How many points may be prescribed

Three numbers are quoted throughout this subject as facts about four-bar linkages: five precision positions for motion generation, five precision points for function generation, nine for path generation.

They are not facts about four-bars. They are facts about what was counted as free, and different books count differently and get different numbers — which reads as disagreement and is not.

The disagreement is worth being clear about, because it is the kind that wastes a reader’s afternoon. A textbook saying three precision points and a paper saying five are not contradicting each other; they have made different decisions about whether the input and output shafts are already located. Neither says so, because to each author the decision was obvious.

So this essay derives them, states the assumptions each derivation makes, and then does the thing the site exists to do: computes the one that has never been computed here, and finds that a four-bar which passes exactly through five prescribed angle pairs is unique, and that finding it means tracking a hundred and twenty-eight paths.

The counting #

Every one of the three is the same arithmetic. Count the free numbers, count the net constraints each prescribed condition imposes, and divide.

Function generation. The output angle is to be a prescribed function of the input. Position and orientation of the whole mechanism are irrelevant, and so is its scale — a linkage twice the size computes the same function. What is left is three ratios of link lengths, plus where the input’s zero sits and where the output’s zero sits. Five free numbers. Each prescribed pair gives one scalar equation, Freudenstein’s. Five precision points.

Motion generation. A dyad is a moving pin (two numbers) and a fixed pivot (two numbers). Four free numbers. N prescribed poses give N − 1 equations, because the condition is that the pivot distance is the same in every pose rather than any particular value. Four equations at N = 5. Five precision positions.

Path generation. Two ground pivots, three link lengths and a coupler point: nine free numbers. Each prescribed point gives two equations — the tracing point must be at a given place — but also introduces one new unknown, the crank angle at which it gets there, which nobody prescribed. Net one constraint per point. Nine precision points.

The two fives are unrelated. One is four unknowns against N − 1 equations; the other is five unknowns against N equations. They land on the same number by arithmetic accident and are constantly presented as though they were the same fact.

A fourth count is worth adding because it is the one this site has actually used most. Three positions for motion generation leaves two of a dyad’s four numbers unspent, which is why the three-position construction has a two-parameter family of answers — any point of the coupler plane will serve as a moving pin. That family is the reason the three-position defect survey can sample 1,176 of them, and the reason five positions cannot be sampled at all.

So the same arithmetic explains both the abundance at three and the scarcity at five, and a designer’s experience of the two problems is completely different for a reason that is one subtraction.

Where the disagreements come from #

Every alternative count in the literature is a different left-hand column, and the two common ones are worth naming.

Function generation with the offsets fixed. If a designer has already decided where the input’s zero and the output’s zero sit — because a shaft is already there — then only three numbers are free and the answer is three precision points. That is the same mechanism and a different problem, and it is the version with a linear solve.

Function generation with the ranges free. Some treatments count the input and output scale factors as design variables too, giving seven. That is a defensible thing to count and it gives seven precision points.

None of those is wrong. The count is a property of the problem statement, and the way to read a quoted count is as a summary of what its author considered adjustable. Three, five and seven are all correct answers to different questions about the same linkage.

That is why precisionPointCounts returns the unknowns as a list of names beside the number. A count without its column is not checkable.

The fifth point, and what it costs #

Three precision points make function generation linear, and that linearity is the whole appeal: a 3 × 3 solve, no iteration, no guess.

Take the fourth and fifth, and the two angle offsets become unknowns. They enter through their cosines and sines, and Freudenstein’s relation with offsets is

$$K_1\cos(\psi_i + \psi_0) - K_2\cos(\theta_i + \theta_0) + K_3 = \cos\big((\theta_i + \theta_0) - (\psi_i + \psi_0)\big).$$

Expand every offset so that the unknowns are $\cos\theta_0$, $\sin\theta_0$, $\cos\psi_0$, $\sin\psi_0$ rather than nested angles, and the system is seven unknowns — the three $K$’s and the four trigonometric ones — with seven equations: five prescribed pairs and two identities $c^2 + s^2 = 1$.

Every one of them is quadratic. Bézout’s number is $2^7 = 128$.

The linear solve is gone. What replaces it is a polynomial system, and counting its solutions is exactly the machinery this phase’s other field built.

One hundred and twenty-eight, and then one #

128 paths tracked. 12 arrive at a finite solution — 116 leave for infinity. 4 of the twelve are real. And 1 of the four is a linkage.

That last step is the one worth dwelling on, because it is where an algebraic solution stops being a mechanism.

A real solution gives three numbers $K_1$, $K_2$, $K_3$ and two offsets. Turning them into link lengths needs $r_2 = r_1/K_1$ and $r_4 = r_1/K_2$, which fail when a $K$ is zero or negative, and

$$r_3 = \sqrt{r_1^2 + r_2^2 + r_4^2 - 2r_2r_4K_3},$$

which fails when the expression under the root is negative. Three of the four real solutions fail one of those tests. They are perfectly good roots of the system and there is no four-bar corresponding to them.

The survivor, with the ground link taken as 1: crank 4.981, coupler 0.850, rocker 5.050, input offset −3.45°, output offset −34.95°. It reaches all five of its prescribed angle pairs to exactly zero — the forward solver, driven to each precision point, returns the prescribed output angle with no residual at all at double precision.

Where the 116 paths went #

The over-count here is larger than anywhere else in the phase relative to the answer — 128 paths for twelve solutions — and its cause is the same one the algebra field’s essay on wasted paths identifies: structure the degree cannot see.

Two of the seven equations are the identities $c^2 + s^2 = 1$, and an identity of that form is a circle. A circle passes through the two circular points at infinity, so each identity contributes intersections at infinity that Bézout counts and that correspond to no offset angle whatever. Two circles, four such points, and the leg equations interacting with them account for most of the loss.

The rest comes from the trigonometric structure. The four unknowns $\cos\theta_0$, $\sin\theta_0$, $\cos\psi_0$, $\sin\psi_0$ are not four independent numbers; they are two points on two circles, which is two degrees of freedom dressed as four. Bézout has no way to know that, and every equation it counts as quadratic in four unknowns is really something smaller in two.

A bound that saw the sparsity would give a much lower number. Naming that is better than implying 128 is the intrinsic difficulty of the problem: 128 is the difficulty of the formulation, and the formulation was chosen for being easy to write down correctly rather than for being tight.

The gap between four and one is the finding #

There is a strong temptation to report “four solutions” and move on, and it would be the same error the three-position defect survey exists to prevent.

The three-position survey generates 1,176 exactly correct syntheses and finds 176 that a machine could use. The five-position one generates six and finds two. Here the algebra generates four and yields one.

In every case the construction is exact, the failures are exact too, and the thing that distinguishes them is a test the construction does not contain. The tests differ — a circuit sweep for motion generation, a positivity check for function generation — and the shape is identical: a solution of the equations is a candidate, not an answer.

That is the discipline this whole field has been built on, arriving at its last rung in the smallest possible form. Four numbers, one of which is a linkage.

One linkage is a design with no choices in it #

The result that five precision points give exactly one four-bar has a consequence for practice that is worth separating from the arithmetic.

At three precision points there is a two-parameter family of solutions, because two of the five free numbers are unspent — a designer picks the offsets and gets a linkage, and picking differently gets a different one. There is room to reject an answer whose proportions are unusable and take another.

At five there is one. Its crank is 4.98, its coupler is 0.85 and its rocker is 5.05 on a ground link of 1, which is a mechanism with two long thin links either side of a very short one. Whether that fits in the space available is not a question the construction was asked and not one it can be asked afterwards. There is nothing else on the shelf.

So the fifth precision point is expensive twice over. It costs the linear solve, replacing it with a hundred and twenty-eight tracked paths; and it costs every degree of freedom that would have let a designer trade accuracy for proportions. The remedy, as at five positions, is to change the specification rather than the design — prescribe four points and keep one number free, or accept an approximation and optimise over the range.

Which is the argument that the whole of rungs five and six make from the other direction, and it is worth noticing that they meet here. Exact synthesis at the maximum number of precision points and approximate synthesis over a range are the two ends of one trade, and the counting is what says where the ends are.

Nine points, and what this site does not do #

Path generation’s nine is the one with the famous answer and this site does not compute it.

The system for a four-bar passing through nine prescribed points is large, and its solution count was first obtained in the 1990s by exactly the method used here — homotopy continuation — on a scale far beyond anything tracked on this machine. The published count runs to several thousand solutions, organised into triples by Roberts’s cognate theorem, since each coupler curve is drawn by three different linkages and the algebra finds all three.

That is a result this site quotes and has not reproduced, and it is named as such rather than presented alongside the numbers that were computed. Reproducing it would need a formulation this library does not have and a run time this build cannot pay for.

What is computed is the parameter count that says the number should be nine and not eight or ten, and the five-point function generator, which is small enough to track completely. Between them they establish the pattern; the nine-point case is where the pattern goes and it is over the horizon.

What the counts do not tell anybody #

A parameter count says when the equations and the unknowns balance. It says nothing about four further questions, and all four decide whether a synthesis is usable.

Whether any solution is real. The count guarantees a finite number of solutions over the complex numbers. How many of them have no imaginary part is a property of the prescribed data, and it can be zero. Five poses can have four real Burmester points, or two, or none, and the count is four either way.

Whether a real solution is a mechanism. Three of the four real five-point generators here are not, because their link lengths come out negative or imaginary. The count does not know that link lengths must be positive.

Whether a mechanism can be assembled through its own conditions. This is the branch and circuit question, and it disqualifies four of six five-position linkages and a thousand of 1,176 three-position ones.

Whether it will fit. No count anywhere is about proportions, and the five-point generator’s are extreme.

So the counting is a statement about the problem, in the sense of how much can be asked before the asking becomes over-determined. It is a genuinely useful statement and it is upstream of every question a designer has.

The shape of the whole field #

This is the last rung of the synthesis ladder, and the field has a shape worth setting out now that all of it is present.

Rungs 1 to 3 were exact and constructible. Three positions and a circumcentre; the fourth position and its curve; cognates. Compass work, in principle and largely in practice.

Rung 4 was exact and not constructible. Five positions needs two cubics intersected, which is algebra. The answers are still exact; the method stopped being drawable.

Rungs 5 and 6 gave up exactness. Approximate synthesis over a range, an objective that has to be chosen, an optimiser whose answer depends on where it started. And six bars, where the interesting mechanisms are the ones whose dimensions are themselves computed from a curve.

Rung 7 is the counting. How many conditions each problem takes, why the numbers are what they are, and what happens at the last one a four-bar will accept.

Read in that order, the field is a steady retreat from the classical position — from a construction anybody can follow to a computation nobody can check by eye — and each step is forced by a specification the previous method could not express. Nothing here was given up for convenience. The fifth position needs cubics because four positions leave a curve; the range needs an optimiser because a range is not a finite set of equations; the fifth precision point needs 128 paths because two angle offsets stopped being constants.

One more thing holds across all seven, and it is the reason the field is worth having as a ladder rather than as a set of techniques. Every rung is a specification the previous rung could not express, and the method changes because the specification did — not because somebody found a better way to do the old thing. That is a different history from the one usually told, in which numerical methods replace classical ones. Nothing here was replaced. The constructions still do exactly what they did, and are still used, including as the seeds that make the numerical methods work.

The thing that does not change across all seven rungs is the last step. However a linkage is arrived at, it is handed to the forward solver and swept, because no construction and no objective contains a statement about whether the mechanism can get from one prescribed place to another in one piece.

The three numbers being properties of what somebody counted as free is the finding, and it has a practical form worth stating. Before quoting a maximum, write down the parameter list. Five poses is the answer when the four-bar is described by its two dyads with the coupler’s attachment fixed; nine points is the answer when the coupler point is free as well; and neither is wrong, because they are answers about two different design spaces. So a disagreement between two textbooks about how many precision points a four-bar admits is nearly always a disagreement about the parameterisation, and it can be settled by counting parameters rather than by argument. That also says how to extend the question to a mechanism nobody has tabulated: count the free numbers, count the conditions, and the maximum is where they meet — which is the parameter count the topology field computes from a graph, used for the purpose it was built for. The classical numbers are then instances of one arithmetic rather than three facts to be remembered.