The frame seen from the coupler

The curvature field describes one thing throughout: a plane moving over a plane. The mechanism drops out was its opening claim — near any instant, a four-bar’s coupler and a pair of gears are indistinguishable if their first few derivatives agree — and every construction since has been a statement about the moving plane’s points as seen from the fixed one. The circle of points going straight found where the coupler’s points momentarily travel in straight lines. Every point has a centre found, by Euler–Savary, the centre of curvature of any coupler point’s path.

Nothing in that description privileges either plane. A four-bar’s frame is a link like any other, and whether the coupler moves over the frame or the frame over the coupler is a choice of which link a person stands on. The linkage put back from two curves ended by naming that swap — the inverse motion — as nearly free: every quantity in the field has a counterpart obtained by holding the other link, and a few of those counterparts are famous under other names. The apparatus for both halves was to hand and unjoined.

This essay joins it, and the answer is short enough to state first. Held the other way, the inflection circle is replaced by its mirror image through the pole, and a point and the centre of curvature of its path trade places.

Holding the other link #

Put a frame on the coupler with its origin at the crank pin $A$ and its x-axis along the coupler. At crank angle θ the coupler has turned through φ, and a point $X$ of the frame has coupler coordinates

$$\zeta = e^{-i\varphi}\,(X - A).$$

As θ changes, a fixed frame point’s coupler coordinates change, so seen from the coupler the frame is a moving plane. Its reference point — the frame’s origin — is at $\bar A = -e^{-i\varphi}A$ and its angle is $-\varphi$. That is all the inverse motion is, and its derivatives follow from the ordinary motion’s by the product rule: $\bar A' = -(e^{-i\varphi})'A - e^{-i\varphi}A'$ and so on up to third order, with $(e^{-i\varphi})' = -i\varphi' e^{-i\varphi}$.

The curvature field’s routines take any moving plane described by a reference point, its angle and their derivatives. Handed the inverse motion, they compute its pole, its inflection circle and the path curvature of any of its points with no change at all.

There is a second, independent route, and it is the one that makes the result a claim about machines rather than about notation. Holding the coupler is choosing a different link of the chain as the frame. The four-bar with ground $g$, crank $a$, coupler $b$ and rocker $c$, held by its coupler, is a four-bar with ground $b$, crank $a$ pivoting at the old crank pin, coupler $g$ and rocker $c$ pivoting at the old rocker pin — one of the chain’s inversions. Solved as a four-bar in its own right, in its own frame, its moving plane is the old frame, and its derivatives come out of the loop equation with nothing borrowed from the first route. One care is needed: that four-bar has two assemblies at a given crank angle, and its rocker pin must be seeded at the old ground pivot so that it is the configuration the original linkage is actually in; solved from the usual starting guess, it lands on the other assembly, 5.3 units away, and reports a different circle.

The first route has a check of its own that costs nothing. Swapping the planes twice must give back the original motion, and it does: the swap of the swap reproduces the crank pin’s position and its first three derivatives to $2 \times 10^{-16}$ and the coupler’s angle and its derivatives exactly. A sign error in the product rule — the likeliest mistake, since every term of the third derivative carries one — would fail that test at the first derivative it touched.

The circle on the other side of the pole #

Each route gives an inflection circle for the inverse motion, and both can be compared with a prediction that uses neither: the ordinary inflection circle, carried into the coupler’s frame and reflected through the pole.

At seven crank angles from 20° to 320°, on the four-bar with ground 4, crank 1, coupler 3.5 and rocker 3, the inverse circle found by differentiating the swap is on the reflected circle to $3 \times 10^{-16}$ of the radius, and the one found by solving the coupler-held four-bar to $5 \times 10^{-14}$. That holds where the radius is 5.8 and where it is 400, near the positions at which the coupler momentarily translates and the circle runs off towards infinity. The pole of the inverse motion is the pole of the ordinary motion to $10^{-15}$, as it must be: the point of the coupler that is not moving relative to the frame is the point of the frame not moving relative to the coupler.

The reflected circle is not merely close to the original. The two touch at the pole and lie on opposite sides of the pole tangent, and their centres are one full diameter apart at every angle in the table — so the agreement is between the inverse circle and a circle a whole diameter away from the one a careless reading would have predicted.

Which points go straight in each #

The reflected circle has a physical reading, and it is checked directly rather than inferred.

Frame points are taken every five degrees round each circle, placed in the coupler’s frame, and handed to the inverse motion’s path-curvature routine. Round the reflected circle every one of them has curvature nought, to $3 \times 10^{-9}$ in units of the coupler’s length: those are the frame’s points travelling straight when the coupler is held. Round the ordinary circle the curvature is nowhere nought except at the pole, and away from it at least 0.8.

So the two circles belong to the two directions of one relative motion. The ordinary inflection circle is where the coupler’s points go straight over the frame; the reflected one is where the frame’s points go straight over the coupler. The classical literature calls the second the return circle, and its usual definition — the locus of centres of curvature for coupler points at infinity — is the same statement read through the swap that the next section measures.

A point and its centre trade places #

Euler–Savary is usually read in one direction: a coupler point, and the centre of curvature of the path it traces over the frame. The centre is a point of the frame. The swap says it should also be read the other way.

Six coupler points are chosen at a crank angle of 225°, and each point’s centre of curvature is computed in the ordinary motion. Each centre is then treated as a point of the frame and its own path, as the frame moves over the held coupler, is given a centre of curvature by the inverse motion’s routine — a separate computation with its own derivatives. Every one lands back on the coupler point it started from, to $7 \times 10^{-16}$ at worst.

That is the involution behind the word conjugate. A coupler point and its centre of curvature are a pair, and which of the two is “the point” and which “the centre” is decided only by which link is held. It is also why the line through the pair passes through the pole in both directions: Euler–Savary’s geometry is symmetric in the two planes, and only the signs of its distances change when they are exchanged.

The inflection circle is the special case. A coupler point on it has its centre at infinity. Swapped, a frame point at infinity — a direction — has its centre on the return circle, and a point on the return circle has its centre at infinity in the inverse motion, which is what “travelling straight” means. The mirror relation between the circles is the conjugate relation applied to points at infinity.

Why a mirror #

The mirror follows from Euler–Savary in one line. In the ordinary motion, a coupler point at distance $r$ from the pole and its centre at distance $r^*$ along the same ray satisfy

$$\Big(\frac{1}{r} - \frac{1}{r^*}\Big)\sin\psi = \frac{1}{\delta},$$

with ψ the angle from the pole tangent and δ the inflection circle’s diameter. Exchanging the planes exchanges $r$ and $r^*$, which changes the sign of the left side. The inverse motion’s inflection circle is where $r^* \to \infty$ in the exchanged equation, $r\sin\psi = -\delta$, which is a circle of diameter δ through the pole with the opposite sign of ψ: the same circle on the other side of the tangent.

What the measurement adds to that line is that it is true of the machine and not only of the formula. Two computations with no common step — the product rule on the swapped plane, and a Newton solve of a different four-bar — agree with it at every angle tried, including those where the diameter runs to hundreds of coupler lengths.

The same circle from three prescribed positions #

The mirror circle has already appeared in the collection once, under a different name and without being recognised. Where a pin becomes a slide asked which fixed pins a slot cut in a moving body can pass over in three prescribed positions, and found them on a circle: the circumcircle of the three poles. It also found that the corresponding circle of body points wanting a slide converges on the ordinary inflection circle as the three positions are brought together.

The inverse motion predicts where the other one goes. A fixed pin that a slot in the coupler passes over in three nearby positions is, in the limit, a frame point that travels in a straight line relative to the coupler — a point of the return circle. Measured on the same four-bar at a crank angle of 60°, with the three positions taken at 60° − h, 60° and 60° + h, the pole triangle’s circumcircle has its centre 0.091 from the return circle’s at h = 0.2, 0.023 at h = 0.1, and $2.3 \times 10^{-4}$ at h = 0.01 — a factor of four for every halving of the spread, the same square-law convergence the slider circle showed towards the inflection circle.

So the finite construction that has one of the two circles as its limit has the other as its limit too, with the planes exchanged, and the pole triangle and the image-pole triangle that are mirror images for three finite positions become the two circles that are mirror images at an instant. A designer choosing a slot on a coupler to pass over a fixed pin — a common way to guide a link — has the return circle as the place to look for pins that will stay in a straight slot longest.

The inverse motion is another machine #

The second route has a consequence that deserves its own statement: the inverse motion of a four-bar is the ordinary motion of the four-bar whose ground is the old coupler. So every inverse quantity is also an ordinary quantity of a different machine, and the correspondence between curves is a correspondence between mechanisms.

Whether the machine changes kind depends on where the shortest link is, which is the classification by the three signed sums at work. On the three crank-rockers in the table whose crank is the shortest bar, the crank is adjacent to both the ground and the coupler, so holding either leaves a crank-rocker. With the coupler as the shortest bar, the ordinary machine is a double rocker and the coupler-held one is a double crank; with the ground shortest, it is the other way round. The mirror relation holds on all five, to $3 \times 10^{-14}$ of the radius.

That gives the inverse motion a practical use. A designer who needs the inflection circle of a double crank — a drag link, whose centrodes and inflection circles are awkward to draw because the coupler turns all the way round — can compute it for the double rocker on the same chain, where the coupler is the shortest link and barely turns, and reflect.

Where both circles run away together #

The table’s largest radii, 400 at a crank angle of 100° and 303 at 320°, are near the instants at which the coupler stops turning for a moment and translates. There the pole runs off to infinity along with both circles, and every construction of the field that is measured from the pole loses its footing.

The swap does not rescue those instants, and it could not. If the coupler is not turning relative to the frame, the frame is not turning relative to the coupler: the inverse motion translates at the same instant, its pole is at infinity in the same direction, and its inflection circle is as large as the original’s. The mirror relation still holds as the radius grows — the measured agreement at a radius of 400 is as good as at 5.8 — but it relates two circles that are both becoming straight lines, the two sides of a pole tangent that is itself receding.

What survives a translation is the conjugate relation. A point and its centre of curvature are defined without reference to the pole’s distance, and the swap exchanges them whether or not the pole is finite. A designer who needs the curvature of a frame point’s path over a coupler near a translating instant can compute it from the coupler point it is conjugate to, which is the one statement of this essay that does not degrade where the others do, and it is the one to reach for first when a mechanism works close to such an instant.

What this does not settle #

Only the second-order constructions are swapped here. The inflection circle, the pole and the conjugate centres use derivatives up to the second. The cubic of stationary curvature and the Ball point need third and fourth derivatives, and the swap’s fourth derivative is not computed; whether the inverse motion’s cubic is the original’s under some equally simple map is not measured.

The centrodes are not redrawn. The inverse motion’s fixed centrode is the ordinary motion’s moving centrode and vice versa, which is the swap in its most visible form, and it is stated here rather than drawn.

It is one family. Every measurement is on four-bars. The derivation is general, but a slider-crank’s inverse motion — the connecting rod held, the frame and slide moving — has a different second route and is not checked.

Still open: rolling two curves that came from nowhere #

The inverse motion completes a symmetry the field has been using without stating: every planar motion is one centrode rolling on another, and swapping which is fixed swaps every derived construction in the way measured here. What it does not do is leave the four-bar.

The demonstration that would, and the one the linkage put back from two curves named first, is to take two curves nobody derived from a mechanism, roll one on the other, and ask what the resulting motion’s constructions are. Its distinct argument would be that computation for a pair of rolling polygons, the simplest case with no four-bar behind it: the pole jumps from corner to corner instead of sliding, the centrodes’ curvature is infinite at the corners and nought between them, and the question is whether the inflection circle and its mirror degenerate into something still usable — a circle that switches at each corner — or whether the field’s second-order apparatus refuses a motion whose pole does not move continuously. Two things would come out of it: a limit the constructions have to obey as a smooth pair of centrodes is sharpened into polygons, and a measurement of whether the mirror relation survives when there is no machine whose inversion could explain it.