The two numbers are the curves' own

The whole of this field’s second-order theory runs on two numbers. The inflection circle — a diameter — and the pole tangent — a direction. Given those two, every point of the moving plane has its path’s curvature by one relation, the cubic of stationary curvature follows one derivative later, and the Burmester points follow after that.

Every essay has obtained them from the motion’s derivatives, which means from the mechanism. They are properties of the two centrodes, and this measures them off the curves.

The relation, and where it comes from #

Rolling gives the whole of it. The motion is the moving centrode rolling on the fixed one, so near any instant the motion is approximately one circle rolling on another — the two osculating circles at the pole — and a circle rolling on a circle is a motion whose second-order behaviour can be written down.

The result is Euler and Savary’s relation in its rolling form:

$$\frac{1}{\delta} = \frac{1}{\rho_m} - \frac{1}{\rho_f},$$

with $\delta$ the inflection circle’s diameter and $\rho_f$, $\rho_m$ the two centrodes’ radii of curvature at the pole, both signed along the common normal. The pole tangent is the common tangent, which needs no relation at all — it is the direction the two curves share.

So the two numbers the field runs on are: one curvature difference and one tangent direction, both belonging to the two curves rather than to any mechanism.

Why a second-order statement needs only circles #

There is a step in that derivation worth slowing down on, because it is why the relation is about circles at all rather than about the curves in full.

The inflection circle is a statement about second derivatives: it is the locus of points whose path has zero curvature at this instant, and a path’s curvature is a second derivative of position. Two curves that agree to second order at a point produce motions that agree to second order — so replacing each centrode by its osculating circle changes nothing the inflection circle can see.

That is the whole content of “near any instant the motion is one circle rolling on another”. It is not an approximation being made for convenience; it is an exact statement about what the quantity in question depends on. A third-order quantity would see the difference between a centrode and its osculating circle, which is why the last section of this essay is about what happens when the question is asked one derivative further on.

It also explains why the field’s second-order results are so uniform across mechanisms. A four-bar, a slider-crank and a trammel have wildly different centrodes; at any one instant each is approximated by two circles, and the inflection circle knows only the two radii. Two mechanisms with the same two centrode curvatures at an instant have the same inflection circle at that instant, whatever else is different about them — which is a much cheaper statement of the field’s opening claim than the one about whole curves.

Measuring it #

A circle is fitted to twenty-five samples of each centrode either side of the contact, in each curve’s own plane; the moving one’s centre is then carried into the fixed frame; both radii are signed along the normal; and the combination is compared against the inflection circle the motion’s derivatives give.

The agreement is 1.6 parts in a thousand at the median and 6.7 in a thousand at the worst well-conditioned sample. That is not machine precision, and it should not be: a circle fitted to sampled points is a least-squares estimate with a truncation error, and 10⁻³ is what such a fit gives on a curve of this curvature at this sampling. The claim being tested is the relation, and the instrument is the fit.

The sign, which is not a convention #

The sign was got wrong first, and the way it was wrong is worth recording because it is the useful part of the result.

Written as $1/\delta = 1/\rho_f - 1/\rho_m$, the relation reproduced the inflection circle’s diameter with the sign reversed at every one of thirty-five configurations, and the magnitudes agreed to a part in a thousand throughout. That is the signature of a consistent sign error rather than a geometric disagreement: a wrong relation is wrong by varying amounts, and a right relation with a mis-set normal is wrong by exactly one factor everywhere.

The normal here runs from the pole toward the fixed centrode’s own centre of curvature, and with that choice the moving radius comes first. Fixing it is a transposition; finding it needed the comparison to be made between signed quantities across a whole turn rather than between magnitudes at one configuration.

And the sign genuinely moves. Over one turn the two curves change which side of their common tangent they touch on eight times: sometimes the moving centrode curves the same way as the fixed one and sometimes the other way, so $\rho_f$ and $\rho_m$ have the same sign and then opposite signs. The inflection circle flips with them — its diameter is a signed quantity and it passes through infinity, which is the configuration where the two curvatures are equal and the motion is momentarily a translation.

The configurations where the fit gives up #

Nine of the thirty-five samples are ill-conditioned, and they are worth reporting rather than dropping.

Near a dead centre the pole runs off to infinity, and a centrode heading for infinity is locally almost straight. A circle fitted to an almost-straight arc is the standing example of an ill-conditioned fit: the three or more points become nearly collinear, the normal equations become nearly singular, and the radius comes out with an error that grows as the curve flattens rather than shrinking as the sampling improves.

Measured: the disagreement on those samples reaches 3.5, a relative error of three hundred and fifty per cent, on configurations where the fitted radii run past a thousand on a mechanism four units across.

Dropping those samples would have made the measurement agree with itself, which is the one thing a measurement must never be allowed to do. They are drawn in the warning colour and the two error figures are reported separately — the median over everything, and the worst over the samples whose fit is conditioned.

Where the pole tangent went #

The relation above accounts for one of the field’s two numbers and is silent about the other, which looks like an omission and is not.

The pole tangent is the direction the two centrodes share at their contact. It needs no relation at all — it is not combined from anything, it is simply read off either curve, because the two agree there by definition of rolling. So of the field’s two second-order data, one is a difference of curvatures and the other is a tangent, and only the first required any work.

That asymmetry is worth carrying because it explains a feature of Bobillier’s construction that otherwise looks lucky. That construction obtains the pole tangent from two lines already on the drawing and gets it exactly, with no limit and no approximation, while the inflection circle it then builds is a genuine second-order object. The tangent was cheap because it is a first-order quantity wearing a second-order name.

There is a check hiding in that which has not been made. The pole tangent from the motion’s derivatives, the pole tangent from Bobillier’s lines, and the common tangent of the two fitted centrodes are three routes to one direction, and only the first two have ever been compared here. The third is available from the fits already computed above, and it would close a triangle rather than a pair.

What this changes about the field’s own construction #

Bobillier’s theorem gets the pole tangent out of two lines already drawn on the mechanism, with no derivative anywhere, and the inflection circle follows from three points and a pair of compasses. That essay’s claim was that the field’s second-order data is obtainable by construction rather than by calculus.

This essay says something adjacent and stronger: the data is not merely obtainable without derivatives, it is not about derivatives. The inflection circle is a statement about two curves’ curvatures, and a curvature is a second derivative of a curve rather than of a motion. The difference matters because a curve is an object a designer can draw, measure or manufacture, and a motion’s second derivative is not.

What the numbers look like across the turn #

Reading the sweep as a table rather than as a plot makes the behaviour concrete, and three rows are worth quoting.

At 30° the two radii are 49.09 and 10.73, both positive: the moving centrode is curving the same way as the fixed one and lying inside it, and the inflection circle comes out at 13.73 against a measured 13.72.

At 20° they are 23.10 and −23.33, almost equal and opposite: the two curves touch as two circles side by side, nearly the same size, and the inflection circle is 11.61 — small, because two nearly-equal opposite curvatures add rather than cancel.

At 300° they are 3,930 and −4,782: both centrodes are nearly straight, the inflection circle is over two thousand across, and the fit reports 2,157 against a measured 2,051 — five per cent, on the pair of samples where the fit has nearly nothing to work with.

Those three are the whole range of behaviours, and what they show is that neither radius on its own predicts anything. The inflection circle is small when the curvatures differ most and large when they agree most, and the quantity that governs it is the difference of their reciprocals rather than either of them.

The degenerate cases, which are the interesting ones #

Three configurations make the relation say something a formula would hide, and all three are reachable on an ordinary four-bar.

Equal curvatures. When $\rho_f = \rho_m$ the right-hand side vanishes and $\delta$ is infinite: the inflection circle becomes a straight line. That is the instant at which the moving plane is momentarily translating — every point’s path is momentarily straight, which is a circle of infinite radius through every point rather than through a circle’s worth of them.

A straight fixed centrode. $1/\rho_f = 0$ gives $\delta = \rho_m$: the inflection circle is the moving centrode’s own osculating circle. That is the case of a wheel rolling on a rail, and it says the inflection circle of a rolling wheel is the circle of diameter equal to the wheel’s radius through the contact point — which is the classical result for a cycloid’s inflections, arriving here as an instance rather than as a separate fact.

Contact on the same side. When the two curvatures have the same sign the curves touch as a circle inside a circle, and $\delta$ is the harmonic difference. When they have opposite signs the curves touch as two circles side by side, and the difference becomes a sum in magnitude. The sign of $\delta$ is which of those it is, and the eight flips counted above are the motion passing between them.

What was checked, and what a single configuration would have missed #

Four requirements, and two of them are about the sweep rather than about any one number.

The relation must hold at the median to better than a part in a hundred — it holds to 1.6 in a thousand. The worst disagreement among well-conditioned fits must be smaller than the worst overall, which is the statement that the failures are the instrument’s rather than the relation’s; a requirement that only bounded the median would have passed on a relation that failed badly somewhere.

The two curves must change sides at least twice in a turn. That is the requirement the first version of this work did not have and needed: a check made at one configuration fixes a sign, and a sign that is fixed by the check cannot be found wrong by it.

And both centrodes must be fittable at enough configurations to sweep at all, which is a statement about the window the centrodes are traced in rather than about the geometry.

What a designer could do with it #

The practical reading is worth separating from the theoretical one, because they point at different things.

Theoretically the result says the field’s second-order apparatus is apparatus about two curves. Practically it says something narrower and more usable: if the two centrodes are the design variables, the inflection circle is a design output with a closed form in them.

That is the situation in exactly one family of mechanisms met so far, and it is the family where the exact results live. A pair of rolling circles has constant $\rho_f$ and $\rho_m$, so its inflection circle has constant diameter and the whole second-order theory of the motion is two numbers rather than two functions. A designer choosing the two radii is choosing the inflection circle directly, which is why an exactly straight line falls out of the choice rather than out of an optimisation.

For a four-bar the centrodes are not design variables — they are consequences of four lengths, and consequences that run to infinity twice a turn. So the relation is an instrument for understanding rather than for designing, and the honest summary is that its practical value rises as a mechanism’s centrodes get simpler.

Which suggests a question this field has not asked: which mechanisms have bounded centrodes? A four-bar’s are unbounded because two of its link lines become parallel. A drag-link, where both cranks turn fully, may not have that configuration at all — and if it does not, its centrodes are closed curves, its inflection circle never passes through infinity, and its second-order theory is a good deal better behaved than the four-bar drawn throughout. That is a measurement the apparatus here could make and has not.

Two routes, and why the second one is worth its cost #

It is worth being explicit about what the second route buys here, because the first one is already a complete calculation.

The inflection circle from the motion’s derivatives is exact. It comes from a closed form, the closed form is derived from the loop equation, and the loop equation is solved to the last bit — so there was nothing wrong with the number this field has been using. Recomputing it from two fitted circles produces a worse number: three decimal places instead of fifteen, with nine samples out of thirty-five where the fit falls apart entirely.

What the worse number buys is a check that the better one cannot perform on itself. A closed form has one failure mode that no amount of precision detects: being the wrong closed form. A sign, a factor of two, a radius where a diameter belongs — all of those produce a perfectly precise answer, and everything downstream inherits it. That is not hypothetical in this field: the inflection quadratic’s polar form is exactly the kind of expression where a factor can hide, and it is used by every essay that follows.

An independent route at three decimal places rules that out. It cannot improve the number and it can catch the class of error that precision is blind to, which is why a rough second measurement is preferred here to a refined first one.

The habit has a cost and it is worth naming too. Two routes take twice the work, they disagree at the third decimal place, and explaining why they disagree — a circle fit’s truncation, an ill-conditioned window — takes more words than either calculation. What it buys is that the numbers in this field are checkable by somebody who does not trust the algebra, which is the position a reader is in.

Still open: the third derivative, in the same terms #

The field’s next two constructions — the cubic of stationary curvature and the Ball point — are one derivative further on, and the same question can be asked of them.

A cubic of stationary curvature should be expressible in terms of the two centrodes’ third-order data: their curvature derivatives at the pole, or equivalently the rate at which each osculating circle’s radius changes along the curve. The classical literature carries such a form and it is not derived here, and the reason it is worth deriving is exactly the reason above — a statement about two curves is a statement a designer can act on and a statement about a motion’s third derivative is not.

The measurement it would need is harder than the one here by a whole order: fitting a curvature rate to sampled points is what the ill-conditioning above becomes when it is squared. That is a real obstacle rather than a formality, and naming it is the honest form of leaving the question open.