Exact because two circles roll

Two things are called a straight-line mechanism on this site and they are different in kind rather than in degree. Watt’s linkage traces a curve that departs from its chord by nine per cent of its span, Chebyshev’s by twelve, and Peaucellier’s cell traces a line exactly — straight to $9.8\times10^{-16}$ of its span, which is fourteen orders better and is not an improvement so much as a change of category.

There is a third case, older and much simpler than Peaucellier’s, and it makes the reason for the distinction visible in a way the inversor does not. Take a rod with its two ends running in two perpendicular slots. Each end travels in a perfect straight line, obviously, because a slot is a straight line.

The interesting part is what the rest of the rod does, and why.

The pole, first #

The rod’s two ends have known velocity directions: along their own slots. The pole is where the perpendiculars to those directions meet, which is the corner of a rectangle with the rod as its diagonal.

So if the rod has length $L$ and makes angle $\theta$ with the horizontal, its ends are at $(-L\cos\theta, 0)$ and $(0, L\sin\theta)$, and the pole is at $(-L\cos\theta, L\sin\theta)$. That point is at distance $L$ from the origin at every $\theta$: the fixed centrode is a circle of radius $L$, measured to $8.9\times10^{-16}$ over a full turn.

In the rod’s own frame the pole’s body coordinate traces a circle of radius $L/2$ centred at the rod’s midpoint — measured to $6.7\times10^{-16}$. The moving centrode is a circle of half the radius, and since the motion is the moving centrode rolling on the fixed one, this is a circle rolling inside a circle of twice its diameter.

What a circle rolling inside a circle of twice its size does #

That arrangement has a name — the Cardan circles, after Gerolamo Cardano, though the construction predates him by a long way and is often called the Tusi couple after Nasir al-Din al-Tusi, who used it in the thirteenth century.

Its property is that a point on the rolling circle’s rim traces a diameter of the fixed circle. Exactly. A hypocycloid with a radius ratio of two is a degenerate hypocycloid: it collapses onto a straight line segment, traversed back and forth.

The rod’s two ends are rim points of the rolling circle, which is why they run in the slots. And by the cusp argument, a rim point is a point of the moving centrode, so its path should be cusped — and it is: a point running back and forth along a segment has a corner at each end, which is exactly what a degenerate hypocycloid’s two cusps are.

Every other point of the rod’s plane traces an ellipse, which is why the mechanism is called an elliptic trammel and why it was sold as an ellipsograph.

Straight to nothing #

The rod ends stay on their axes to zero. Not a small number: the measured worst departure over 360 positions is exactly $0$, because the coordinates come out as $-L\cos\theta$ with an exact zero in the other component, and there is no arithmetic left to lose.

Set against the site’s other straight-line numbers:

The gap between the second and third rows is the one the exact-and-approximate thread exists for. The gap between the third and fourth is arithmetic rather than kinematics: Peaucellier’s cell reaches its line through a solve and a product of two lengths, so it carries rounding, and the trammel’s answer is a coordinate that was never computed.

Why the exactness is not surprising once the centrodes are in view #

The site’s usual framing of the straight-line problem is that it is hard: Watt spent years on it, the approximations are what everyone used for a century, and Peaucellier’s exact solution took until 1864. Against that, a rod in two slots looks like cheating.

The centrode picture explains why it is not. A motion is completely determined by its two centrodes, and the question “is there a motion with a point that travels in an exact straight line” is the question “is there a pair of centrodes with that property”. The Cardan pair answers it immediately.

What is hard is not producing the motion; it is producing it with pin joints only. A slot is a prismatic pair, and a prismatic pair is a straight line that somebody had to make. Watt’s problem was to get a straight line out of hinges, on a beam engine, in 1784, where a long accurate slideway was expensive and wore out. Peaucellier’s achievement is that his cell has no slot in it and no straight edge anywhere; the line comes out of an inversion.

So the ranking above is not a ranking of cleverness. It is three answers to three different questions, and the trammel’s is the easy question.

The check that had to refuse something #

The assertion behind this essay measures three quantities over a full turn: the rod ends’ departure from their axes, the fixed centrode’s departure from a circle of radius $L$, and the moving centrode’s from a circle of radius $L/2$. All three come back at or below $10^{-15}$.

Three measurements at rounding is exactly the pattern that ought to make a reader suspicious, because it is also what a check that is not testing anything looks like. So it is worth being explicit about where each number could have gone wrong.

The rod-end measurement reads coordinates out of the motion machinery rather than out of the closed form — the same loop-derivative code that draws every other figure in this field, given a loop with two sliders in it instead of two revolutes. If that code mishandled a prismatic pair, this is where it would show, and the value would not be zero.

The two centrode measurements come from the general pole expression $iA'/\varphi'$, evaluated on this motion. They are the same lines of code that place the pole on a four-bar. A sign error there would put the fixed centrode’s radius at something other than $L$.

And separately, the trammel’s four derivatives of $A$ are compared against their closed forms — $L\sin\theta$, $L\cos\theta$, $-L\sin\theta$, $-L\cos\theta$ — and agree to $2.2\times10^{-16}$. That is the check that the general loop machinery, fed a loop with sliders, reproduces a case where the answer is known in advance.

Getting the Cardan motion out of pins after all #

There is one more twist worth having, because it turns the trammel into a piece of hardware that appears in real machines.

Since the motion is a circle rolling inside a circle of twice the radius, it can be produced by gears: an internal ring gear with twice the teeth of a planet, with the planet rolling inside it. A pin on the planet’s pitch circle then travels in an exact straight line, and the whole thing is made of pin joints and teeth with no slot anywhere.

That is a Cardan gear or a hypocycloidal straight-line drive, and it has been used for exactly that purpose — reciprocating motion from rotation without a slideway. It is also the same arrangement as a cycloidal drive’s in a different ratio, and the same as one of the epicyclic trains the transmission field builds with the ring held.

Two mechanisms, then, that share nothing in their construction — two sliders and a rod, or a ring gear and a planet — and produce the same motion because they have the same centrodes. That is this field’s opening claim with hardware attached to both ends of it.

What the trammel does to the rest of this field’s machinery #

The trammel is the field’s degenerate case and it breaks two of the field’s measurements, which is worth recording because both breakages are informative.

Its angular rate is exactly constant. The rod’s angle is the input, so $\varphi' = 1$ and $\varphi''$, $\varphi'''$ and $\varphi''''$ are all exactly zero. That is a motion with no third-order content at all.

The consequence is that the condition defining the points with fifth-order circle contact becomes an identity along the whole cubic rather than a condition satisfied at isolated points. A search for those points returned 346 of them before the machinery was taught to notice, and four is the most a planar motion can have. The count is now reported as undefined with the reason attached, because a large number here is not a count of anything.

Its acceleration field is purely centripetal. With $\varphi'' = 0$ the map from a point’s offset to its acceleration is a negative real scalar, so every point of the rod accelerates straight towards the acceleration pole at a rate proportional to its distance. That looks exactly like a body in steady rotation about a fixed point, and the rod is not in steady rotation about anything. It is a case where a correct general formula evaluated on a special motion produces a familiar picture that would be badly wrong if read back into the general case.

The inflection circle, which comes out as the rolling circle #

One more coincidence that is not one. The trammel’s inflection circle — the locus of points momentarily going straight — has diameter $L$ and radius $L/2$, which is the moving centrode’s radius.

That is forced. The two rod ends are on the inflection circle at every instant, because they are travelling in exact straight lines and a point travelling in a straight line has zero path curvature. Both ends are also on the moving centrode. A circle through the pole and both rod ends is determined, and it is the rolling circle.

For a general motion the inflection circle and the moving centrode are different curves that happen to touch at the pole. For this one they coincide, and the reason is that every point of the moving centrode momentarily goes straight — which is what a degenerate hypocycloid family means.

So the trammel is simultaneously the field’s simplest example and its most degenerate one, and both facts have the same cause.

Where the exactness comes from, in one sentence #

An approximate straight-line linkage keeps a tracing point near the inflection circle over a stroke. An exact one puts a point on it and keeps it there.

Watt’s point is inside the circle at the middle of the stroke and drifts across as the mechanism turns, so its curvature passes through zero twice and is small in between — nine per cent of the span. Ball’s point is exactly on the circle at one instant and drifts off, which raises the order of contact at that instant and improves the stroke to 3.8 per cent. The trammel’s rod ends are on the circle at every instant, because the circle moves with them.

That is the whole hierarchy: near the circle sometimes, on it once, on it always. The three cases are approximate, higher-order approximate, and exact, and nothing in between the last two exists.

Three mechanisms, one motion, and what that is worth #

It is worth collecting what has been shown to produce this same motion: two sliders and a rod; a planet gear rolling inside a ring with twice its teeth; and — a third, which the site’s inversion machinery supplies for free — the inverted slider-crank obtained by fixing a different link of the same four-bar chain the trammel belongs to.

Three arrangements with different parts, different joints, different failure modes and different manufacturing costs. Identical motion, because identical centrodes.

That is the practical payoff of describing a motion rather than a mechanism, and it is a designer’s payoff rather than an analyst’s. A wanted motion does not determine a machine. It determines a family of machines, and the choice within that family is made on grounds the kinematics does not see: which parts can be made accurately, which joints wear, which arrangement fits in the space, which one has a transmission angle that stays workable.

The site has met the same structure once before, in Roberts’s cognate theorem: three different four-bars trace the same coupler curve exactly. That is the finite version of this — three mechanisms sharing a whole curve — and the centrode statement is the stronger one, because sharing centrodes means sharing every point’s path at once rather than one point’s.

The straight line is one end of a family of ellipses #

The mechanism is called an elliptic trammel and the essay has so far used only the degenerate member of the family. Taking the whole family at once explains the exactness better than the Cardan circles do, and it shows why the straight line is not the achievement it looks like.

Put a tracing point on the rod at a distance $p$ from one end and $q$ from the other, so $p + q = L$. As the rod runs, that point traces an ellipse with semi-axes $p$ and $q$, aligned with the two slots. One rod therefore draws every ellipse whose semi-axes sum to its own length, and which one is drawn is chosen by sliding the tracing point along the rod.

The two ends of that family are the interesting members and both are already in this essay. At $p = q = L/2$ the semi-axes are equal and the ellipse is a circle — which is the point at the middle of the rod, and it is the pole’s own path. At $p = 0$ one semi-axis vanishes and the ellipse degenerates to a straight segment of length $2q$ — which is a rod end, running in its slot.

So the straight line is the degenerate ellipse, and the circle is the symmetric one, and everything in between is a proper ellipse traced by an ordinary point of the rod. The mechanism is not a straight-line linkage that happens to draw ellipses; it is an ellipse-drawing mechanism whose family contains a straight line at its boundary.

That is the clearest account of why the exactness costs nothing here. Watt’s and Chebyshev’s linkages produce a curve that is nearly straight because a sextic has been arranged to have a flat stretch, and the flatness is a local property bought by choosing four lengths well. The trammel produces a straight line because a straight line is a member of the family its motion produces, exactly, at a particular tracing point — no arrangement, no working range, and nothing to optimise.

It also says what the mechanism costs to build in the currency that matters. The straight line and the circle are the two members needing no accuracy in the tracing point’s position along the rod, since both are at distinguished places — the end and the middle. Every proper ellipse needs the tracing point placed accurately, and an error in $p$ moves both semi-axes, one up and one down. So the family is drawn to whatever accuracy the rod is marked to, except at its two ends, where the geometry does the marking.

Which is a small instance of the pattern this field keeps producing. A motion is the object; the mechanisms are ways of producing it; and the curves are what points of the moving plane happen to trace. Asking for a straight line and getting a family of ellipses is not a bonus — it is the consequence of having described the motion rather than the curve.

What is not claimed #

The trammel is exact and it is not a good machine, and the site’s habit is to say why rather than to leave the impression that the classical designers were missing something obvious.

Two sliding pairs is two slideways, and a slideway is the component Watt did not want. It also has a transmission problem: near the ends of the stroke the driving force is nearly perpendicular to the direction the driven end can move, and the mechanism is close to jamming — the same geometry the site’s toggle essay is about. And a rod in two slots has clearance in both of them, so the built version’s rod ends do not stay on the axes to zero, they stay on them to whatever the slot clearances are.

The exactness is a fact about the kinematics of an ideal mechanism, which is what this whole field computes. It is not a claim that the resulting machine is straight, and the site has a field about the difference.