Five positions, and what is left

The synthesis field has been counting down.

Three poses. Any point of the moving body will serve as a pin. Its three images are three points, three points have a circumcentre, and the circumcentre is the ground pivot. A two-parameter family of solutions — the whole coupler plane to choose from — and the construction is six lines of code.

Four poses. Four images of a point are concyclic only for points on a cubic, so the choice collapses from a plane to a curve. That cubic is what Burmester’s name is on, and it is still a construction: a point can be walked along the curve and a linkage read off.

Five poses. The curve becomes a finite set of points, and this is where the compass stops.

Why the compass stops #

The reason is worth being precise about, because it is not that the problem gets harder in some vague sense.

Five positions can be split into two overlapping sets of four, and each set of four gives a centre-point curve — a cubic. A point that works for all five must be on both cubics. So the five-position solution set is the intersection of two cubics, and intersecting two cubics is not an operation available to a compass and straightedge. Two circles, yes. A circle and a line, yes. Two curves of degree three, no.

Bézout says two cubics meet in nine points. Not all nine are useful: a circular cubic passes through the two circular points at infinity, both cubics do, and there are further common points forced by the construction. What is left is a small finite number, and how small is a fact about the poses rather than about the method.

Rather than working through that subtraction on paper, the site does what this phase built the machinery for and counts it.

There is a second reason the construction stops, and it is less often stated. Even where two cubics can be intersected — with a computer, trivially — the classical framework has nothing to say about which of the intersections is worth having. A construction produces a solution and the reader accepts it because the steps were sound. A set of four solutions of which two are unbuildable needs a criterion the construction does not contain, and supplying that criterion means abandoning compass work for a sweep. So the fifth position pushes the problem out of drawing in two ways at once: the solving becomes algebraic, and the selecting becomes empirical.

Four unknowns and four equations #

The system is smaller and prettier than the two-cubics description suggests, and the prettiness is the reason it is written this way.

A dyad is a moving pin k, given in the body’s own frame, and a fixed pivot m in the ground frame. Four unknowns. The condition is that the pin’s distance from the pivot is the same in every pose, which for five poses is four equations — one for each pose after the first.

Expand one of them. The pin’s image in pose $i$ is $p_i + R_i k$, so

$$|p_i + R_i k - m|^2 = |k|^2 + |p_i - m|^2 + 2(R_i k)\cdot(p_i - m),$$

using $|R_i k| = |k|$ because $R_i$ is a rotation. Subtract the same expression for the first pose, and the $|k|^2$ cancels. So does the $|m|^2$ hiding inside $|p_i - m|^2$.

What is left is degree one in k and degree one in m — bilinear, total degree two. Four quadratics in four unknowns, Bézout’s number sixteen.

Twelve paths go to infinity and four arrive. Those four are the Burmester points, and for these five poses all four are real.

Six linkages from four points #

A dyad is half a mechanism. Any pair of Burmester points is a four-bar: two moving pins on the coupler, two ground pivots, and the four link lengths determined by the distances. Four points give $\binom{4}{2} = 6$ candidate linkages.

Every one of them reaches all five poses. That is a theorem and it holds exactly: the verification drives each linkage to each pose with the forward solver and measures the miss, and the worst across all six is $10^{-14}$.

Then the same question the field has been asking since the three-position essay: can any of them be built?

Two of the six. The other four reach their five poses on more than one circuit — exactly correct, and requiring the linkage to be taken apart and reassembled between poses.

The proportion is worse than it looks #

Two of six is a third, which sounds better than the three-position survey’s 176 of 1,176 — fifteen per cent. The comparison is misleading in both directions and the reasons are instructive.

Five positions leave nothing to choose. The three-position survey has 1,176 candidates because there is a two-parameter family and 1,176 is how many were sampled from it. If most of them are defective, a designer discards them and takes another. At five positions there are six, full stop, and if all six were defective there would be nothing to fall back on.

The failures are the same failure. Both counts fail for branch and circuit reasons: the linkage reaches the poses and cannot travel between them. Nothing in either construction attends to that, because both constructions are statements about distances and circuits are statements about connectivity.

And the sample size is four. Four real Burmester points is what these five poses give. Another five poses give a different number — the finite count is four either way, but how many of the four are real is a property of the poses, and it can be two, or none.

That last point is the one worth stating loudest. A designer who prescribes five positions is not guaranteed a linkage at all. The construction always produces four solutions of the algebra; whether any of them are real, and whether any of the real ones can be assembled, are two further questions that the classical treatment answers with a shrug and this site answers by sweeping.

The degenerate solutions the algebra hands back #

Two of the sixteen paths deserve a note, because the system has solutions that are not dyads and the code has to know it.

A pin coinciding with its own pivot satisfies every equation trivially. If k is placed so that its image is exactly m in the first pose, the distance is zero, and zero equals zero in all five — so a link of no length is a perfectly good solution of “the distance is the same in every pose”. The algebra has no way to object; a length of zero is a length.

synthesiseFivePosition discards those explicitly, by measuring the pivot distance in the first pose and dropping anything below $10^{-6}$. That is a filter on the interpretation rather than on the arithmetic, and it is the sort of step that is easy to leave out and then to explain away later as a numerical artefact. It is not an artefact. It is a solution, of a system that was written down slightly more generously than the question intended.

The same shape appears wherever a condition is expressed as an equality of distances. The three-position construction has it too, and handles it by returning null when three images are collinear — a moving point whose images lie on a line has its pivot at infinity, and the dyad it wants is a slider rather than a crank. Rounding that to a very distant pivot would produce a linkage with a bar a thousand units long and a caption that did not mention it.

Two computations that have to agree #

The five-position result is checked against the four-position one, and the check is the useful kind — two things computed by different routes that must land on each other.

A Burmester point of five poses is, in particular, a point whose images in the first four poses are concyclic. So it must lie on the four-position centre-point curve. The curve is computed by an entirely different piece of machinery — a marching-squares contour of the concyclicity defect over a grid — and the points come from a homotopy tracker that knows nothing about it.

Every one of the four lies on the curve to $3.6 \times 10^{-15}$.

And the concyclicity itself is checked directly: for each point, the distances from its pivot to its five images must all be equal, and the spread across all four points is $1.6 \times 10^{-15}$.

Neither of those is a formality. The first would catch a tracker that had converged to something satisfying a mis-transcribed equation; the second would catch a construction that was correct about four poses and had dropped the fifth. Both are the kind of error that produces a perfectly plausible picture.

The construction, and what replaced it #

It is worth saying what Burmester actually did, because “intersect two cubics” is a modern description of a nineteenth-century result and the difference in method is the point of this rung.

Burmester’s own treatment is projective and constructive throughout. The centre-point curve is obtained as a locus, the intersections are obtained graphically, and the whole thing is executed with drawing instruments on a board. That was not a limitation reluctantly accepted — it was the technology, and the constructions were designed around what a draughtsman could do accurately.

What a draughtsman can do accurately does not include intersecting two cubics, so the five-position case was in practice handled by drawing both curves and reading the crossings off the paper. That works, it is how the problem was solved for a century, and its accuracy is the accuracy of a pencil.

The algebraic route replaces the pencil and changes what can be claimed. Reading a crossing off a drawing gives a pivot good to perhaps three figures and no statement about how many crossings there are. Tracking sixteen paths gives four points good to fourteen figures and the count. The second half is the part the drawing could never supply: a draughtsman who finds three crossings has found three crossings.

There is a cost, and it is the ordinary one for this site. The construction explains itself — every step is a circle or a line with a reason — and the homotopy does not. Nobody looks at 1,458 tracked paths and understands why the answer is four. The site keeps the construction for the three- and four-position cases, where it is both exact and legible, and reaches for the algebra at exactly the point where legibility was already lost.

What a designer actually gets #

The practical shape of five-position synthesis is worth stating plainly, because the classical presentation makes it sound more usable than it is.

Five is the maximum. A dyad has four free numbers and each pose after the first is one equation, so five poses is where the equations meet the unknowns. Six poses is over-determined and generically has no solution at all. This is not a limitation of Burmester’s method; it is a count, and the essay on how many precision points a linkage will take works it out for all three kinds of synthesis.

Exactness is free and buildability is not. The construction gives poses met to fourteen places, every time, and says nothing about circuits. Every solution has to be swept.

And there is no dial. With three poses a designer who dislikes the result picks a different coupler point. With four, a different point on the curve. With five, there are four points and no parameter — so the only remedy for six bad solutions is to change the prescribed poses, which is a change to the specification rather than to the design.

That last is the real cost of the fifth position and it is rarely stated as one. Every position added buys accuracy and spends freedom, and at five the freedom is gone.

Why the six are so unequal #

The six candidate linkages are worth looking at as a set, because their spread is the most surprising thing in the computation.

Their link lengths run from about 1.0 to 28.4. Two of them have a ground link nearly six times the longest moving link; one has a rocker of 24.2 against a crank of 1.0. These are not near-misses of reasonable designs — they are mechanisms whose proportions no engineer would draw, produced by an exact construction from perfectly ordinary prescribed poses.

The reason is that a Burmester point can sit anywhere in the plane, including a long way away, and a pivot far from the moving body gives a long link. Nothing in the construction penalises distance, because distance is not part of the condition being satisfied. The condition is that five images are concyclic, and a very large circle through five nearly-collinear points satisfies it as exactly as a small one.

So a practical five-position synthesis has a filtering step that is not in the mathematics: discard the solutions whose pivots are outside the machine. That is a design constraint rather than a kinematic one, and it interacts badly with the fact that there are only six candidates to begin with. Of the six here, two are usable and both of those are also compact — which is luck rather than a pattern, and the site’s own three-position survey rejects a pose set precisely because its three usable solutions all had pivots outside the machine.

Where this leads #

Two directions open from here, and the rest of this ladder takes both.

The first is to stop demanding exactness. A linkage that passes exactly through five poses is one thing; a linkage that passes near a hundred is another, and the second is what most design problems actually want. That is approximate synthesis, and it trades the theorem for an optimiser.

The second is to stop demanding a four-bar. If four free numbers per dyad is the constraint, more links means more numbers — and six bars can do things four cannot, including holding still.

Both are departures from the discipline that has carried the field so far, which is that everything is a construction and every construction is exact. It is worth marking the boundary: everything in this field up to and including this essay is a theorem with a picture. Everything after it is a measurement with an error bar.

The boundary is not a decline in rigour, and it would be easy to read it as one. An exact construction that produces six linkages of which four cannot be built is not more useful than an optimiser that produces one linkage which is wrong by a fifth of a degree and works. Exactness is a property of the answer to the question that was asked, and the question “pass exactly through five poses” is one a designer chooses to ask, usually because it is the one with a classical answer rather than because it is the one the machine needs.

The unevenness of the six is worth one more sentence, because it says what a solution count is worth as a measure of design freedom. Four Burmester points give six four-bars, and six sounds like a comfortable set to choose from; in practice several are degenerate, several have their pivots in impossible places, and the usable count is often one or none. So the algebraic count is an upper bound on the design freedom and not a measure of it, in exactly the way a census of chains is a bound on the mechanisms available rather than a list of usable ones. The useful number is what survives filtering, the filters are engineering rather than algebra — pivot location, link length ratios, transmission angle, branch — and none of them is visible in the count. That is why a synthesis method that returns a number of solutions has not finished, and why the next step is always a filter rather than a choice.