray constant would break: a construction with a hard-coded tolerance, a fixed grid spacing or a default length in it would fail to commute with a scaling, and the failure would be a wrong answer at some sizes and not others.

The construction commutes with a scaling, and that is what it means for the field’s figures to be figures of families: enlarge the drawing and it is a valid drawing of a larger problem with a larger answer.

The demand carries the size #

That has a consequence for what a design specification is actually specifying.

Three prescribed positions in absolute coordinates are nine numbers: three positions and three orientations. Of those, the similarity group of the plane accounts for four — two translations, a rotation and a scaling — which do not change the problem, only where it is written down.

The rigid part of that has always been understood: nobody thinks moving a demand across the page changes it. The scaling has not, and it is the same kind of thing. A construction that commutes with a transformation cannot distinguish demands related by it, so the four together are a gauge freedom of the specification.

So a three-position demand has five numbers that decide the answer’s shape and four that decide where and how big it is. A designer adjusting the demand’s absolute size is not changing the design problem at all; a designer adjusting the relative placement of the three poses is.

That is worth knowing because the two look identical on a drawing. Three poses moved further apart and three poses scaled up are different edits and one of them changes nothing.

That has a consequence for what a design specification is actually specifying.

Five numbers of design content #

The count is worth doing carefully because it is the essay’s most usable result.

Three poses of a moving plane in the plane are nine numbers: (x, y, φ) three times. Apply a rigid motion to all three and the design problem is the same problem written in different coordinates — that is three numbers. Apply a scaling and it is the same problem at a different size, which the construction commutes with — one more.

Nine minus four is five. Five numbers of genuine design content in a three-position specification.

Those five are not any particular five of the nine; they are five combinations, and the natural ones are the two relative displacements between successive poses expressed as ratios and angles. A designer who fixed the first pose at the origin with zero orientation, and scaled so the first displacement had unit length, would be left with exactly five numbers.

A specification written that way cannot be adjusted in a way that does nothing, which is the practical benefit: every edit changes the answer.

A specification written in absolute coordinates can be, and often is. Moving all three poses a hundred millimetres to the left is a change to nine numbers and to no part of the design.

What the field’s defects are #

The field’s most useful results are about the ways a synthesis can succeed and produce something unusable, and every one of them sorts by the same probe.

A branch defect — the linkage reaches all three poses on different assembly branches, so it cannot be driven through them — is a statement about signs and about which intersection is taken. Dimensionless. A defect at one size is a defect at every size.

A circuit defect and an order defect are likewise conditions on which configurations are reachable in what sequence. Dimensionless.

The size defect — two pins closer together than the material round them — is not. It compares a returned link length against a boss radius, which is a length chosen independently of the demand, so it is a comparison of two things only one of which scales.

So three of the field’s four defect classes are shapes and one is a size, and the one that is a size is the one the bodies field found.

A Burmester curve and a dyad chosen from it are the construction working as intended. What the construction leaves undetermined is the part that matters here, because the freedom a designer is handed at this step is precisely the direction a measurement would afterwards have to pin down.

Why the size defect is different #

That one deserves its own look because it is the field’s newest defect and its behaviour is the one a designer would get wrong.

Of 1,176 exactly correct three-position syntheses, 116 put two pins closer than one boss diameter at a boss radius of 0.2, and 313 do at a radius of 0.4. Those counts depend on the boss radius, which is a property of the hardware rather than of the demand.

Scale the whole demand up by two and re-run: the returned linkages are twice the size, their shortest links are twice as long, and the same boss radius now rules out far fewer of them.

The size defect can be cured by making the machine bigger and none of the other three can. That is a genuinely useful asymmetry and it is invisible unless the defects are sorted this way: three of them are properties of the demand’s shape and cannot be escaped by scaling, and one is a comparison against a fixed length and can.

There is a matching statement in the other direction and it is the one that bites. A design shrunk to fit a smaller envelope acquires size defects it did not have, at a rate the boss radius decides, and acquires no branch, circuit or order defects at all. Shrinking a mechanism can only break it in one way, which is a reassuring thing to know and a specific thing to check.

That one deserves its own look because it is the field’s newest defect and its behaviour is the one a designer would get wrong.

The structural error is a shape #

The field’s other characteristic quantity behaves as the pattern predicts and it is worth confirming, because a reader might expect an error to be a length.

Structural error is how far a synthesised function generator departs from its demanded function between the precision points. For a function generator the demand is a relation between two angles, both dimensionless, so the departure is an angle and is dimensionless.

Scale the linkage and the departure does not move. So the whole of what the field says about precision-point placement — that Chebyshev spacing beats uniform, by how much, and where the worst departure falls — transfers between designs of any size.

For a path generator the demand is a curve in the plane and the departure is a distance, which is a length. There the error scales with the machine, and quoting it as a fraction of the path’s own extent would make it a shape.

Function generation’s error is a shape and path generation’s is a size, which is not a distinction the field currently draws and which decides whether a quoted accuracy transfers. A structural error of 0.3° means the same thing on any machine; a path error of 0.3 units means something only about a machine of a stated size.

What an identification of a synthesised linkage recovers #

Turning the field round to face the identification question gives a tidy answer.

A synthesised linkage is a four-bar, so an angle-only measurement of it recovers its three invariants and not its size.

Which means: a measurement recovers whether the linkage is a crank rocker, its transmission angles, its dead centres, and whether it has a branch defect — all shapes. It does not recover whether it has a size defect, because that compares its lengths against a boss.

So the three defects that survive a scaling are recoverable from a protractor and the one that does not is not. A measurement can check a synthesised linkage for every defect except the one a scaling could have cured, which is a neat and slightly perverse pairing.

What a designer should specify #

The counting turns into an instruction and it is a change to how a demand is written rather than to anything computed.

Specify the demand up to a similarity. Fix the first pose at the origin, unrotated, and scale so that a chosen reference distance is one. The remaining five numbers are the design, and the size is a separate decision made afterwards for whatever reason sizes are chosen — the envelope, the loads, the standard parts available.

Doing it that way has three benefits and one cost.

It makes every edit meaningful. It makes two demands comparable: two specifications in normalised form can be looked at side by side and their difference is a difference in design. And it makes the synthesis’s answer reusable — one solved shape, instantiated at whatever size a particular application needs.

The cost is that the demand no longer reads like a drawing. A designer with a machine to fit into a housing has the poses in millimetres because the housing is in millimetres, and normalising is an extra step that separates the design problem from the packaging problem.

Which is exactly what it is for. The packaging problem is a size and the design problem is a shape, and running them together is why a specification has four numbers in it that decide nothing.

Three prescribed pairs, and the ground length #

The function-generation version makes the point most sharply because there the construction is explicit about it.

Freudenstein’s relation takes three prescribed input–output angle pairs and returns three coefficients. The routine’s signature is freudenstein(pairs, { ground = 1 }), and a ground length has to be supplied because the three coefficients do not contain one.

That is the scaling freedom made explicit in an argument list. The construction computes a shape and then reports one representative of it at whatever size the caller asked for, and the default of 1 is a normalisation rather than a convenience.

The site has been calling that function for years of essays with the ground length passed in as a matter of course. Reading it as a normalisation changes not one line of arithmetic and a great deal about what the routine is understood to do.

The same is true of the geometric constructions and it is less visible there, because they take their size from the demand’s coordinates rather than from a named argument. A Burmester construction handed three poses in millimetres returns a linkage in millimetres and never mentions that it could have returned one in inches with the same shape. A construction that takes its scale from its input’s units is doing the same thing as one that takes it from an argument, and only the second says so.

Five positions and the same argument #

The counting extends to the field’s harder problems and the answer keeps the same shape.

Four positions are twelve numbers, less four for the similarity: eight of design content. Five positions are fifteen, less four: eleven.

Five is the most a four-bar can be asked for, and the field’s own counting of why is done in free parameters of the linkage rather than in the demand. Putting the two counts beside each other is instructive: a four-bar has nine free numbers as a path generator, of which the similarity group accounts for four, leaving five of shape — which is exactly the content of a three-position demand.

So three positions determine a four-bar’s shape and leave a two-parameter family, five positions over-determine it, and the arithmetic works out in the normalised coordinates as cleanly as in the absolute ones.

Counting in normalised coordinates removes four numbers from both sides and changes no conclusion, which is the check that the normalisation is doing nothing sneaky.

A size defect is a repair and the others are not #

The field’s four defects sort three to one under the scaling question, and the one that sorts differently is the one a designer can do something about.

A branch defect means the linkage reaches its prescribed poses on two different assemblies and cannot move between them. A circuit defect means it reaches them on two different closed circuits of the same assembly. An order defect means it reaches them all, on one circuit, in the wrong sequence. Every one of those is a statement about which configurations connect to which, all three are unchanged by scaling the whole linkage, and none of them can be escaped by building the thing bigger.

A size defect is not like that. It means a returned link is too short to be made — shorter than the boss the pin sits in, thinner than stock, smaller than the bearing it must carry — and it is a comparison between a length the synthesis returned and a length the workshop fixed. Scale the linkage up and the first grows while the second does not. The defect goes away, and nothing about the design changes.

That asymmetry is worth acting on, because a synthesis routine that reports a size defect is not reporting a bad answer. It is reporting a good answer at the wrong size, and the repair is one multiplication. A designer who discards the solution and re-prescribes the poses has thrown away a correct piece of design work over a units problem — and the field’s own defect survey lists all four side by side, in the same table, with no indication that one of them is a rescale and three are rejections.

There is a second consequence for how a demand should be written. Three prescribed positions are nine numbers, and four of them decide nothing about the mechanism’s shape: two of translation, one of rotation, and one of scaling. A three-position demand contains five numbers of design content, and a designer adjusting a demand that produced an unbuildable linkage is well advised to know which four of the nine are free. Moving all three positions bodily, turning them together, or scaling them about any point whatever produces a different linkage that is the same design.

So the honest advice to somebody specifying a synthesis is: state the shape of the motion, and state the size separately as a constraint on the returned links rather than as a property of the prescribed poses. The construction cannot use size information in the demand — it is a similarity, and a similarity discards it — so putting size into the poses only conceals the one place it genuinely belongs.

The construction knows the shape and the workshop knows the size, and a demand that mixes them makes the routine answer a question it was never asked.