A degree counted on a line

This field now has two ways of saying that a four-bar’s coupler curve has degree six, and they have something in common that is easy to miss because it looks like the point of both.

The first fits it. Six hundred traced positions, a least-squares problem at each of four degrees, and a decision read off the gap between two singular values. The second eliminates it: two circle conditions that turn out to be linear in the coupler’s orientation, Cramer’s rule, and a determinant whose degree can be counted. One is a measurement and the other is a derivation, and they agree.

Both of them produce the curve’s equation. The fit returns its twenty-eight coefficients as a vector; the elimination returns them as expressions in the four lengths. The degree is read off an object that has to be built first.

This essay measures the degree without building that object at all.

What degree means before it means anything else #

The degree of a plane curve is usually introduced as the highest power in its equation. There is an older and more useful definition that does not mention an equation: the degree is the number of points in which a general line meets the curve. A circle meets a line in two points, a cubic in three, and a coupler curve, if the textbooks are right, in six.

Everything turns on the word general, and it carries three separate conditions.

A real line can miss the curve entirely, or cross it at four points and pass the other two by. So the count has to be taken over the complex numbers, where the missing meetings are still there.

A line tangent to the curve meets it twice at one point, and a line through a crossing of the curve does the same. So the line has to avoid every special position, and the reliable way to do that without knowing where the special positions are is to choose its coefficients at random. The set of bad lines has measure zero, and a random line avoids it with probability one.

And some of the meetings may be at infinity rather than in the plane. That condition matters more for a coupler curve than for most, because this curve is tricircular, and the essay comes back to it once the count is in.

The question written as a machine #

The line is α·x + β·y + γ = 0, with α, β and γ three complex numbers drawn from a seeded generator — seeded so that the same line is drawn every time the figure is, and random in the only sense that matters, which is that nothing about the four-bar was consulted in choosing them.

The unknowns are the four-bar’s two moving pins: the crank pin’s two coordinates and the coupler pin’s two. Three conditions say the machine is assembled. The crank pin is its crank length from the first ground pivot, the coupler pin is the rocker length from the second, and the two pins are the coupler length apart. Each of those is a quadratic.

The fourth condition says the coupler point is on the line. The coupler point is a fixed combination of the two pins, 0.45 of the way along the coupler and half a coupler length across it, so the condition is linear in the unknowns.

Four equations of degrees two, two, two and one in four unknowns: Bézout’s number is eight. That is how many paths the continuation has to track, starting from a system whose eight solutions can be written down and deforming it into this one.

Six arrive. Every one of them is a configuration of the four-bar, in complex coordinates, whose coupler point lies on the random line, so between them they are six points of the coupler curve on that line. Nothing in the computation is a polynomial in the plane’s own coordinates, nothing was traced, and nothing was fitted.

The two that leave, and where they are going #

The count of eight against six is not a defect of the method, and the field has seen its shape before. A four-bar with its crank held still is two circles, Bézout’s number is four, and two of the four paths leave, for the same reason every pair of circles in the plane loses two: all circles pass through the two circular points at infinity, and a pair of them meets there as well as in the plane.

The two paths that leave here can be asked where they are going, since each is a vector of coordinates growing without bound, and what matters is the direction it grows in. For both, the crank pin’s second coordinate divided by its first comes out at −i on one path and +i on the other, to four figures, and the coupler pin’s ratio does the same. Those are the directions of the two circular points. The machine’s pins are running off along the isotropic lines, which is where every circle condition is satisfied at infinity at once.

So the six-and-two split is the two-circles split carried one level up, and the pair of paths lost is the same pair. What the extra freedom and the extra line buy is four more finite meetings. That is the curve’s degree beginning to show through.

Six on every line tried #

One line is one measurement, and a count that happened to come out at six on one line could be a coincidence of that line. The claim is that it does not depend on the line, which is exactly what makes it a property of the curve, so the same machine was sliced by four different random lines, each solved from scratch.

Every one gave eight paths, six configurations, and six distinct points of the plane.

The last column of that table is worth a moment. None of the six solutions is real. That is not a statement about the curve, which is a perfectly good real curve drawn by a perfectly good real machine; it is a statement about the line. A line with complex coefficients has, in general, no real points at all, so nothing on it is real. The count is of complex meetings, which is the only count that does not depend on where the line happens to lie.

A real line is a worse instrument #

The figure below is the same computation with a real line: a line drawn across the site’s standard crank-rocker at about 53° to the ground line. It crosses the traced curve at four places. Solved completely, it meets the curve in six points, and the other two are a complex-conjugate pair that no drawing can show.

Sweeping that line across the plane makes the difference between the two counts plain.

The real count is 0 while the line is clear of the curve, rises to 2 and then 4 as the line enters the ovals, and falls back to 0 once it is past. Nowhere in this sweep does it reach six. That is one family of parallel lines coming up short, not a proof that no real line meets this curve six times, and it is recorded as exactly that.

The real count is a fact about where the line is; six is a fact about the curve. A measurement made with real lines only would report a maximum of four, and be wrong about the degree by two, for the same reason that a machinist looking for four assemblies of a planar platform finds two: the missing ones are complex, and complex solutions are not optional extras of a polynomial system.

Configurations are not points #

A solution of the system is a configuration of the machine, and the degree of a curve is a count of points in the plane. For the coupler point those two counts agree, and it is worth checking that they must rather than assuming it, because on the other bodies of the same four-bar they do not.

A point on the crank traces a circle, so a line meets its curve in two points. The system nevertheless has four solutions, because each of those two crank positions can be completed in two ways — the coupler pin on one side of the line of centres or the other — and both completions put the crank point in the same place. The rocker is the same with the roles exchanged.

So the tracker’s count is the number of points times the number of configurations that draw each one, and the degree is the first factor. For the crank and rocker the second factor is two. For the coupler point it is one: each point of the coupler curve is drawn by exactly one configuration, which is the statement that the curve is not traced twice over.

Nothing here is specific to the four-bar, and it will matter much more on a six-bar, where the configurations outnumber the points by as much as six to one.

The six are all of them, and a sum says so #

A total-degree homotopy reaches every isolated solution with probability one, and the field has already measured what that probability-one statement is worth when a platform’s count was in doubt. It is still a statement about the method rather than about the answer. A second check that looks only at the six points, and would fail if a seventh existed or one of the six were spurious, is worth having.

There is one, and it is old: the trace test.

Hold the line’s direction fixed and slide it: the lines α·x + β·y + t = 0 for varying t form a pencil of parallel complex lines. Along the pencil the six meetings move. Substitute the line into the curve’s equation — which exists, even though nothing here computes it — and the meetings are the roots of one polynomial of degree six in the position along the line. The sum of a polynomial’s roots is the ratio of its top two coefficients. The top coefficient comes only from the curve’s highest-degree terms and does not involve t at all, and the next coefficient is at most linear in t.

So the sum of the six witness points is an affine function of t. Not approximately: exactly, for every curve and every pencil, provided every root is in the sum.

With all six, the middle value is the mean of the outer two to 1.3 × 10⁻¹⁵ of the sum’s change across the pencil — which is the arithmetic’s noise and nothing else. Leave out any one of the six and the sum of the other five is not straight: it bends by between 3.6 × 10⁻⁴ and 7.5 × 10⁻³, eleven to thirteen decades above the noise. The gap between those two outcomes is the test.

It is worth being precise about what the test established. It does not prove that the curve has degree six in the abstract; it proves that these six points are every point in which this line meets the curve they lie on. Combined with the four random lines all giving six, that is as strong as a numerical statement about a degree can be made.

One curve, not two ovals #

The sweep above showed the crank-rocker’s coupler curve as two separate ovals in the plane, which is what a crank-rocker’s two assemblies draw: each circuit of the machine traces one closed loop, and nothing joins them. A reasonable reading of that picture is that the machine draws two curves.

The witness points can settle it. Take the line’s constant round a loop through the complex numbers and bring it back to where it started. The six witness points are carried along, and when the loop closes they must return to the same six points as a set — but not necessarily each to its own place. A loop can permute them.

If the six points lay on two different curves, no loop could carry a point of one onto a point of the other, because each curve’s meetings with the moving line would stay on that curve throughout. So if the loops reach every point from every other, the points lie on a single irreducible curve.

The first loop moves nothing; the second carries three points round a cycle and leaves four orbits; by the seventh there are two; the tenth joins those. From then on the six are one orbit, and the remaining loops only shuffle within it.

The two ovals are one curve. They are the two real pieces of a single irreducible sextic whose complex points join them, and the machine traces one piece on each circuit because the real part of that curve happens to come apart into two. That is the same conclusion the component count reached about the configuration space from the other side. There the two circuits were two real components; here they turn out to be one complex object seen through the reals.

It is also a test that can fail, and on a special four-bar it does. A parallelogram — ground equal to coupler, crank equal to rocker — has the same eight paths and the same six witness points, but under thirty-two loops its points settle into one orbit of four and one of two and never merge further. A parallelogram’s coupler curve really is two curves, of degrees four and two, and the loops say so without either of them being written down.

Three routes, and what each can be asked #

The degree of the four-bar’s coupler curve has now been established three times, and the three share nothing.

The fit needs traced points and returns coefficients. It can be run on a curve that arrived as data from anywhere, and it pays for that generality in conditioning: its answer is a gap between singular values, and the gap depends on how much of the curve was traced and how the columns were scaled.

The elimination needs the equations in closed form and a way of combining them symbolically. It returns the degree as an integer and the coefficients as formulas, which is the most anybody could want, and it depends on the combination being short enough to carry out. For the four-bar it is three lines; for most mechanisms nobody has found the lines.

The slice needs neither. It needs the machine’s constraints as polynomials, which every mechanism in these essays already is, and one random line. It returns a count, a completeness certificate in the trace test, and an irreducibility certificate from monodromy. What it does not return is the equation, and nothing about the curve’s shape beyond where it meets the lines that were tried.

Having all three for the same curve is what makes the third trustworthy on its first real outing. It reports six where the other two report six, and a method that agrees with two independent routes on a known case can be used on a case where neither of them works.

Why the line costs almost nothing #

The price of this route is worth setting against the other two, because the comparison runs the opposite way to what the sizes of the objects suggest.

A fit’s cost is set by the curve. A polynomial of degree d in two variables has (d + 1)(d + 2)/2 coefficients, which is twenty-eight at six and a hundred and ninety at eighteen, and every one of them is a column the traced points have to pin down. Doubling the degree roughly quadruples the unknowns and makes the columns more nearly dependent at the same time.

The slice’s cost is set by the machine. The three closure conditions are quadratics whatever the curve does. The line is linear because a point on a body is a linear combination of that body’s pins, and a linear equation multiplies Bézout’s number by one. So eight paths are tracked here because the four-bar is three quadratics, not because the curve is a sextic. A six-bar with five quadratic closure conditions costs thirty-two paths whichever of its bodies is sliced — a point whose curve is a circle and a point whose curve is far more complicated alike.

The elimination’s cost is set by neither, and that is its difficulty. It depends on whether a short combination of the equations exists, which for the four-bar it does, and which is a question with no general answer.

The whole four-bar count, eight paths of which six arrive, takes tens of milliseconds. Its completeness certificate is three more short homotopies per witness point, and its irreducibility certificate is two per point per loop. Nothing in any of it grows with the degree of the curve, and that is the property the harder case below depends on.

What the count leaves unsaid #

Four limits, stated because the count is clean enough to be overread.

It is probability one, not certainty. The line’s coefficients are fixed draws from a seeded generator. A line that happened to be tangent to the curve would give a double meeting and five distinct points, and nothing inside that one run would announce it. The protection is the four independent lines giving the same six, not any single run.

Distinct means distinct to a tolerance. Two witness points are counted as one when they agree to one part in a million. Every pair here is far further apart than that, and the tolerance is a decision rather than a fact about the curve.

It is the degree of the image. A body whose curve is drawn twice over, like the crank point, shows up as a configuration count double the degree, and the essay reads the degree off the distinct points. Where that multiplicity is not constant along the curve — which does not happen on any body measured here — the count would need more care.

It says nothing about the real picture. The largest number of real meetings a line can have with this curve is not settled by anything above; the search found four.

What comes next: a curve with no equation anybody has written #

The four-bar is the easy case, and deliberately so. Its coupler curve has a fit that works and an elimination that fits on a page, which is exactly what made it the right place to check a third route against the first two.

The six-bar is the reason the third route exists. A six-bar whose second dyad hangs from the coupler of a four-bar draws, with a point on that dyad, a curve that nobody eliminates by hand. Fitting it at the degree the literature quotes would need a hundred and ninety coefficients, and the fit’s gap closes long before that. The slicing method above runs on it unchanged: thirty-two paths instead of eight. What it finds, what the fit fails to find, and why a six-bar’s configurations outnumber the points they draw, is the curve nobody eliminates.

A second question sits beside it. The witness points joined the two ovals into one curve, and the component count says a four-bar never has more than two circuits. Why the number is two, and why a six-bar’s is not, is a question about the genus of the configuration curve, and it is never three circuits.