The straightest point there is

A point of a moving plane whose path is momentarily straight is on the inflection circle. A point whose path curvature has momentarily stopped changing is on the cubic of stationary curvature. A point on both has a path that is straight and staying straight — zero curvature, and a curvature that is not about to become nonzero.

There is one such point, at nearly every position of a mechanism, and it has a name. Robert Stawell Ball described it in 1871 and it is called Ball’s point, or sometimes the point of undulation.

What four-point contact means, measured #

The claim is that Ball’s point’s path holds its tangent line one order longer than a neighbour’s does. The measurement that establishes it is deliberately blunt.

Take the point’s tangent line at the instant. Drive the mechanism away from that instant by a step, and measure the perpendicular distance from the point to that fixed line. Repeat over a decade of step sizes and fit a slope on log-log axes.

For Ball’s point the fitted exponent is 3.95. For a point on the same ray from the pole but at 60 per cent of the distance — off the circle, off the cubic, ordinary in every way — it is 1.87.

The second number deserves a moment. A point that is not on the inflection circle has nonzero path curvature, so its distance from its own tangent grows like the square of the step, and $1.87$ is that. A point on the circle but not on the cubic has zero curvature and nonzero $\kappa'$, so the distance grows like the cube. Ball’s point has both zero, so it grows like the fourth power. The exponent is a count of how many derivatives of the offset vanish, and it reads directly off the plot.

Finding it, and the three searches that were wrong #

Both curves are known in closed polar form about the pole, so the obvious search is to compare their radii ray by ray and look for a sign change in the difference. That was the first version and it returned two points at every position of the crank, which is a problem, because the classical statement is that there is one.

Checking the two candidates settled it. At one crank position the first had a path curvature of $-1.3\times10^{-15}$ and a curvature rate of $-6.3\times10^{-14}$: on both curves, genuinely. The second had a curvature of $2.3\times10^{-16}$ — on the circle — and a curvature rate of twelve. It was not on the cubic at all.

The reason is the cubic’s asymptote. Its polar radius runs to infinity at the four ray directions where its cubic coefficient passes through zero, jumping from $+\infty$ to $-\infty$, so the difference of the two radii changes sign there without the curves meeting. A bisection cannot tell that from a root; it converges neatly on an angle where nothing happens.

The second version walked along the inflection circle instead and looked for a zero of $\kappa'$, which removes the cubic’s radius from the search entirely — every point visited is on the circle by construction, so $\kappa$ is zero at all of them and only one condition is left. That was better and it still missed the point at seven of 360 positions, because it parameterised the circle by ray angle from the pole. The circle’s radius along a ray vanishes at the two ray directions along the pole tangent, $\kappa'$ runs to infinity there, and a Ball point close to the pole is squeezed between the root and that asymptote.

The third version parameterises the circle by its own arc. Uniform spacing in distance rather than in a ray angle, one singularity left instead of three, and it is at a place that can be computed exactly — the pole’s own position on the circle — so the samples can be crowded there rather than guessed at.

Found at 358 positions of 360, and the two exceptions #

With that search, over 360 crank angles of the site’s four-bar: one Ball point at 358 positions and none at two.

The two exceptions are not failures of the search. At 119° the coupler’s angular rate is $9.5\times10^{-5}$ per radian of crank and at 308° it is $8.2\times10^{-4}$ — the coupler is instantaneously translating — and the inflection circle’s diameter at those positions is $6.4\times10^{10}$ and $2.2\times10^{8}$ coupler lengths. The point is real and it is not on any page: it has gone to infinity along with the circle it sits on.

That is a better statement than the textbook’s exactly one, because it says where the object exists and where it does not, and the boundary is a measurable property of the position rather than a caveat. The check that goes with it requires the count to be one at more than 95 per cent of positions, and requires every exception to be a position with $|\varphi'| < 0.01$ and $\delta > 10^5$ coupler lengths — so a miss for any other reason fails rather than being absorbed.

Where the classical designers put their tracing point #

Watt’s linkage and Chebyshev’s both exist to make a point travel in a nearly straight line, and both trace with the midpoint of the coupler. That choice comes from the mechanism’s symmetry: the two arms are equal, the configuration is symmetric about a vertical, so the midpoint is the obvious place.

The question this essay asks is whether it is the right place, and the answer is no in both cases, by margins that differ by an order of magnitude.

The comparison is made at the middle of each linkage’s working arc rather than at its symmetric pose. That distinction is not a detail. The foundation of this site recorded that the symmetric configuration of both linkages is a limit of the travel rather than the middle of it, and it is exactly as misleading here: at Watt’s symmetric pose the coupler midpoint’s path curvature is $4.64$, a radius of a fifth of a unit on a linkage four units across, because the mechanism is nearly at a dead centre. Sampling there would describe the dead centre.

Watt’s, measured #

Watt’s linkage with span 4, arms 2 and coupler 1, over the middle 80 per cent of what it can reach:

The first row is a number this site already publishes. The straight-line essay reports Watt’s approximation at nine per cent of its span, computed by a completely different route — a traced polyline and a chord — and this field’s independent machinery reproduces it. That agreement is worth as much as anything else in the table: a new field arriving at an established number by a new route is the cheapest available check that the new field is describing the same mechanism.

The second row is the finding. Ball’s point is 0.27 coupler lengths from the midpoint and it more than halves the error over the whole stroke, from 8.98 per cent to 3.79. The worst absolute departure falls from 0.187 to 0.087.

That is a substantial improvement available for free — a hole in a different place on the same bar — and Watt did not take it. He could not have: Ball’s point is a third-derivative object and Ball published in 1871, eighty-seven years after Watt’s patent. The point being made is not that Watt missed something available to him; it is that the criterion he was optimising and the criterion the motion answers are different, and the difference is measurable.

Chebyshev’s, where the answer changes character #

Chebyshev’s lambda linkage, same treatment:

The relative departure improves by a factor of 3.6. The worst departure improves by a factor of 1.34, which is almost nothing.

The reason is in the numbers that are not in the table. Ball’s point for Chebyshev’s linkage is 3.26 coupler lengths from the midpoint — a point that would need a rigid extension three times the length of the coupler bolted to it — and the stroke it traces is 10.24 units long against the midpoint’s 3.83. So the relative measure is being computed over a much longer span, and dividing a slightly smaller absolute error by a much larger span produces most of the improvement.

That is not a defect of the measurement; it is the reason both columns are reported. A relative error over a span is the right measure for comparing two mechanisms of different sizes, which is what the straight-line essay needed it for. It is the wrong measure for asking whether moving a pen improves a machine, because moving the pen also changes the span.

Chebyshev was answering a different question, and said so #

The deeper reason Chebyshev’s point is not Ball’s point is that Chebyshev was not looking for a point of high-order contact at one instant. He was a founder of approximation theory and he was minimising a maximum over a range.

Those two criteria pull in opposite directions and the shape of the error curve shows it. A point with high-order contact has an error that is very small near the instant and grows quickly at the ends of the stroke. A minimax point has an error that oscillates, touching the same worst value several times, with nothing special happening at the middle. The first is a Taylor criterion; the second is a Chebyshev criterion, and the name is not a coincidence.

The site has this distinction elsewhere in a sharper form. The essay on where the precision points go is about exactly this trade in the synthesis field, and the one on where an optimiser starts records the measurement that a least-squares fit beats an interpolant by 18 per cent on root-mean-square error and is 75 per cent worse on the maximum. An optimiser wins on the measure it was given, and so does a designer.

How far the two criteria can be pulled apart #

The comparison so far is between one point chosen for symmetry and one chosen for contact order. It is worth asking how much of the gap is about the criterion and how much about the particular linkages, and the answer comes from the two rows of the table read against each other.

Watt’s Ball point is a quarter of a coupler length from the midpoint and improves everything. Chebyshev’s is three and a quarter coupler lengths away and improves one column dramatically and the other hardly at all. So the two criteria are close together on one linkage and far apart on the other, and what separates them is not the criterion but how big the inflection circle is relative to the mechanism.

Watt’s $\delta$ at the middle of its arc is 8.24 on a linkage whose coupler is 1 unit long — the circle is eight times the coupler. Chebyshev’s is 5.08 on a 2-unit coupler, so about two and a half times. A large circle relative to the mechanism means the region of nearly straight points is broad and the midpoint is already inside it; a small one means the midpoint may be well off it and the correction is a long way.

That is a rule of thumb rather than a theorem and it is stated as one. What it is good for is knowing when the question is worth asking: on a linkage whose inflection circle is many times its own size, the tracing point’s exact position is a fine adjustment, and on one where the circle is comparable to the mechanism it is a design decision.

Two ways to be nearly straight, and only one of them is local #

Underneath the whole comparison is a distinction the site keeps meeting and which is worth stating once in its own right.

There are two ways for a mechanism to draw a nearly straight line. The first is to have a point whose path agrees with a line to high order at one instant, so the departure is tiny near the middle and grows quickly. The second is to have a path that wanders back and forth across a line by a small amount over a long stretch, never agreeing to high order anywhere. Ball’s point is the first kind and Chebyshev’s linkage is the second, and no amount of measurement of one will produce the other.

The site has the same pair in the synthesis field under different names. A three-position synthesis makes a linkage pass exactly through three prescribed poses and says nothing about what it does between them; an approximate synthesis makes it pass near many poses and exactly through none. The first is interpolation and the second is fitting, and a designer who wants a machine usually wants the second while a designer who wants a proof usually wants the first.

The reason this field ends up on the interpolation side is not a preference. It is that a derivative is a local object, and everything obtainable from the derivatives of a motion at an instant is a statement about a neighbourhood of that instant. Getting from there to a claim about a stroke requires integrating, and integrating is what the stroke measurements in the tables above do.

What Ball’s point is not #

Three things, because each of them is a plausible over-reading.

It is not a property of the linkage. It is a property of the motion at one position, and it moves as the crank turns. A tracing point drilled at Ball’s point for one position is Ball’s point only there; a few degrees later Ball’s point is somewhere else and the drilled point is ordinary again. Every number in the tables above is what happens when a point is fixed at the Ball position for the middle of the stroke and then dragged through the whole of it.

It is not the straightest point over a stroke. Nothing in its definition mentions a stroke. It happens to be much better over Watt’s stroke and only somewhat better over Chebyshev’s, and the essay reports both because the definition does not predict either.

It is not exact. The path through Ball’s point is not a straight line; it is a curve that agrees with a line to one order further than usual. Peaucellier’s cell draws an exact line and the difference between those two situations is fourteen orders of magnitude, which is the site’s standing example of a difference in kind rather than degree. Ball’s point is firmly on the approximate side of it.

The one motion where the point is exactly what it claims #

There is a case where the high-order contact becomes complete, and it is worth naming because it is the boundary of the whole idea.

If a motion’s centrodes are a circle rolling inside a circle of twice its radius — the Cardan arrangement, produced by an elliptic trammel or a pair of gears — then a point on the rim of the rolling circle traces an exact diameter of the fixed one. Not four-point contact with a line: the line, exactly, to the last bit of a double.

At that instant Ball’s point is that rim point, and the contact order measurement runs out of anything to measure because the departure from the tangent is zero at every step. The exponent fit has no signal in it. So the field’s most-quoted measurement degenerates on the one motion where the thing it measures is perfect, which is a reasonable place for a measurement of approximation quality to break down.

The check, and the half of it that is a refusal #

The assertion behind this essay has four parts and the ordering is deliberate.

One: at most one Ball point is found, the number of sign changes it was chosen from is reported, and any position that returns none has to be one where the coupler is instantaneously translating. Printing the candidate count is what makes the asymptote problem visible rather than silently handled.

Two: the point’s path curvature is at rounding — $-3.8\times10^{-16}$ — and its curvature rate likewise. Being on both curves is checked directly rather than inferred from having been found by a crossing search.

Three: its order of line contact is $3.95$ and a neighbouring point on the same ray gives $1.87$. The gap is required to exceed one, so the check fails if the exponent measurement has lost its ability to discriminate.

Four: Watt’s published nine per cent and Chebyshev’s twelve are reproduced within a percentage point. That one is the refusal in disguise: if the machinery in this field were describing a different mechanism from the one the straight-line essay describes — a different assembly branch, a different working arc, a different tracing point — the two numbers would not agree, and no amount of internal consistency in this field would reveal it.

There is a fifth, and it is the one that keeps the essay honest about Chebyshev. It requires the improvement in the worst departure to be less than a factor of 1.6. If a future change made Ball’s point improve Chebyshev’s worst error dramatically, this essay’s central caveat would be wrong, and the check would say so rather than the prose quietly becoming false.

What comes after four-point contact #

The obvious next question is whether there is a point with five-point contact with a line, and the answer is that there is not, generically. Five-point line contact would need three conditions — $\kappa = 0$, $\kappa' = 0$, $\kappa'' = 0$ — on two coordinates, which is one condition too many. Such points exist only at isolated positions of the mechanism, where a special coincidence occurs.

Five-point contact with a circle is a different matter, because a circle has a free radius and the counting works out. Those points exist, there are up to four of them, and the essay on them finds that two of the four are always the mechanism’s own pins and the other two come and go as the crank turns.