A point the machine never reaches

The essay on the two ovals ended on a ring. A crank-rocker whose two ovals cross twice turned out to have a third special point, at (3.771, 0.555), that satisfies the sextic, lies in the real plane, and is visited by no configuration of either assembly.

It is not a quirk of that linkage. Every four-bar’s coupler curve has at least one real point of that family, and on the site’s standard crank-rocker, the linkage whose curve has been fitted, eliminated and compared across cognates, the only one it has is of the unvisited kind. This essay says what those points are, where they have to be, how many there are, and why the count cannot come out even.

Two configurations through one point #

A double point of a curve is a point where the curve passes twice. For a curve drawn by a machine that has a direct reading: two different configurations of the linkage put the coupler point in the same place.

The elimination that produced the sextic makes this exact rather than pictorial. Fix a candidate position P for the coupler point and ask which orientations of the coupler put it there. The crank pin must then sit a crank’s length from its pivot and the rocker pin a rocker’s length from its own, and after using cos²φ + sin²φ = 1 both conditions are linear in the pair (cos φ, sin φ):

$$\alpha_1 \cos\varphi + \beta_1 \sin\varphi + \gamma_1 = 0, \qquad \alpha_2 \cos\varphi + \beta_2 \sin\varphi + \gamma_2 = 0,$$

with every α and β linear in the coordinates of P and every γ quadratic. Two linear equations in two unknowns have one solution when their determinant is not zero, and the sextic is precisely the statement that this solution lands on the unit circle.

So at an ordinary point of the curve the orientation is fixed: one configuration, one pass. The only way to get two is for the two equations to stop being independent, so that they agree on a whole line of candidate orientations rather than meeting at one. That needs the determinant α₁β₂ − α₂β₁ to vanish, and both of Cramer’s numerators with it. What happens next depends on where that line sits relative to the unit circle, because an orientation has to be on the circle to be an orientation.

A crossing, an isolated point, and the case between #

The line of candidates can meet the unit circle in two real points, in none, or touch it.

Two real orientations means two real configurations of the linkage place the coupler point at P. The curve genuinely passes through P twice with two different tangents, and a drawing shows it crossing itself. On the linkage with ground 4, crank 1.4, coupler 2.5 and rocker 3, the double point at (0.368, −1.759) has its line 0.677 from the centre of the unit circle, well inside, so it is cut twice. That is the crossing of the two ovals found in the essay on the two ovals, where the machine is solved at −16.4° on one assembly and −140.0° on the other.

No real orientation means the line passes outside the unit circle, and both orientations are a complex-conjugate pair. P is still a solution of the sextic, because the sextic only asks that the orientation system be consistent with some point on the circle, and complex points of the circle count. It is still a singular point, because the gradient of the polynomial vanishes there. But no real linkage has its coupler point there. The double point at (3.771, 0.555) on the same linkage has its line 1.203 from the centre, and it is exactly the ring drawn there.

A line that touches the circle would make the two orientations coincide, and P would be a cusp: the one configuration in which the coupler point has momentarily stopped. That is the classical case already on the site as the coupler point at the instant centre, and it is a boundary between the other two rather than a third region: move a linkage’s dimensions slightly and a cusp opens into a small loop or closes into an isolated point.

The classification is made twice, and the two ways share nothing. The first is the line and the circle, as above. The second is the sign of the determinant of the sextic’s second derivatives at P, a negative value for a crossing and a positive one for an isolated point. Every real double point reported here is classified the same way by both.

The circle through the three pivots #

Double points need the determinant α₁β₂ − α₂β₁ to vanish, and that determinant is a quadratic polynomial in the coordinates of P. Its quadratic part is exactly −4b²v(x² + y²), with the coupler point’s position along the coupler cancelling out in the same way it cancels in the tricircularity argument. So for any coupler point off the coupler line, the set where the orientation can be undecided is a circle.

It is a specific circle, and it turns out to be a familiar one. It passes through both fixed pivots of the linkage, and through the third fixed pivot that Roberts’s construction places for the cognates. For the site’s crank-rocker its centre is at (2.000, 0.010) and its radius is 2.0000, which is why it looks so nearly centred on the ground link: the ground is 4 long, and the third pivot lies just above the line.

That makes a geometric claim out of an algebraic one. All three finite double points of every coupler curve lie on the circle through the three cognate pivots. It is a strong constraint on what a coupler curve can look like. A crossing, a loop’s knot or an isolated point cannot be placed anywhere a designer likes: wherever it is, it and the three pivots are concyclic, and the three cognates, which share the pivots, share the circle.

When the coupler point lies on the coupler line, v = 0, the quadratic part vanishes and the circle degenerates into the ground line, with the three pivots collinear on it. The construction below uses the circle’s parametrisation, so it refuses that case rather than handling it silently; it is excluded from everything counted here, and it is recorded as not covered.

Six roots, of which three are double points #

With the circle known, the double points can be found by construction rather than by search.

Parametrise the circle rationally, $x = c_x + R(1 - t^2)/(1 + t^2)$ and $y = c_y + 2Rt/(1 + t^2)$, so that every real t is a point of the circle and so is the limit t → ∞. Pull the cubic β₁γ₂ − β₂γ₁, one of Cramer’s numerators, back along it and clear the denominator. The result is a real polynomial of degree six in the single variable t, and its roots are every place on the circle where the numerator vanishes as well as the determinant.

Not all six are double points, and each of the other three can be identified by what it is.

Two roots are t = ±i, which the parametrisation sends to infinity along the two isotropic directions. These are the circular points, through which every circle passes, and they are roots because the circle and the curve both meet the line at infinity there.

One root is the third pivot itself. At that point both β’s vanish together, so the numerator vanishes for a reason that has nothing to do with the curve, and a check of the sextic there shows the pivot is not on it.

The remaining three are the double points. A real one is then checked against the sextic directly, requiring the polynomial and both of its first derivatives to vanish: on the site’s crank-rocker the value is below 10⁻¹¹ and the gradient below 10⁻⁹ of the polynomial’s size, and a figure that ever drew a real double point failing that test would refuse to draw.

Here is the arithmetic that fixes the parity. The polynomial in t has real coefficients, because every length and the coupler point’s position are real. A real polynomial’s non-real roots come in complex-conjugate pairs. The two circular points are one such pair, and the third pivot is real. That leaves three double points, and three objects whose non-real members pair off must include one real or three real. There is no way to have none and no way to have two.

Where three comes from #

The number three is not an accident of this construction, and it is worth connecting to the classical count, which is quoted here rather than reproduced.

An irreducible plane curve of degree six can have at most ten double points, counted with multiplicity; any more and it would have to split into pieces. The coupler sextic is tricircular: it passes through each of the two circular points three times, and a triple point absorbs three double points’ worth of that allowance, so the two circular points take six. That leaves room for four more, and a coupler curve has three, which leaves a deficit of one. The deficit is the curve’s genus, and a genus of one is the reason the curve can have two ovals and never three, which is the subject of the essay on circuits.

The site’s measurement is the finite count, three, on every curve examined. The genus is a consequence stated in the classical theory, and it is used here only to explain why the count comes out as it does.

The standard crank-rocker’s one real double point #

On the site’s crank-rocker the construction returns three double points, one real. The real one is at (3.964, −0.368), and it is isolated: its line of candidate orientations misses the unit circle, and the second-derivative test agrees.

It sits 0.628 from the nearer oval, just under the rocker pivot at (4, 0) and outside both of the closed curves the two assemblies draw. The other two double points are a complex-conjugate pair and have no position in the real plane at all.

So the four-bar curve analysed most thoroughly in these essays has a real point of its equation that is on neither of its ovals. The drawing of the curve that every earlier essay has shown is correct and complete as a drawing of where the coupler point goes, and incomplete as a drawing of the equation. The equation includes a dot.

That dot is not a numerical artefact and not a point that some unusual configuration reaches. It is the place where the coupler point would be if the coupler could take either of two orientations whose cosines and sines are complex numbers. No solver asked for real configurations will ever put the machine there, and the reason is that the machine is not there.

What a contour plot cannot see #

The obvious way to draw an implicit curve is to evaluate the polynomial on a grid and mark every cell whose corners disagree in sign, since a continuous function that changes sign between two points is zero somewhere between them. Every contour plotter works this way, and on the site’s crank-rocker it works well.

Every one of the 348 flagged cells lies within 0.028 of a traced oval, which is inside one cell’s diagonal. None is anywhere near the isolated double point. The sextic takes the same sign all round that point, as a polynomial must at an isolated zero of even order along every line through it, so there is no sign change for the scan to catch. The nearest flagged cell is 0.614 away.

This is a general property rather than a limitation of this grid. Refining the grid does not help, because the difficulty is not resolution: the polynomial touches zero at a single point and rises again on every side. A method that finds a curve by where a function changes sign finds the ovals and cannot find the dot.

It is the same kind of blindness met elsewhere, where a real count and a complex count disagree and only one of them is visible to the instrument being used. The isolated point is a real solution that behaves, to any sign-based method, like no solution at all.

Never none, never two #

One crank-rocker is an example and the parity argument is a claim about all of them, so it is checked over a census.

Four hundred coupler curves are drawn from a seeded generator, with the ground fixed at 4, the crank between 0.5 and 3.5, the coupler and rocker between 1 and 4, and a coupler point anywhere off the coupler line. Each has its double points found by the construction above and each real one classified.

All 400 have exactly three. Eighty-seven have one real double point and 313 have three. None has none, and none has two, which is the parity argument holding on every curve it was tested against.

The kinds are more varied than the count. Of the curves with one real double point, 37 have it as a crossing and 50 as an isolated point. Of those with three real, 56 have three crossings, 124 have two crossings and an isolated point, 97 have one crossing and two isolated points, and 36 have all three isolated.

Two of those groups are worth dwelling on: the 50 whose only real double point is isolated, and the 36 whose three are. On those 86 curves no assembly of the machine ever passes the same point twice. Each oval is a simple closed loop, no oval crosses another, and a drawing of everything the machine does shows no double point at all; for the 50 that is measured directly on the traced ovals, and for the 36 it follows from every real double point being isolated. Every one of them still has a real double point in its equation. More than a fifth of the census draws a curve with no visible double point and an equation with at least one, so the site’s standard crank-rocker is typical rather than special.

What a tolerance accepted #

The construction is a count. A search is what most people write first, and it is worth seeing what a search reports on the same curves, because it reports something different and plausible.

The search is Newton’s method on the two first derivatives of the sextic, started from a grid of points, accepting any place where the gradient and the value are both below 10⁻⁹ of the polynomial’s size. It knows nothing about orientations or circles. Run on the first 120 curves of the census, it can be compared with the construction point by point.

It agrees with the construction on 114 of the 120 curves. It accepted 302 genuine double points, at relative values no larger than 2.2 × 10⁻¹⁶ and located to 3.6 × 10⁻⁶. It also accepted five points that are not double points, at relative values from 1.3 × 10⁻¹² to 5.2 × 10⁻¹⁰, and it missed one genuine crossing.

The five are not errors in the arithmetic. Each is a genuine stationary point of the polynomial, within 0.10 of a real double point whose line of orientations passes within 0.010 of tangency to the unit circle, which is to say a double point close to becoming a cusp. Near such a point the polynomial is very flat over a small region, and a second nearby place where both derivatives vanish has a value small enough to pass any tolerance a reasonable person would set. On a curve where it accepts one of those, the search reports an even number of real singular points, which the parity argument forbids, and nothing in the search’s own output says which of them is the impostor. Only the construction does.

That is why the construction is the count and the search is the check. A tolerance cannot establish a parity, for the same reason a residual could not establish tricircularity: both are claims about exact zeros, and an instrument that accepts small numbers accepts the wrong small numbers too.

What this essay does not establish #

The coupler point on the coupler line is excluded. There the circle becomes the ground line and the rational parametrisation fails; the double points still exist, and a different construction would be needed to count them.

Linkages at a change point are excluded as well. On a chain where the sum of the shortest and longest links equals the sum of the other two, the census count is wrong in a specific way: such curves have four or five double points rather than three, and at least one of them is not on the circle. That is the subject of a sextic that comes apart, where the extra points turn out to be the places the machine chooses between two motions.

No cusp appears in the census. A cusp needs the line of orientations exactly tangent to the circle, which a random draw of lengths reaches with probability zero; the five near-tangent cases found by the search are the closest the census came. And the genus of one is quoted from the classical theory to explain why the finite count is three. The count itself is what is measured.

What comes next #

The linkages whose sextic falls apart. A parallelogram chain and a kite each have a coupler sextic that factors into a circle and a quartic, one factor per circuit. Their extra double points are where the two factors meet, and some of those meetings are configurations belonging to both. A sextic that comes apart divides the polynomial exactly and finds which meetings are real change points.

The circle as a synthesis constraint. Every double point of a coupler curve is on the circle through the three cognate pivots, so a designer who wants a crossing at a stated place has, in effect, prescribed a circle through that place and the two pivots. Whether that turns the synthesis field’s precision-point problems into anything simpler is a question with a measurable answer: how many of the prescribed-crossing linkages a search finds satisfy the circle condition exactly, and how the cognates of one are the cognates of all.

An isolated point as a measurement hazard. The metrology field recovers linkages from their traced paths. A curve fitted to a traced path contains its isolated double point whether or not anything traced it, so a fitted equation carries information about a place no measurement visited. Whether that point’s position is well determined by the fit, or is exactly the kind of extrapolated quantity the fit determines worst, is the next thing to measure about it.