What the fourth position costs

Three prescribed positions leave the whole coupler plane to choose from. Any point at all has three images, three points determine a circle, and the circle’s centre is a fixed pivot that works.

Ask for a fourth position and that stops being true, immediately and completely. Four points do not generally lie on a circle. Three of them determine one, and the fourth is somewhere else.

The difference is not one of degree. Three poses give a designer a two-parameter family and the problem of choosing among too many exact answers; four give a one-parameter family confined to a curve that the designer did not choose and cannot move. It is the same shift in character that the mobility count undergoes when a mechanism leaves the plane — the arithmetic is continuous and what it describes is not.

The collapse #

A dyad has four numbers to spend: two for the fixed pivot and two for the moving pin, with the link length then determined by the geometry.

The first prescribed pose is free — it only says where the mechanism sits relative to the body, and any dyad can be positioned to satisfy it. Every pose after that imposes two conditions: the moving pin’s image must be on the circle, which is one equation, and there is one such equation per additional pose.

So:

• Two poses: four numbers, one condition. A three-parameter family, which is the trivial hinge answer.
• Three poses: four numbers, two conditions. A two-parameter family — choose the moving pin anywhere in the plane and the pivot follows.
• Four poses: four numbers, three conditions. A one-parameter family — a curve of solutions.
• Five poses: four numbers, four conditions. Finitely many solutions, and the classical count is four dyads, giving up to six four-bars.
• Six poses: five conditions on four numbers. Generally none at all.

That is the whole arithmetic of the field, and it is the reason exact synthesis stops at five. Nothing about the method breaks at six; there is simply nothing left to solve for.

The two curves #

For four poses, the moving points that work are those whose four images are concyclic. That set is a cubic curve in the moving body’s plane and is called the circle-point curve.

For each such point, the circle’s centre is where the fixed pivot goes, and the set of those centres is a second cubic in the fixed plane, the centre-point curve. Both are named after Ludwig Burmester, who worked them out in the 1870s and 80s.

A four-bar for four poses is then any pair of points from the circle-point curve, with their two centres taken from the corresponding places on the centre-point curve. The curve is one-dimensional, so pairs from it form a two-parameter family — but that is a family of pairs, not of individual dyads, and the difference matters. With three poses each dyad could be chosen freely and independently. With four, each dyad must land on a curve, and the freedom left is only in which points of that curve to take.

Drawn as a measurement #

The circle-point curve has a closed form. It is a cubic, its equation is in the literature, and plotting it would take a dozen lines.

The figure does not do that, and the reason is this site’s habit rather than difficulty. A plotted equation is a drawing. A contour of a measured quantity is a measurement.

So the quantity measured is this: for each point of a grid over the moving body, place it in all four prescribed poses, fit a circle through the first three images, and record how far the fourth image falls from that circle. That number is a length in the drawing’s own units, so it can be reported to a reader as “this point misses by two millimetres” rather than as an algebraic residual with no interpretation.

The curve is the zero contour of that number, traced by marching squares.

What this buys is a check the closed form could not supply. Points taken off the traced contour are refined onto it by bisection and handed back to the circle fit, which has to agree that their four images are concyclic — and it does, to about 10⁻¹⁴. Points six grid cells away from the contour are handed to the same fit, and they miss by 0.7, which is five orders of magnitude worse. A contour drawn in the wrong place would fail the first test; a check that only looked at points on the curve would pass with the contour drawn anywhere.

Two things about that refinement are worth recording because both were wrong first.

The bisection has to search across the contour, not along a fixed axis. Refining along x̂ works where the curve runs steeply and fails where it runs flat, because there is no sign change to bracket along a direction the curve is parallel to. Nine of thirteen sample points refused to refine for that reason alone, and the failure looked like the contour being wrong rather than the search being pointed the wrong way. The normal to the local contour segment always crosses.

The defect has poles, and a cell straddling one flips sign without a root between. Where three images are nearly collinear the fitted circle’s radius runs off to infinity and so does the miss distance, so the sign changes across the pole with no zero in between. Marching squares does not know that and will happily draw a contour segment there. Cells whose corner values jump by more than the grid can account for are skipped, which is why the traced curve has gaps rather than spurious branches.

Why the curve is a cubic #

The degree is not an incidental fact and it is worth a paragraph, because it explains both the shape of the curves and the count of solutions at five positions.

Concyclicity of four points is a determinant condition. Four points lie on a common circle exactly when a certain 4 × 4 determinant vanishes — the one whose rows are (x² + y², x, y, 1) for each point. Substituting the four images of a single moving point, each of which is a linear function of that point’s coordinates, gives a polynomial condition in two variables. The x² + y² terms are quadratic, the rest linear, and the leading quadratic parts cancel between rows because all four images are rigid images of the same point. What survives is cubic.

Two consequences follow directly. A cubic curve is what it is: it can have one branch or two, it can have a node or a cusp for special pose sets, and it has at most three intersections with any line. And two cubics — the circle-point curve for one set of four poses and for another — meet in at most nine points, which is the projective count behind the classical result that five poses give up to four real Burmester points once the degenerate intersections at the poles are removed.

That is also why the field stops where it does rather than continuing with diminishing returns. The degrees do not grow gently; the problem stops having solutions.

A worked reading of the figure #

The hero figure repays being read slowly, because two curves in two different planes are drawn on one set of axes and that is a liberty worth flagging.

The circle-point curve lives in the moving body’s frame. A point on it is a place on the coupler — a location to drill a hole. It is drawn here in the fixed frame with the body in its first prescribed pose, which is the usual convention and is the reason the curve appears to sit among the poses rather than on the part.

The centre-point curve lives in the fixed frame proper. A point on it is a place on the machine — a location to bolt a bearing.

The two are in correspondence: each circle point has exactly one centre point, and the correspondence is the circle fit. So a designer reads the figure by picking a point on the circle-point curve, following the correspondence to the centre-point curve, and asking whether both locations are acceptable. That is two packaging questions per dyad and four per mechanism, all of which must be answered yes at once, which is a more demanding condition than it sounds.

Drawing them together is a compromise. It shows the correspondence at a glance and it invites the mistake of measuring a distance between a point on one curve and a point on the other, which means nothing at all — they are in different frames. The alternative, two panels side by side, loses the correspondence. Neither is right, and the figure’s description says which frame each curve is in for that reason.

The shape of the curves #

Burmester’s cubics have some structure worth knowing, because it explains what a designer is choosing among.

They generally come in two branches — a closed oval and an unbounded piece — and which branch a solution comes from tends to decide what kind of mechanism it is. Solutions from the compact branch give compact mechanisms; solutions from the unbounded branch run off toward the distant pivots and long links that the three-position essay warns about.

The curves also pass through the poles of the four displacements, which is not an accident: a pole is a point that does not move between two of the poses, so it has fewer distinct images than a general point and the concyclicity condition is satisfied trivially. Those points are solutions in the formal sense and useless in every other, and they are a good example of why the construction cannot be trusted without a check downstream.

And the curves can be empty in the region that matters. There is no theorem saying a four-pose specification has solutions anywhere near the machine, and a designer who prescribes four positions somewhat arbitrarily will often find that both branches lie outside the space available. That is the fourth position’s real cost: not the loss of exactness, which is preserved, but the loss of the ability to place the pivots where there is room for them.

The grid, and what it can resolve #

The contour is only as good as the grid it was traced on, and the figure states the cell size for that reason.

A cell is the span divided by the number of samples, and a contour segment found by linear interpolation along a cell edge is accurate to something like the cell size times the local curvature. For the figure’s grid that is a fraction of a percent of the span — fine for drawing and not fine for building a linkage from, because a dyad synthesised from a point one cell off the curve misses the fourth pose by about a cell.

So the two uses of the contour are separated. Drawing uses the marching-squares output directly. Building uses a refinement: bisect across the contour until the concyclicity defect changes sign to the last bit, which takes about sixty halvings and lands at 10⁻¹⁴. The synthesis then reports the miss it actually achieved rather than assuming it is zero, which is the same discipline the three-position construction is held to even though there the answer is exact by construction.

This is a general point about contoured quantities and it applies to any figure on this site that draws a level set: the contour is a picture of where something vanishes, and the picture’s resolution is not the computation’s. Quoting one for the other is how a figure comes to claim more than it measured.

What a fourth position is worth asking for #

Since the fourth position is expensive, it is worth saying when it earns its cost, and there is a clean answer.

It earns it when the fourth position is a requirement of the machine rather than a description of the motion. A hatch that must be flat at one end of its travel, upright at the other, and clear of an obstruction at a specific intermediate point has three real requirements and one more that is also real. A designer who adds a fourth position because the coupler seemed to wander is not adding a requirement; they are trying to control the path, and the path is what the atlases were for.

The distinction shows up in what happens when the synthesis fails. If the fourth position is a genuine requirement and the curve lies outside the machine, the specification and the packaging are in conflict and something has to be renegotiated. If it was added to tidy up the motion, the right response is to drop it and spend the recovered freedom on the transmission angle instead.

Five, and then none #

At five prescribed poses the family collapses to points. The classical result is that there are generally up to four Burmester points — moving points whose five images are concyclic — and any two of them give a four-bar, so up to six mechanisms.

“Up to” is doing real work. The number of real solutions depends on the poses, and for many specifications it is two, or zero. Unlike the three- and four-pose cases, where a solution always exists and the question is whether it is usable, five poses may have no exact four-bar solution at all.

At six the count says none, and the count is right for a generic specification. What a designer does at that point is one of three things: use a six-bar, which has more dimensions to spend and correspondingly more theory; give up exactness and optimise instead; or reduce the specification, which usually means admitting that some of the six positions did not really need to be exact.

The six-bar route deserves a sentence, because it is what industrial practice actually does. A Watt or Stephenson six-bar has more links, more pivots and correspondingly more dimensions to spend, and the counting extends: the same dyad arithmetic applied to a chain with more of them pushes the exact-synthesis limit out to eight or nine positions depending on the type. What it does not do is make the problem easier — the equations are higher degree, the solution sets are harder to characterise, and the branch and order questions get worse rather than better because there are more ways for a longer chain to fold up.

What the fourth position does not buy #

There is a temptation, having gone to the trouble of prescribing a fourth position, to believe that the resulting mechanism is better controlled than a three-position one. It is more constrained. Whether it is better is a separate question and often the answer is no.

Three positions leave four free parameters, and those parameters can be spent on where the ground pivots go, on the transmission angle, on making the input link a full rotator, and on avoiding the branch problems that disqualify most solutions. Four positions leave two, and the two remaining are constrained to a curve that may not pass anywhere useful.

So the fourth position competes with every other design requirement for the same budget, and it should be added only when it is genuinely a requirement rather than a way of pinning down a motion that felt under-specified. The characteristic mistake is to prescribe four positions because three “did not seem like enough”, and then to discover that every solution puts a bearing where the frame is.

None of this is visible in the construction. The circle-point curve is computed from the four poses and nothing else — it has no idea where the machine is, what the packaging constraints are, or whether the resulting linkage can pass between the poses without being dismantled. That last question, in particular, is answered nowhere in this essay, and it disqualifies more solutions than everything above put together.