The circle a point stays on longest

The ladder so far: a point’s path curvature is fixed by one relation; the points whose curvature is zero form a circle; the points whose curvature has stopped changing form a cubic.

One condition further is the natural next question. Which points have $\kappa' = 0$ and $\kappa'' = 0$ — a curvature that has stopped changing and whose stopping is itself not about to reverse? Those points’ paths hold their osculating circle to fifth order rather than fourth, and they are the infinitesimal counterpart of Burmester’s five-position points.

The counting says there should be finitely many. Two conditions on two coordinates leaves isolated points, and the classical bound is four.

Two of the four are always hardware #

Before counting anything, two of the four have to be set aside, and the reason is embarrassingly simple.

The crank pin travels on an exact circle about its fixed pivot. Its path curvature is not merely stationary — it is constant, so $\kappa'$, $\kappa''$ and every higher derivative vanish identically. The pin satisfies the fifth-order condition, and the sixth, and every one after that. The same for the rocker pin.

So a four-bar’s coupler plane has two points with infinite-order circle contact at every position of the mechanism, they are the two pins, and a search that does not know about them reports four points and spends two of them on hardware.

That is worth separating out rather than absorbing, because the interesting question is about the points that are Burmester points because of the motion rather than because somebody put a pin there. The search classifies its hits — pole, pin, or point — and only the last are counted.

The count changes as the crank turns #

With the pins removed, the count over 360 positions of the site’s four-bar:

• two Burmester points: 241 positions
• none: 111 positions
• one: 8 positions

The eight single-point positions are the transitions. The two points approach each other, merge, and vanish — which is a pair of real solutions becoming a complex conjugate pair, the same bifurcation that runs through the assembly-mode counts in the parallel field and the real-root counts in the algebra field.

Nothing about the mechanism changes across a transition. The lengths are the same, the joints are the same, the mobility is the same. What changes is a count of real solutions to a pair of polynomial conditions, and it changes because the coefficients of those conditions depend on the position.

At the merging position the two coincident points have contact of one order higher again — six-point contact with a circle — which is a genuinely exceptional point of a genuinely exceptional position, and it exists for the same reason a double root exists.

The window, stated rather than hidden #

A Burmester point can be a long way from the mechanism. The search looks within a stated distance of the pole and the count depends on that distance, so the dependence is published rather than absorbed into a default:

The count rises and flattens. Going from 6 to 30 coupler lengths finds 46 more positions; going from 100 to 400 finds 4. So the answer is not sensitive to the window once the window is large, and the figures use 100.

The reason for publishing the table rather than the single number is the fleet’s standing rule about silent caps: a bounded search that does not say what it bounded reads as a complete one. A Burmester point 200 coupler lengths from a four-bar is not a useful design object — a pivot that far away is not a pivot — but it is a real point of the moving plane, and reporting 241 without saying 245 exist further out would be a claim about the geometry rather than about the search.

Finding them, and the crossings that are not crossings #

The search walks the cubic — which is parameterised by ray angle from the pole, one point per ray — and looks for sign changes of the numerator of $\kappa''$ along it.

That is the same shape of search that finding Ball’s point used, and it has the same trap. The cubic’s polar radius runs to infinity at four ray directions, jumping from $+\infty$ to $-\infty$, so a quantity evaluated along the cubic changes sign there without anything happening.

The remedy here is different from the one used for Ball’s point, because there is no second curve to check membership of. What distinguishes a root from an asymptote is continuity: at a genuine root the quantity is small on both sides of the crossing, and at an asymptote it is large on both sides. So each candidate is checked at a small offset either way and required to be small on both, relative to the largest value the quantity reaches over the whole scan.

Without that test the count was noisy — spurious single points appearing at eight or fifteen of 360 positions depending on the window — and the noise looked exactly like the genuine transitions, which are also single points. Two effects producing the same observable is the situation where a check earns its place.

The contact order, measured #

The claim that these points hold a circle to fifth order is checked the same way as everything else in this field: replace the point by a crank pivoting at its path’s centre of curvature, drive the mechanism away from the instant, and measure the geometric distance from the point to the fixed circle.

At a crank angle of 1.6 radians, where two Burmester points exist:

Measured at four crank positions and eight points, the Burmester exponents run from 4.96 to 5.03. Five.

Nothing in the measurement knows which kind of point it was given. It traces a path with the solver, measures distances to a circle, and fits a slope. A wrong $\kappa''$ would produce a point that is not a Burmester point, whose exponent would come back at four, and the check would fail — which is the property that makes the exponent worth reporting rather than the condition.

What a fifth-order point is worth #

The practical reading is the same as the cubic’s, one order stronger. A Burmester point is where a single pivot can stand in for the whole linkage for longest.

Concretely: replace a four-bar’s coupler point with a crank of the right length pivoted at the right place, and the two mechanisms agree to some order. For an ordinary point the disagreement grows as the cube of the crank step; for a Burmester point as the fifth power. At a step of a tenth of a radian that is a factor of a hundred in the error, and at a step of a fiftieth it is a factor of two and a half thousand.

That is what makes the points useful in synthesis. A designer with a wanted motion who has to approximate it with a simpler mechanism wants the approximation to hold as long as possible, and the fifth-order points are where it does. Burmester’s five-position construction is the finite version of the same idea, and the site’s count there is the same: up to four points, of which some are real and some are not, and the usable ones are fewer still once branch and order defects are applied.

Where an arc replaces a machine, in practice #

The abstract statement is that a pivot can stand in for a linkage. The place that matters most on this site is not a linkage at all.

A cam profile is a curve that has to be cut, and cutting an arbitrary curve is expensive while cutting a circular arc is cheap. A cam designer who can replace a stretch of profile with an arc, and know how much error that costs, has bought something real. The order of contact between the profile and the arc is exactly the quantity this essay measures, applied to the motion the profile is an envelope of.

The same argument runs in the gear field, where an involute flank was for a long time approximated by a circular arc for manufacturing reasons — the Willis odontograph and its relatives were tables for choosing the arc — and where the error of that substitution is a contact-order question.

And it runs in reverse in approximate synthesis: a designer replacing a wanted motion with a four-bar’s coupler motion is making the same kind of substitution one level up, and the same distinction between a criterion at an instant and a criterion over a range applies.

Two points, and whether they are anywhere useful #

A count is not much use without knowing where the points are, so it is worth reporting the range.

Over the positions where two exist, their distances from the pole run from under one coupler length to well over a hundred, and at most positions one of the pair is close in and the other is far out. The close one is at a plausible place for a coupler point — a few tenths to a few multiples of the coupler length — and the far one usually is not.

That is the practical asymmetry. A motion has at most one Burmester point in a useful place at a time, and whether it has even that depends on the position. A designer who wanted a coupler point whose path holds a circle to fifth order over a chosen part of the stroke would be choosing among one candidate, not four, and would be checking that the candidate is somewhere a hole can be drilled.

It is also why the two-of-360 style of count is the right summary rather than a map. The map would be four numbers per position with three of them uninteresting, and the count is the thing that changes.

The finite version, and a gap this field leaves #

The relation to the finite construction ought to be the same as the one measured for the cubic: bring five prescribed positions together and Burmester’s five-position points should tend to these.

This site has not measured that. The four-position limit was measured, at an exponent of 2.02, and the five-position one is named here and left. It is a harder measurement for a specific reason: the fifth-order concyclic condition’s conditioning is worse than the fourth’s, so the range of spreads over which a convergence rate can be fitted is narrower, and the site’s experience with three-point circle fits getting worse as the sampling gets finer says that the narrow range is where such a measurement goes wrong quietly.

Recording that as a gap rather than doing it badly is the right trade. It is the same decision the algebra field made about the BKK bound and the same one the depth phases made about Hunt’s three-systems.

The degenerate motion, and the number that was not a count #

One motion in the field’s ledger returns no count at all, and the way it failed first is worth recording.

The elliptic trammel’s angular rate is exactly constant, so every derivative of its angle above the first is exactly zero. Put that into the $\kappa'' = 0$ condition and it holds identically along the whole cubic rather than at isolated points, because the terms that would have made it a condition are all zero.

The search, which looks for sign changes, found 346 of them — a number produced by rounding noise crossing zero repeatedly along a curve where the quantity is identically zero. Four is the most a planar motion can have.

346 is worse than an error, because it is a number, and a reader shown a number has no way to know it is not a count. The machinery now refuses: more than twelve hits means the condition has degenerated, and the ledger reports the motion as degenerate with the reason attached rather than printing a figure.

That is the same shape as the site’s other refusals — a curvature quoted at the pole, a mobility formula applied to a paradoxical linkage — and the rule behind all of them is that a general expression evaluated on a special case returns something, and the something is not always worth printing.

The name, and what it is borrowed from #

The points counted here are usually called Burmester points, which is a borrowing from the finite problem rather than a name coined for the infinitesimal one. Ludwig Burmester’s five-position construction gives at most four points of a moving body whose five prescribed positions lie on a circle; the infinitesimal ones are what those become as the five positions coalesce, so the name travels with the object.

It is worth being alert to the borrowing, because the finite and infinitesimal objects have different failure modes. A finite Burmester point can be perfectly real and completely unusable, because the linkage built on it turns out to have a branch or order defect — the site’s own survey found that of 1,176 exactly correct three-position syntheses only 176 could actually be built. An infinitesimal Burmester point has no such filter attached to it: it is a point of the coupler plane, and whether a mechanism built to exploit it works is a separate question this field does not ask.

Two real roots becoming two complex ones #

The count changing while the mechanism does not is the strangest thing in this essay, and it has an entirely ordinary explanation once the four points are read as roots rather than as objects.

The conditions defining a Burmester point are polynomial, so the four points are the roots of a system, and roots come in real ones and conjugate pairs. Two of the four are the mechanism’s own pins and are real always. The other two are a conjugate pair: real together, complex together, and never one of each. That is why the count goes two, none, two, none and never one — except at the transitions.

At a transition the pair is a double root. Two real points slide together, coincide, and leave as a complex conjugate pair; run the crank backwards and they arrive the same way. So the eight single-point positions are not positions with one Burmester point in the ordinary sense; they are positions where the two are the same point, which is exactly why the contact order there is one higher.

That also explains the arithmetic of the counts. 241 positions with two, 111 with none, and 8 transitions between them, summing to 360 — and eight transitions means the pair appears and disappears four times in a revolution, in four intervals of existence separated by four of absence. A count that changed once would be a mechanism with one bounded window; four is a mechanism whose discriminant crosses zero eight times.

Which puts the phenomenon in a family this site keeps meeting from other directions. Assembly modes merging at a singularity is two real roots becoming complex, in the solutions of a closure. An odd count of assemblies is the same event caught in a histogram. Here it is two Burmester points, and in every case the mechanism is doing nothing unusual — the arithmetic of its solutions is crossing a discriminant.

The lesson worth carrying is the negative one. A count of real solutions is a property of a configuration and not of a mechanism, so any quantity reported as a count needs the configuration attached, and a survey that reports a single number for a mechanism has averaged over a discontinuity. That is why this essay’s summary is 241/111/8 and not “a four-bar has two Burmester points”, which is the sentence the classical result invites and which is true at two thirds of the positions.

Where the ladder stops #

Six-point circle contact would need three conditions on two coordinates, which is one too many, so it happens only at isolated positions of the mechanism rather than at points of the plane — the transitions counted above.

Beyond that the ladder stops being about points at all and becomes about which mechanisms have positions with exceptional contact, which is a question about the four lengths rather than about the moving plane. That is a synthesis question and it belongs to the field that asks what a fourth position costs rather than to this one.

The natural end of this field, then, is fifth order: the last order at which the answer is a set of points of the moving plane rather than a condition on the machine.