A curvature is a size with a minus sign

The curvature field is about how a moving plane’s points turn as well as move: where a point’s path is momentarily straight, where its curvature is momentarily stationary, and which circle its path is hugging at this instant.

Every construction in it is a similarity. Every value it produces is a reciprocal length.

Two things a scaling can do to a field #

Before the measurements, the distinction the whole essay turns on, because scale-invariant gets used for two different properties.

A construction is scale-invariant when the scaled input produces the scaled output — when the picture enlarges correctly. Every ruler-and-compass construction has this, because a similarity carries lines to lines, circles to circles and intersections to intersections.

A quantity is scale-invariant when its numerical value does not change. That is a much stronger requirement and most quantities fail it: a length changes, an area changes, a curvature changes.

The curvature field has the first everywhere and the second almost nowhere. Its constructions are similarities and its numbers are reciprocal lengths, and running the two together produces the natural mistake — that a figure valid at any size implies a number valid at any size.

A similarity carries a curvature to a different curvature, which is the whole of what this essay is about and is the one thing a synthetic proof never mentions, because a synthetic proof does not produce numbers.

Every construction here is a similarity #

Take a moving plane, its instant centre, its inflection circle and its cubic of stationary curvature, and scale everything by a factor.

The instant centre moves to the scaled position. The inflection circle’s centre moves and its diameter scales. The cubic scales. Every point that was on a locus is still on the scaled locus, and every angle in every construction is unchanged.

That is what a similarity means, and it is worth having as the first statement because it says the field’s pictures are size-free. A figure of an inflection circle beside a coupler curve is a figure of a family of machines, and enlarging the drawing produces a valid drawing of a larger member.

The geometry transfers between sizes exactly. Euler–Savary relates four distances along a ray through the instant centre and its statement is a relation among ratios; scaling all four leaves it satisfied.

The values are not #

Now the numbers, and here the field parts company with most of the site.

The curvature of a coupler point’s path — one over the radius of the circle the path is hugging — has a fitted exponent of −1.000000 under a uniform scaling. Double the machine and every curvature halves.

That is not a shape and it is not an ordinary size either. It is a reciprocal length, and it is the only quantity class on this site with a negative exponent.

The consequence is worth stating in the direction a designer notices. A mechanism scaled up traces a gentler path: the shape of the path is identical, every radius is doubled, and the sharpest corner is half as sharp. Anything that depends on the curvature — a follower’s ability to track it, a roller’s minimum radius, an accelerating mass’s demand — improves proportionally with size, from geometry alone.

What the figures already assumed #

Worth noticing that the field’s own drawings have been relying on the similarity property since the field opened.

An inflection circle is drawn beside a coupler curve, with both scaled to fit a canvas. The canvas scale is arbitrary — it depends on the figure’s width and the content’s extent — and the picture is correct because the whole construction is a similarity. If it were not, every figure in the field would be a picture of a different mechanism from the one in the caption.

That is a dependence nobody would think to check and it holds for a reason worth having explicitly. A drawing routine that scales its content to fit is applying a similarity, so any construction it draws must be similarity-invariant for the picture to mean anything.

The site’s scene frame preserves aspect for exactly this reason and its docstring says so — distorting the axes independently would change every angle, and the angles are quantities the essays quote. The reason curvature figures survive that treatment is one step further: the radii in them are all scaled by one factor, so their ratios, which is what the construction is about, are preserved.

Where the minus sign comes from #

Not mysterious and worth writing out, because it says which of the field’s quantities share it.

A curvature is a rate of change of direction with respect to arc length. Direction is dimensionless and arc length is a length, so curvature is one over a length by construction. Anything defined that way inherits the exponent.

The radius of curvature is the reciprocal, exponent +1, an ordinary length. The inflection circle’s diameter is a length. The distance from the instant centre to a point is a length.

So the field’s quantities come in reciprocal pairs, and which member of a pair gets quoted is a convention. The site quotes radii in some places and curvatures in others, and the exponents differ in sign — which is harmless when it is noticed and is the kind of thing that produces an inconsistent table when it is not.

The measurement, and how it was made #

The exponent is fitted rather than argued, which is the survey’s method and is worth describing here because the quantity is awkward to compute.

A coupler point’s path curvature is obtained by a second difference: solve the mechanism at three nearby crank angles, take the first and second derivatives of the point’s position by central differences, and combine them as (ẋÿ − ẏẍ) divided by (ẋ² + ẏ²) to the power three halves.

That is a ratio of a second derivative to a three-halves power of a first, so the arithmetic is delicate: too small a step and the second difference is noise, too large and it is measuring the wrong thing. The step used is 4 × 10⁻³ radians, which sits comfortably between the solver’s floor at 10⁻¹³ and the scale over which the curvature itself varies.

The fitted exponent comes back at −1.00000000000225 over six scale factors from 0.8 to 1.6. That last figure is the differencing, and it is worth quoting because an exponent that came back at −0.98 would have been evidence of a step-size problem rather than of geometry.

A fitted exponent is a test of the computation as well as a statement about the quantity, which is the third job the survey’s method does and the one that matters most for a quantity computed this way.

What an angle-only measurement recovers #

Applying the field’s own question to the curvature field’s objects gives a clean answer with one surprise in it.

Where the inflection circle is, relative to the machine — a ratio of lengths, exponent zero, recovered.

Which coupler point traces a momentarily straight path — a location on the coupler expressed as a fraction, exponent zero, recovered. That is the whole content of the straight-line problem as far as which point goes.

The curvature of that point’s path — exponent −1, not recovered. A protractor on the output link cannot say how sharply a coupler point turns, because it cannot say how big the machine is.

And the ratio of two curvatures at the same instant — exponent zero, recovered exactly, because the two reciprocal lengths divide.

That last one is the surprise and it is the practically useful case. A designer asking is this point’s path five times gentler than that one’s is asking a question an angle-only calibration answers, and one asking what radius does it hug is not.

Euler–Savary is a relation among ratios #

The field’s central relation deserves the same treatment, because seeing why it is scale-free says why the whole field is.

Euler–Savary relates four distances measured along one ray through the instant centre: to the moving point, to its path’s centre of curvature, and the corresponding pair for the centrodes. The relation is a statement that a certain combination of their reciprocals is constant along the ray.

Scale everything and every distance is multiplied, every reciprocal is divided, and the relation holds unchanged. It is homogeneous of degree −1 in the distances, so multiplying them all by k multiplies both sides by 1/k and the equality survives.

That is the general pattern for the field’s relations and it is stronger than saying they are dimensionally consistent. A relation among reciprocal lengths is automatically scale-free, which is why every construction in this field is a similarity and why the classical proofs, which are all synthetic, never need to mention a size.

The one thing the relation cannot do is supply a size. Given the ratios it determines every other ratio, and a machine of any size satisfies it — which is the same statement, arriving from the geometry, that an angle-only measurement makes about the parameters.

The centrodes scale too #

The fixed and moving centrodes — the paths the instant centre traces in the two frames — are curves, so their positions are lengths and their shapes are similarity invariants.

Scale the mechanism and the two centrodes scale together. They still roll on one another without slipping, because rolling is a condition on arc lengths and both scale equally, and the rolling condition is a statement about their ratio.

So the centrodes’ rolling is scale-free and their sizes are not. The radius of the fixed centrode at the current instant is a length; the ratio of the two centrodes’ curvatures there is a shape and appears in Euler–Savary.

That is a good example of the field’s general pattern: the relations are dimensionless and the ingredients are not, so every classical construction transfers between sizes and every number it produces has to be given one.

Where the exponent matters in practice #

Two places, both real.

A cam follower’s roller. A roller cannot follow a concave profile whose radius is smaller than the roller’s own — the profile undercuts. Scale the cam up and the profile’s radii all scale while the roller’s radius is chosen separately, so a design that undercuts at one size may not at another. The cam field computes the undercut margin at one size, and the margin’s exponent decides whether the result transfers.

A coupler curve used as a dwell. A circular-arc stretch of a coupler curve is exploited by hanging a link of that stretch’s radius on the coupler point. The radius is a length, so the link’s length scales with the machine, and the quality of the dwell — how nearly circular the stretch is, measured as a fraction of the radius — is a shape and does not.

In both cases the design decision involves comparing a curvature against something that was chosen independently, and the comparison moves with size while neither the geometry nor the choice does.

Acceleration is where it stops being simple #

One quantity in the neighbourhood does not follow the pattern, and it is worth naming because it is where the field’s boundary is.

An acceleration is a length per unit time squared. Scale the machine and hold the crank’s rate fixed and every acceleration scales as the length — exponent 1. Scale the machine and hold the rim speed fixed instead and the crank’s rate falls as 1/k, so accelerations scale as 1/k. Two different exponents for one quantity, decided by what is held constant.

That is not a defect of the quantity. It is that a mechanism’s motion has a time in it and the scaling as defined here touches only the lengths, so anything with a time in it needs a second convention before it has an exponent at all.

The survey states its convention — a fixed crank rate — and the coupler point’s speed comes out at exponent one accordingly. Curvature escapes the problem entirely because it is a property of the path rather than of the motion along it: the same curve traced quickly or slowly has the same curvature everywhere.

Curvature is a geometric quantity and a speed is a kinematic one, and only the first has an unambiguous exponent. That distinction runs through the whole site and this is the field where it is sharpest.

The one quantity with a negative exponent #

Worth pausing on, because the survey has exactly one row at −1 and it comes from here.

Nothing else the site computes is a reciprocal length. Angles are dimensionless, positions and speeds are lengths, areas are squares, counts are nothing. Curvature is the only quantity whose value falls as the machine grows.

That singularity in the table is why the curvature field’s results have to be quoted with a size and why they are the easiest of the site’s results to misread. A curvature of 0.4 means nothing without knowing that the machine’s frame bar is 4 units; the same machine at 40 units has a curvature of 0.04 at the corresponding point, and neither number is more correct.

Quoting a curvature as a multiple of the frame bar’s reciprocal — a dimensionless number — would fix it, and no field on this site does that. Recorded as an inconsistency rather than an error: every figure here draws one machine at one size and the numbers are correct for it.

A dimensionless curvature would fix it #

The inconsistency named above is worth turning into a proposal, since it costs nothing.

Every curvature this field computes could be quoted as the curvature times a reference length — the frame bar, say, or the coupler. That product is dimensionless, exponent zero, and transfers between sizes.

A coupler point whose path curvature is 0.4 on a machine with a frame bar of 4 has a dimensionless curvature of 1.6. The same machine at any size has the same 1.6. Two designs can be compared, a table can be read without knowing which machine it is about, and the number tells a reader something about the shape of the path rather than about the units.

Nothing on this site does that, and the reason is the ordinary one: every figure draws one machine at one size, so the convention never bit. It would bite the moment somebody compared two of the site’s own machines, and the comparison would be wrong by their size ratio.

Multiplying a reciprocal length by a length is the cheapest normalisation there is, and the choice of which length is a convention that has to be stated. The frame bar is the natural one for a four-bar and there is no natural one for a mechanism without a frame, which is presumably why nobody has standardised it.

The sign is the whole of it #

One exponent on this site is negative and every practical consequence in this field follows from it.

A curvature is one over a length. Scale a machine up by k and every curvature it produces is divided by k: the paths get gentler, the inflection circle grows, the centres of curvature move outward, and each of them moves reciprocally rather than proportionally. That is the only class of quantity on the site that behaves this way, and it inverts two habits a reader brings from everywhere else.

The first is about error. Everywhere else a size measured one per cent too large gives a result one per cent too large. Here it gives a curvature one per cent too small — same magnitude, opposite direction — so a systematic size error and the curvature error it produces have opposite signs, and a calibration that corrects one over-corrects the other if the sign is dropped anywhere along the way. The magnitude of the mistake is small and the direction of it is wrong, which is the harder kind to notice.

The second is about where the sensitivity lives. A quantity with exponent −1 is most sensitive where it is largest, because d(1/L)/dL = −1/L². A tight curve’s radius is badly determined by a size measurement and a gentle curve’s is well determined, which is the opposite of the intuition that says a big feature is easy to measure. On a coupler curve the sharpest turns are exactly where the geometry is least certain.

And it explains why the field’s practically useful results are all ratios of curvatures rather than curvatures. Which coupler point traces the straightest path, by how much it beats its neighbours, where the inflection circle sits relative to the machine: every one of those is a ratio of two reciprocal lengths, so the exponents cancel, so an angle-only measurement recovers all of it. What no protractor recovers is what radius anything actually hugs — and that is one number, obtained with a rule, exactly as it is everywhere else on the site.

So the field is recoverable and its units are not, which is the site’s standing result with a minus sign attached, and the minus sign is the part worth carrying. Euler–Savary is a relation among ratios and holds unchanged at every size. The centrodes are curves and scale like the machine. Only the values on them go the other way.

A dimensionless curvature — the radius as a multiple of the ground link, say — would fix every one of these at the cost of one division, and the field prints the dimensional one throughout. That is the recommendation the survey has made in five other fields, arriving here with the extra reason that a reciprocal quantity is the one a reader is most likely to scale the wrong way.