Three positions, and a circumcentre

A body has to occupy three prescribed positions. Find a four-bar that puts it there.

The answer is shorter than the question deserves, and the shortness is the point of the essay.

The construction #

Pick a point of the moving body — any point at all, specified in the body’s own frame so that it travels with the body.

In pose one it is somewhere. In pose two it is somewhere else. In pose three, somewhere else again. Three points.

If a link is going to connect that point to a fixed pivot, the link has constant length, so the point must stay a constant distance from the pivot. It must be on a circle centred on the pivot — which is the same constant-distance constraint the solver imposes on every bar, read as a requirement on the pivot rather than on the pin. It is on that circle in all three poses, so the circle passes through all three of its images — and three points determine exactly one circle.

The fixed pivot is the circumcentre of the three images. The link length is the circumradius.

That is the whole construction. There is no iteration in it, no tolerance, and nothing that could converge to the wrong answer, because three points determine a circle exactly and a circumcentre is a rational function of six coordinates.

The pair of a fixed pivot and a moving pin, joined by a link, is called a dyad. Two dyads make a four-bar: the two moving pins are both on the coupler, so the coupler is the link between them, and the two fixed pivots are the ground.

Every point works, which is the difficulty #

Notice what the construction did not require. It did not require the chosen point to be special, or well placed, or anywhere near the mechanism. Any point of the moving body’s plane has three images and therefore a circumcentre.

So the solution set is the whole plane, twice: choose any point for the first dyad and any other for the second. That is a two-parameter family for each dyad, four parameters in all, and every member of it satisfies the three prescribed poses exactly.

This is not the abundance it sounds like. A designer who asks for three positions and is handed a four-parameter family of exact solutions has not been given an answer; they have been given the same problem with different variables. The real work is choosing which solution, and the criteria for choosing have nothing to do with the three positions:

Where the ground pivots land. A dyad whose circumcentre is three metres off the machine is exact and unbuildable. This is the commonest disqualification and it is easy to test: compute the circumcentre and look at it.

How long the links are. A dyad whose three images are nearly collinear has a circumcentre very far away and a very long link. The construction returns it happily. In the limit — three images exactly collinear — the circumcentre is at infinity, and the honest thing to return is not a very distant pivot but the information that this point wants a slider rather than a crank. The library returns nothing in that case rather than a large number, because rounding an infinity to a finite pivot produces a linkage with a bar a thousand units long and a caption that does not mention it.

Whether the mechanism can be driven. Grashof’s condition decides which link of the resulting four-bar can turn fully, and a synthesis whose input link only rocks needs a different input arrangement — often a different link of the same chain driven instead. Nothing in the construction attends to this.

Whether the linkage can get between the poses. This is the one that disqualifies most of them and it has an essay to itself.

Choosing the two points well #

Since every point works, the interesting question is which to choose, and there are some usable rules that fall out of the construction rather than out of experience.

Points far from the poles give short links. The circumradius of three images grows as the point moves away from the region where the rotations are centred, so a coupler point out at the edge of the body tends to want a long link and a distant pivot. Points near the pole triangle give compact mechanisms.

Points near the line through two images give distant pivots. This is the collinearity degeneracy approached rather than reached, and it is the failure mode to watch for when the three prescribed poses are nearly a pure translation — a translation moves every point along nearly parallel lines, so nearly every point of the body has nearly collinear images and nearly every dyad wants a slider.

The two chosen points should be well separated. The distance between them becomes the coupler length, and the coupler is the link a tracing point rides on, so choosing two points close together gives a short coupler and a mechanism whose two dyads are nearly fighting over the same constraint. It also makes the four-bar sensitive: a small error in either pivot is a large fraction of the coupler.

A point can be chosen to put the pivot somewhere specific. This is the most useful move and it inverts the construction. Rather than choosing the moving point and accepting the pivot, decide where the ground pivot must be — where there is room for a bearing — and ask which points of the body have their circumcentre there. That is a well-posed question with a one-parameter answer, and it is how the freedom actually gets spent in practice.

What the poles say #

There is a classical way to see the same construction that is worth having, because it explains where the freedom comes from.

Moving a rigid body from one position to another can always be done by a single rotation about a single point, the pole of that displacement. Three poses give three poles — one for each pair of positions — and they form the pole triangle, which is the object most of Burmester’s classical results are stated in terms of.

The perpendicular bisector construction is the same thing seen locally. The circumcentre of three points is where the three perpendicular bisectors meet, and each bisector is the locus of centres for a rotation carrying one image to another. So the circumcentre is the point that can rotate the body from pose one to pose two and from pose two to pose three, which is exactly what a fixed pivot must do.

The nineteenth-century treatment builds with compasses and the modern one computes a circumcentre, and the two agree because they are the same statement. What the compass construction makes visible and the formula hides is the degeneracy: when the three images are collinear the bisectors are parallel and there is no meeting point, and the drawing shows it immediately.

The dyad is the unit, not the four-bar #

It is worth being explicit about why the construction is stated in terms of dyads rather than four-bars, because it is what makes the method extend.

A dyad is one fixed pivot, one link, one moving pin. It is a constraint on the moving body: “this point of the body must stay on this circle”. It knows nothing about the rest of the mechanism.

A body in the plane has three freedoms. Each dyad removes one. So two dyads leave one freedom, which is a mechanism with one input — a four-bar. Three dyads would leave zero, which is a structure, and that is why nobody synthesises three dyads for a rigid-body guidance problem.

Thinking in dyads has three payoffs. It makes the construction independent of what the other dyad is doing, so the two can be chosen for completely different reasons. It makes the extension to other mechanism types straightforward — a dyad whose fixed pivot is at infinity is a slider, and the same three images that give a circumcentre also give the line the slider must run along, which is the direction of the perpendicular bisector when the images are collinear. And it makes the counting transparent: each prescribed pose after the first costs a dyad two of its four numbers, which is the arithmetic behind the whole field.

That last point is the one to carry. A dyad has four numbers — two for the pivot, two for the moving pin, with the link length then determined. Pose one is free (it just places the mechanism). Poses two and three each impose two conditions. Four numbers minus four conditions leaves zero for a specified pin, which is why choosing the pin determines the pivot rather than leaving a choice. Adding a fourth pose imposes two more conditions than there are numbers, and something has to give.

Doing it with compasses #

The construction predates computation by a century, and the hand method is worth describing because it makes the degeneracies visible in a way the arithmetic does not.

Draw the moving body in its three prescribed positions. Choose a point on it and mark its three images. Join image one to image two and construct the perpendicular bisector; join two to three and construct that bisector. Where they cross is the fixed pivot; open the compasses from there to any image and that is the link length. Repeat for a second point, and the four-bar is drawn.

Everything that can go wrong announces itself on the paper. If the two bisectors are nearly parallel, the intersection wanders off the drawing board — that is the near-collinear case, and the draughtsman sees a long link before computing anything. If the pivot lands somewhere the machine cannot have a bearing, it is visibly in the wrong place. If the resulting four-bar looks as though it would fold up, that too is apparent from the drawing in a way it is not from four numbers.

The modern version computes the circumcentre in closed form and loses all of that, which is why the library reports a collinearity measure alongside every dyad: how close the three images came to being in a line, scaled so that the number means something. It is the numerical stand-in for the draughtsman noticing that the bisectors were nearly parallel.

Checking a theorem #

The construction is exact. That is a mathematical statement, and this site does not print mathematical statements without a measurement beside them.

So every synthesis here is handed back to the forward solver. For each prescribed pose, compute where the crank pin should be, drive the crank to that angle, close the loop with Newton–Raphson, and compare the rocker pin’s solved position with where the pose says it should be.

The agreement is at the 10⁻¹⁵ level, which is arithmetic noise — the same level the four-bar’s own closure residual sits at. That is the expected answer and it would be worthless on its own, because a check that only ever sees correct inputs cannot distinguish a working check from one that always returns true.

So the same comparison is run on a linkage with one ground pivot moved by a hundredth. It fails, by about 10⁻². The tolerance therefore has three orders of headroom below it and thirteen above, and the check is measuring something.

That negative case was not decoration. The first version of it compared the pose error against a tolerance of 10⁻³ on a mechanism whose links are of order one, and passed on a four-bar that was visibly not the right one.

What the verification is not #

There is a boundary here that matters and is easy to blur.

The verification above drives the mechanism to each pose independently, seeding the solve at the pose itself. It confirms that each prescribed configuration is one the mechanism has.

It says nothing about whether the mechanism can move from one to another.

Those are different questions and the site keeps them apart deliberately. A four-bar has two assembly branches — the coupler and rocker reflected about the diagonal — and a physical one cannot pass between them without a pin coming out. A synthesis whose three poses land on two different branches is exactly correct and physically useless, and driving to each pose separately will find all three and report success.

Making the verification carry both questions would have been possible and would have been a mistake, because the two failures then become one number and the second one is much more common. They are separated, and the second gets its own machinery: sweep from the first pose in both directions, carrying the branch the way a built mechanism must, and see which poses are met along the way.

The same construction, upside down #

Three-position synthesis is stated above for motion generation — guiding a whole body. The same construction solves function generation, and the trick for converting one into the other is worth knowing because it is how most of the classical results get reused.

In function generation the requirement is a relation between the input and output angles: when the crank is at θ₁ the rocker should be at φ₁, and so on. Nothing is said about where any body goes.

The device is kinematic inversion — the same idea that turns one chain into four mechanisms. Instead of holding the ground still and asking where the coupler goes, hold the coupler still and ask where the ground goes. The relative motion is identical, so a linkage that satisfies the inverted problem satisfies the original one.

Under that inversion a function-generation specification becomes a motion-generation specification: the required input–output angle pairs become prescribed positions of the ground link relative to the coupler, and Burmester’s construction applies unchanged. Three angle pairs are exactly satisfiable, four leave a curve, five leave finitely many. The counting is the same because the geometry is the same problem in a different frame.

That is worth having for its own sake, and it is also a warning about reading synthesis literature: the same theorem appears under three names because the three problem classes reduce to each other, and a result quoted for one is usually available for the others.

Where the freedom goes next #

Three poses leave the whole plane to choose from. That is the most freedom any exact synthesis problem gives, and it is why three-position synthesis is where a designer starts.

It is also why three positions are rarely enough. A specification with only three positions in it usually means the designer has three positions they care about and no opinion about the rest of the cycle — and “no opinion” tends to become an opinion the first time the mechanism is built and the coupler sweeps through something it should not.

The natural response is to add a fourth position, and that is where the counting starts to cost something: the whole plane collapses to a curve, and only points on that curve will do.

The other response is to keep three positions and use the freedom for something else — pick the two coupler points to put the ground pivots where there is room, or to get a favourable transmission angle, or to make the input link a full rotator. That is what the two-parameter family is for, and it is the reason three-position synthesis remains the most used exact method rather than a stepping stone to the harder ones.

Every point works, which is the difficulty is the sentence this rung turns on, and it is worth naming what kind of difficulty it is. Three-position synthesis does not have too few solutions; it has a two-parameter family of them, one for every point of the coupler plane, each giving a legitimate dyad. So the problem is not to solve anything — the circumcentre construction solves it instantly for any candidate — but to choose, and nothing in the three positions helps with choosing. That is the opposite of the situation a designer expects from a synthesis problem, and it is why the interesting work in this rung is downstream of the construction: pivot location, link proportions, transmission angle, whether the mechanism reaches the three positions on one branch. Each of those is a filter over the family. The construction supplies candidates and the filters supply the design, and a method that stopped at the construction would have solved the easy half of a problem whose difficulty is entirely in the other half.