A symmetric curve from a lopsided machine

The curve at the top of this page is drawn by a linkage that is not symmetric in any sense a drawing would show. Its crank is 1 long and its rocker 2.5, its ground is 3, and the coupler carries its tracing point off to one side on a triangular plate with a right angle at the rocker pin. Turn the crank and nothing about the motion looks balanced: the plate lurches, the rocker swings through an arc that is not centred on anything, and the tracing point runs round a long leaning oval.

The oval is a perfect mirror image of itself. Reflect it in the dashed line through the rocker pivot and it lands on itself, point for point, to a few parts in 10¹⁵ — which is the arithmetic, not a small error.

Coupler curves are usually described as sensitive, varied and hard to predict, and all of that is true. This essay is about one of the few things that can be said about a coupler curve’s shape in advance, from three lengths and without tracing it: when it is symmetric, where the axis is, and why. It then asks what the condition does not say, and what the symmetry is worth to somebody building a machine.

Three lengths that meet at one pin #

The condition is that three distances measured from the rocker pin B are equal: B to the rocker pivot O₄, which is the rocker length; B to the crank pin A, which is the coupler length; and B to the tracing point P, which is how far the tracing point sits from the rocker pin on the coupler plate. In the figures those three are all 2.5.

The tracing point can then sit anywhere on a circle of radius 2.5 about B, and where it sits is one angle: the angle at B from the direction of A round to the direction of P. Call it γ. In the hero figure γ is a right angle, which puts the tracing point at (1, 1) in the coupler’s own coordinates, a full coupler length along and a full coupler length across.

The three distances make one geometric fact, and every result below comes from it: the crank pin, the tracing point and the rocker pivot always lie on one circle, centred on the rocker pin. The circle moves as the machine moves, because B moves, but its radius does not change and the three points never leave it.

An angle the machine cannot change #

A chord of a circle subtends the same angle at every point on the circle’s far arc, and that angle is half the angle the chord subtends at the centre. That is the inscribed angle theorem, and it is a fact about circles rather than about mechanisms.

Here the chord is AP. Its length is fixed — the coupler plate is rigid — and so is the angle it subtends at B, which is γ. The rocker pivot O₄ is on the same circle. So the angle between the direction from O₄ to the crank pin and the direction from O₄ to the tracing point is γ/2, at every position the machine can reach.

That is a strong statement about a mechanism in motion. The crank pin goes round, the rocker pin swings, the circle slides and grows no bigger, and seen from the rocker pivot the tracing point is always exactly 45° round from the crank pin. Measured at 360 crank angles, the angle changes by 4.4 × 10⁻¹⁶ radians, which is the last digit of the arithmetic.

Two details keep that exact rather than nearly so. The angle is measured as a line’s direction rather than a ray’s, because O₄ can pass from one arc of the circle to the other, where the inscribed angle is the supplement; as a line’s direction the two readings are one. And the circle is a real circle at every position, because nothing in the condition is approximate: three lengths are equal or they are not.

Every coupler curve is the crank circle, turned and stretched about O₄ #

The inscribed angle becomes a symmetry through a way of reading any four-bar at all.

Given where the crank pin is, the triangle O₄AB is fixed by three lengths — the distance from O₄ to A, the coupler and the rocker — up to which side of the line from A to O₄ the rocker pin sits on, and that side is the assembly. The tracing point is rigidly attached to A and B, so the triangle O₄AP is fixed too. Seen from O₄, then, the tracing point is always the crank pin turned through some angle and stretched along its ray by some factor, and both depend only on how far the crank pin is from O₄ and on the assembly.

The crank pin runs round a circle about O₂, and that circle is symmetric about the ground line. Reflecting the crank pin in the ground line changes the sign of the crank angle and leaves its distance from O₄ alone.

In a general four-bar the turn angle varies as that distance varies, and the reflection is scrambled on its way through. On the condition, the turn angle is the inscribed angle and does not vary. A construction that turns every point through one fixed angle about O₄, and stretches it along its ray by an amount depending only on its distance, carries a mirror line through O₄ onto a mirror line through O₄, turned through the same angle. The ground line, which is the crank circle’s axis of symmetry, becomes the curve’s.

The crank angle is the pairing #

The argument says more than the curve is symmetric. It names which point is the mirror image of which: the point traced at crank angle θ reflects onto the point traced at −θ, on the same assembly. That is a measurement with no search in it — nothing is fitted and no nearest point is sought — and it is why the pairs figure can report a miss of 4.2 × 10⁻¹⁵ rather than a distance to a polyline.

The difference between those two kinds of measurement is worth a moment, because the curves field has been burned by it before. A comparison of two sampled curves that looks for each point’s nearest neighbour on the other is a statement about where the samples happened to fall: two different curves passing close to each other over the sampled stretch pass it, and a genuinely identical curve sampled coarsely fails it by the chord error. A pairing that names its partner in advance has neither weakness. The partner of the point at 37° is the point at −37°, computed by the same solver from the same lengths, and the only thing left to disagree is the arithmetic. That is why the figure’s number can be read as a statement about the curve rather than about the sampling.

Two consequences follow at once. The curve crosses its own axis exactly at crank angles 0° and 180°, the only angles equal to their own negatives, which are the two positions where the crank lies along the ground line. And the axis is the ground line turned about O₄ through half the angle at B, which for γ = 90° is 45°.

Nothing in the argument needs the crank to turn. A linkage with ground 3.4, crank 1.2 and the other three lengths 2.2 fails Grashof’s condition, so its crank only rocks, and with the tracing point at 130° round the circle its curve is symmetric about the ground line turned through 65°, with a worst miss of 2.7 × 10⁻¹⁵. The pairing then runs over the crank’s range rather than its full turn, and the range is itself symmetric about 0°, as every four-bar’s is.

Each assembly is its own mirror image #

A crank-rocker has two assemblies, and each draws its own separate curve. The obvious guess about a reflection is that it would swap them — it swaps sides of a line, after all. It does not.

The reason is in the argument. The turn-and-stretch reading is made on one assembly, and reflecting the crank pin in the ground line preserves its distance from O₄ without changing which side of the line from A to O₄ the rocker pin is on. So θ pairs with −θ within an assembly and never across.

The crossed assembly’s curve in that figure is a very different shape from the open one — it swings down past O₄ and throws a small loop — and it is symmetric about the same line. The axis belongs to the three equal lengths and the angle at B, not to either curve.

A test that is not told where the axis is #

Everything so far has confirmed a symmetry the argument predicted. That is a weaker kind of evidence than it looks, because a test handed the axis can only report how well a curve fits that axis. It cannot say whether some other line would have done better, and it cannot find a symmetry nobody predicted.

So the second route asks the question with no axis in it at all. A curve symmetric about some line has a signed curvature, as a function of arc length, that is even about the point where the line crosses it: a reflection reverses the sense of turning and walking the mirrored half backwards reverses it again. The test samples the curvature exactly from the loop equations, resamples it in arc length, and searches every possible centre for the one about which it is most nearly even. What it reports is the largest remaining difference as a fraction of the largest curvature — an asymmetry about the best line there is.

Two things come out of it. On the condition the test finds nothing above its own floor, so there is no better line than the predicted one and no second symmetry hiding. And off the condition the asymmetry is proportional to the error, with a slope of one on both lengths: a rocker 1% longer than the coupler leaves a curve about 3% asymmetric, and an arm 1% longer about 1%.

That second fact is the useful one for anybody making the part. The symmetry does not degrade gracefully into something nearly as good. It degrades linearly, from the first micron, and the rocker length matters about three times as much as the position of the tracing hole.

Moving the point round the circle turns the axis by half #

The tracing point is free to sit anywhere on the circle about B, and each place draws a different curve with its own axis.

The axes fan out from O₄ at exactly half the angles the tracing point was moved through, which is the inscribed angle theorem read as a design rule: to put a symmetric curve’s axis in a chosen direction, put the tracing hole at twice that angle from the crank pin, measured about the rocker pin. The four curves in that figure differ in size, in shape and in where they sit, and all four are exact.

The same rule runs backwards as a check on a drawing. A coupler plate drawn with its three equal lengths and a curve whose axis is not at half the plate’s angle has been drawn wrongly, whatever it looks like.

The condition is sufficient, and a cognate shows it is not necessary #

It would be natural to read the condition the other way round: a four-bar whose curve is symmetric has three equal lengths at its rocker pin. That is false, and Roberts’s theorem supplies the counterexample without any search.

Every coupler curve is drawn by three four-bars, and that has been checked by overlaying their traces and by fitting one equation to all three. Applied to the symmetric linkage, the construction gives two more.

The cognate grounded on O₄ and the new pivot O₃ has crank 2.5 at O₄, coupler 2.5 and rocker 1, with its tracing point a full coupler length across from its crank pin. That is the same condition with the crank and rocker exchanged — three equal lengths at the crank pin instead of the rocker pin — and its pivot O₄ is on the axis. It is a second instance of the rule rather than an exception to it.

The cognate grounded on O₂ and O₃ is the one that matters. Its lengths are 4.243, 3.536, 1.414 and 3.536, it is a double rocker, and no three of its lengths are equal at either moving pin. Neither of its pivots is on the axis: both are 2.121 away. Every point it traces satisfies the original curve’s exact equation to 2 × 10⁻¹⁸, on the same oval rather than the other assembly’s, and the axis-free test finds its curve’s asymmetry falling towards zero as the sampling is refined — from 2.2 × 10⁻⁴ at 1,024 positions to 4.4 × 10⁻⁶ at 8,192 — where a curve 2% off the condition holds at 0.0385 however finely it is sampled.

So the symmetry is a property of the curve, and the three equal lengths are a property of one way of drawing it. A designer who wants a symmetric curve from a double rocker can have one, and cannot find it by looking for equal lengths.

Symmetric in time as well as in shape #

The pairing is by crank angle, so it says something about the motion that the curve’s shape alone does not. If the crank turns at a steady rate, the tracing point arrives at the axis crossing along one half of the curve exactly as it leaves along the other half, run backwards.

That is the version of symmetry a machine cares about. A pick-and-place motion, a feed that must approach a station and withdraw from it, or a transfer that accelerates and decelerates through a gap all want the second half of a stroke to mirror the first in time. A curve that is merely the right shape does not guarantee it — the speeds along the two halves could differ — and on the condition it is guaranteed, because the two halves are the same crank angles with the sign changed.

What a designer is given: a vertex on the axis #

A symmetric curve’s curvature, as a function of arc length, is even about each point where the curve crosses its axis, so its rate of change is zero there. A point where a curve’s curvature stands still is a vertex, and at a vertex the osculating circle is followed to fourth order rather than third. Those are exactly the points the site’s dwell linkages look for: a link pinned at the centre of curvature holds its far end nearly still while the tracing point passes.

The symmetry locates the vertices without a search: they are at crank angles 0° and 180°, on the axis, for every tracing point on the circle and every set of lengths meeting the condition. Finding a vertex on a general coupler curve means solving for where curvature stands still, and the answer moves when anything moves.

But a vertex is not rare. Every closed curve has several, and a fourth-order dwell is available at each of them whether or not the curve is symmetric. It would be an overstatement to say symmetry buys the dwell. What it buys is sharper and smaller, and it needs its own measurement.

At a generic vertex the deviation off the circle has a fifth-order part that is odd in the crank angle, so the dwell is slightly different on the way in and on the way out: fitted slopes 4.99 on both perturbed curves. On the symmetric curve’s axis that part is zero to 1.8 × 10⁻¹⁵, while the even part grows at the fourth power, fitted at 3.98, as at any vertex. The dwell on the axis is exactly as good arriving as leaving, which for a machine that holds a part still between two motions is the property that makes the two motions interchangeable.

What is not established #

Four limits, each of a different kind.

Necessity is refuted and nothing replaces it. The cognate shows a symmetric curve need not come from three equal lengths. What the full set of four-bars with symmetric coupler curves looks like — whether it is exactly the linkages meeting the condition at either moving pin together with their cognates — is not claimed here.

The axis-free test has a floor, and the floor depends on the route. With exact curvatures it reads 8.7 × 10⁻¹²; for the double-rocker cognate, whose curvature is taken by differences along a circuit that reverses at its limits, the reading falls only as the square of the spacing and was not pushed below 4.4 × 10⁻⁶. The claim that the cognate’s curve is symmetric rests on its points lying on the symmetric curve’s equation, not on that test.

No dwell machine was built. The vertex on the axis is located and measured, and a six-bar using it is not assembled, swept or compared with the searched dwell already built.

Tolerance is measured only one length at a time. The two slopes in the residual figure say how each error acts alone; how the errors combine, and whether a clearance in the pins destroys the time symmetry before it destroys the shape, is not computed.

What comes next #

The next step builds the dwell. The dwell made from a curve found its six-bar by searching a coupler curve for the stretch most nearly a circular arc, and held 146° of crank inside a one-degree band. The symmetric curve offers a different construction with no search in it: pin the dwell link at the centre of curvature where the curve crosses its axis, at crank angle 0° or 180°, and choose the angle at B to shape how long the curve follows the circle. It would sweep that angle, measure the dwell each choice gives against the searched one, and test whether the exact equality of approach and departure survives the second loop of the six-bar — which it should, because the output link’s motion is a function of the tracing point’s alone, and which is exactly the kind of claim that ought to be measured rather than assumed.