Which numbers have a size

A protractor cannot see a machine’s size, which raises a question the whole site has an interest in: how much of what this site computes is a size?

The way to find out is not to argue from units. It is to take every length in a mechanism up and down together and fit the power the quantity follows.

The method #

For each quantity, evaluate it at scale factors from 0.8 to 1.6, fit log of the value against log of the factor, and read the slope. A shape gives zero, a length gives one, a curvature gives minus one, an area gives two.

Two things make that a measurement rather than a restatement of dimensional analysis.

The exponent is fitted, not assumed. Nothing tells the routine what the quantity’s dimensions are; it computes the quantity from the field’s own library at several sizes and reads the slope off the numbers. Every row lands on an integer to 2 × 10⁻¹⁵.

And there is a control. A quantity that does not move under a scaling might simply be insensitive to everything. So each row is also perturbed by changing one length by five per cent, and the reading is only called a shape if the first perturbation moves it by nothing and the second moves it by something. Every row passes both.

The rows #

transmission angle                       0
output angle                             0
velocity ratio                           0
Freudenstein's K₁                        0
Freudenstein's K₃                        0
output band, tolerance in per cent       0
Grashof margin  s + l − p − q            1
coupler point, x                         1
coupler point, speed                     1
coupler path curvature                  −1
area the coupler point encloses          2
output band from ±0.01 on the lengths   −1

Why a fit rather than an argument #

Dimensional analysis would give every one of these exponents in a line, and doing it by fitting is the site’s habit rather than a necessity. The habit earns its keep here in three ways.

A dimensional argument requires knowing how the quantity was computed. Several of these go through a solve, a differentiation and a chain rule, and reasoning about the dimensions of the result means tracking them through all of it. The fit does not care.

A dimensional argument cannot be wrong loudly. It is a piece of thinking, and a piece of thinking that comes out with the answer somebody expected is not evidence. A fit that returns −1.000 where 0 was predicted is a refutation, and one row of this survey is exactly that.

And a fitted exponent tests the implementation. A quantity coded with a stray absolute constant in it — a hard-coded 0.01, a fixed grid spacing, a threshold — has the wrong exponent, and the fit finds it. That is a genuine class of bug and nothing else on this site looks for it.

So the fitted column is doing three jobs: it states the dimensions, it checks the reasoning about them, and it checks the code that computed the quantity.

What is on the zero line, and it is most of the site #

Six of the twelve are shapes, and the list of what that implies is long.

The transmission angle is an angle in a triangle whose three sides all scale together, so it is unchanged. Everything the site says about transmission — the 40° design rule, where the worst angle falls, that a toggle and the worst transmission angle are 222° apart — is a statement about proportions.

The output angle at every crank position, hence the whole input–output relation, hence the velocity ratio, hence the time ratio, hence where the dead centres are.

Grashof’s classification, since it is a comparison of sums of lengths. Note that the margin s + l − p − q is a length and has exponent one; its sign is what the classification uses and a sign is scale-free. That distinction is worth having: a quantity can be a size while the only thing anybody reads off it is a shape.

And the three invariants, by construction.

So a measurement that recovers only ratios recovers the Grashof class, the full motion, the transmission behaviour and the coupler curve’s shape. The size is one number and almost nothing depends on it alone.

The row worth stopping at #

The tolerance band appears twice with two different exponents, and that pair is the survey’s finding.

Held to a fixed ±0.01 on each length, the output band scales as the reciprocal of the machine’s size. Fitted exponent: −1.000. Double the machine and the band halves.

That is obvious once said and it is not what anybody expects. A tolerance of ±0.01 is a length, so on a bigger machine it is a smaller fraction of every link, and a machine whose proportional errors are smaller is proportionally more accurate. A bigger machine held to the same absolute tolerance is a better machine.

Expressed as a percentage — ±0.5% of each length rather than ±0.01 — the same band has exponent zero. The tolerance now scales with the machine, the proportional errors are the same at every size, and the output band is a shape.

One quantity, two ways of writing the tolerance on a drawing, two different kinds of number. That is not a subtlety about units; it is a statement that the tolerance field’s results are size-dependent or not according to a drawing convention, and the field never says which convention it is using.

Reading the exponents #

The four values that appear are worth a sentence each, because the survey is more useful once they read as kinds rather than as numbers.

Zero is a shape: a ratio, an angle, a count, a classification. Recoverable from angles, comparable between machines of any size, and what most of a design conversation is about.

One is a length: a position, a speed at a fixed crank rate, a clearance, a link. Needs one size measurement and then follows.

Minus one is a curvature or a rate per unit length: a path’s sharpness, a band produced by an absolute tolerance. A bigger machine has smaller curvatures, which is why large mechanisms trace gentler paths than their drawings suggest.

Two is an area: the region a coupler point encloses, a machine’s swept area, a footprint. The exponent that amplifies a size error most on this site.

Nothing here has exponent three, because nothing here is a volume — everything is a plane region, which is a boundary the bodies field states and this survey inherits.

What is a size, and what it means for measuring #

The five non-zero rows are the ones an angle-only measurement cannot recover.

Where the coupler point actually is, how fast it goes, how sharply its path curves, how much area the path encloses, and how much slack the Grashof margin has in absolute terms. Every one of those is a real engineering quantity and every one of them needs one reading with a length in it.

The exponents say how much. A quantity of degree n is recovered to n times the relative error of the size measurement — so a length measured to a per cent gives an area to two per cent and a curvature to a per cent. The size error propagates with a factor equal to the exponent, which is the practical use of the survey.

Nothing on this site has an exponent outside −1 to 2, which is worth knowing: the worst amplification of a size error anywhere here is two.

The survey’s own check #

Every row’s expected exponent is stated before the fit and compared afterwards, and the comparison is the assertion the figure carries: worst departure 2 × 10⁻¹⁵.

That number is not impressive on its own — a units argument and a fit of an exact power law should agree to machine precision. What makes it a check is the row that was wrong the first time.

The tolerance band from a fixed ±0.01 was declared to have exponent zero, on the reasoning that a band is a proportional quantity and proportional quantities are shapes. It came back at −1.000, and the assertion failed.

The failure was the reasoning’s, not the fit’s. A units argument is a piece of thinking and can be wrong; a fitted exponent is a measurement and cannot be. Having a row that disagreed with its own prediction is what turned this from a table of the obvious into a survey worth running.

Why this belongs in this field #

The survey looks like a piece of dimensional analysis and it is here for a reason that is about measurement.

The scale null space says one direction of a mechanism’s parameter space is invisible to an angle sensor. That statement is about parameters. This survey is the same statement about outputs: which of the quantities a designer reads off a mechanism survive the same operation.

The two together settle what an angle-only calibration is worth. It recovers three of four parameters and, through them, every quantity on the zero line — which is six of the twelve here and, if the survey were extended, most of what this site computes. It recovers none of the quantities with a non-zero exponent, and each of those needs the one size measurement.

The parameter null space and the quantity survey are two views of one group action. Scaling acts on parameters and on outputs at once; the invariant parameters are the K’s and the invariant outputs are the zero rows, and a measurement that cannot see the group recovers exactly the invariants of both.

That is why the survey is a metrology result rather than a curiosity, and why running it was worth an essay.

What the control column caught #

The two-perturbation test — scale everything, then change one length — is the part of the method most likely to be skipped, and it is worth saying what it earns.

A quantity that reads zero under a scaling might be a shape or might simply be a number that does not depend on the geometry at all. A count of links, a mobility, a number of assembly modes: all of them read zero under a scaling and none of them is a shape in the sense the survey means, because they read zero under everything.

The control separates the two. Every row of the survey moves by between 0.5% and 13% under a five per cent change in one length, so every one of them is genuinely sensitive to the geometry and genuinely insensitive to the size.

Without it the table would be making a much weaker claim. Invariance is only interesting for a quantity that varies, and asserting that a constant is scale-invariant is a sentence with no content in it.

That check is also what would catch a coding error of a particular kind: a quantity accidentally computed from a fixed set of dimensions rather than from the perturbed ones reads zero under both perturbations, and the control column flags it immediately. That happened once while the survey was written.

A row that is neither #

One class of quantity does not appear in the table and its absence is deliberate.

A mobility reads zero under a scaling and also reads zero under the control perturbation — it is an integer and integers do not move when a length changes by five per cent. So the probe declines to classify it, and the declining is correct: a quantity that is unchanged by everything is not a shape, it is a constant.

That is a third class rather than an edge case, and this site is full of it: mobilities, assembly-mode counts, chain enumerations, chromatic numbers, Grashof classes. It gets an essay of its own, and the short version is that a count’s evidence lives in the continuous margin underneath it rather than in the count.

Mentioning it here matters for reading the table. Twelve rows is not the whole site, and a reader who took the survey as exhaustive would conclude that everything this site computes has an exponent. Most of what it computes does; a substantial minority is counts.

The general instruction #

Two sentences, for anybody who is about to write down a quantity.

Before quoting a number as a property of a mechanism, ask what a scaling does to it. If the answer is nothing, it is recoverable from angles alone and it is comparable between machines of different sizes. If not, it needs a size and it is not comparable.

And watch for quantities whose exponent depends on a convention rather than on the quantity. A tolerance band is the instance here; a clearance quoted absolutely or as a fraction of a diameter is another; a positional error quoted in millimetres or in parts per million is a third. In each case the same physical thing sorts onto a different line depending on how somebody chose to write it down.

The two conventions, once more #

The tolerance row is worth one more pass, because the fix is not to choose the right convention.

Both are used and both are correct for what they say. A drawing carrying ±0.01 means a length; a drawing carrying ±0.5% means a proportion. A shop can hold either. A designer choosing between them is making a decision about whether a bigger version of this machine should be proportionally better or equally good, and it is a real decision with a cost attached.

What is not acceptable is doing the analysis in one convention and reading the answer in the other. A band computed from an absolute tolerance is a size and must not be quoted as a property of the design, because a version of the design at twice the scale has half the band and nothing about the design changed.

The site’s own tolerance figures use absolute tolerances throughout, so every band they quote is a band for a machine of that size. That has been true since the site’s first tolerance figure and has never been said. It is not an error — the machines drawn are all one size — and it is the sort of unstated convention that becomes an error the moment somebody scales a design.

What the survey does not cover #

Three limits, so the list is not read as complete.

It is a survey of one mechanism’s quantities. Twelve rows from a four-bar and its coupler point, chosen because they are the quantities the site quotes most often. A spatial loop’s twists, a cam’s pressure angle, a gear’s contact ratio and a screw’s pitch are not here and would each sort onto one of the same lines.

It is about scaling all lengths together. A quantity can be invariant under a uniform scaling and highly sensitive to a change in proportions, and the control column is what distinguishes the two — but the survey says nothing about which proportions matter, which is what the sensitivity analysis is for.

And it says nothing about time. Every quantity here is geometric. A velocity is a length per unit time and the survey scales only the length, so the coupler point’s speed comes out at exponent one — which is right for a machine driven at a fixed crank rate and would be different for one driven at a fixed rim speed. That choice is stated and it is a choice.

And it says nothing about which quantities matter. A survey sorts by a property of a quantity, not by its importance, and the zero line contains both the transmission angle and Freudenstein’s K₃ — one of which a designer reads every day and one of which almost nobody has heard of. Sorting is not ranking, and the survey is a filing system rather than a set of priorities.

What it does do is tell a reader, for any number this site prints, whether that number would be the same on a machine twice the size. That is a question with a yes-or-no answer, it comes up constantly, and until this survey the site had no way to answer it except by thinking about each case.