Nine times through each circular point

The curve nobody eliminates measured the degree of a Stephenson six-bar’s arm curve by cutting it with random lines. Thirty-two paths were tracked, eighteen arrived, and eighteen was the degree. The other fourteen left for infinity, and the essay ended by asking where they went: on the four-bar the two paths that left had gone to the circular points, and the arm curve’s analogue of that was open.

For the four-bar the answer was already known, exactly. Three linkages, one equation worked out the highest-degree part of the coupler sextic and found it to be $4b^2(x^2 + y^2)^3$ for every four-bar and every coupler point, so the curve meets the line at infinity only at the two circular points, three times at each. That route needed the equation. The arm curve has none.

This essay measures the same property without an equation, on every body of five machines, and finds that the four-bar’s three is one case of a rule: a curve drawn by a point of a machine made only of pins passes through each circular point half its degree times. The arm curve does so nine times.

A line that starts at infinity #

The circular points are the two points at infinity through which every circle passes, (1 : i : 0) and (1 : −i : 0). They are complex, they are not in any drawing, and the paths that leave showed why they are unavoidable in this subject: every bar of a linkage is a circle condition, and every circle condition carries those two points with it.

A line that passes through one of them has an equation of the form x + iy = c, and a line through the other has x − iy = c. These are the isotropic lines. Each passes through exactly one circular point, and for a general constant c it passes through nothing else special.

The instrument rests on one fact about how a line meets a curve. A general line meets a curve of degree d in d points. A general line through a point where the curve passes k times meets it there k times over, and so has only d − k meetings anywhere else. Put the point at infinity and the d − k are all finite, so they can be counted by the same method a degree counted on a line used: write the machine’s closure conditions and the line condition as one polynomial system, track every path from a start system, and count the distinct points the arriving configurations put the tracing point at.

So each body needs three runs. A random complex line gives the degree d. A line through the first circular point gives d − k, and a line through the second gives d − k′. The differences k and k′ are how many times the curve passes through each circular point, which is its circularity at each. No coefficient is fitted and no variable is eliminated. The only change from the degree measurement is the line’s coefficients.

The check on a circle is immediate. A circle has degree two and passes through each circular point once, so an isotropic line meets it in one finite point. Solving x + iy = c against a circle’s equation gives exactly one root, the other having gone to infinity. The four-bar’s crank and rocker, and every six-bar’s output, draw circles, and their rows in the table read degree 2, one finite meeting, circularity 1.

Every pin-jointed body at half its degree #

The table holds twelve bodies, and nine of them belong to machines built from pins alone: the four-bar, ground 4, crank 1, coupler 3.5 and rocker 3, with its coupler point 0.45 along the coupler and 0.5 to the side; the Watt six-bar built on that four-bar with a second dyad from its rocker; and the Stephenson six-bar of the dwell linkage, whose arm hangs from the coupler point.

The four-bar’s coupler curve has degree six and three finite meetings with a line through either circular point, so it passes through each three times. That is the elimination’s tricircularity, reached by a route that never saw a coefficient. The Watt six-bar’s first coupler is the same four-bar’s and reads the same. Its second coupler draws a sextic with every point drawn twice, and reads 6, 3, 3. The Stephenson six-bar’s coupler, which belongs to its four-bar, reads 6, 3, 3 too.

The Stephenson arm curve has degree 18. A line through the first circular point meets it in nine finite points, and a line through the second in nine. It passes nine times through each.

Nothing in the arm’s construction says nine. Its degree was measured rather than derived, and its equation, of a hundred and ninety coefficients, has not been written. The circularity comes from the same thirty-two paths the degree came from, with a different line.

The paths show the same thing from the side of the solver. The four-bar’s coupler system has eight paths; six arrive on a general line and three on a line through a circular point. The Watt second coupler’s thirty-two paths give twelve configurations on a general line and six on the isotropic one, and since every point is drawn twice those are six points and three. The Stephenson arm’s thirty-two give eighteen and nine.

Turning the line to pass through a circular point sends half the finite meetings of a pin-jointed curve to infinity, and it does so on every body measured. The number of paths does not change, the start system is built the same way, and the paths that stop arriving are exactly as many as the circularity.

The count belongs to the curve #

A circularity from one line through a circular point could be a property of that line. The claim is that any general line through the point gives the same count, since the count is supposed to be the point’s multiplicity on the curve and not an accident of where the line crosses.

The arm curve was cut by five lines through each circular point, with five different random offsets, each solved from scratch. Every one of the ten counted nine.

The five general lines were solved at the same time, and they show why a count from a single run is a lower bound rather than a measurement. Four of them counted eighteen. The fifth counted seventeen, and the solver’s own bookkeeping says why: of its thirty-two paths, fourteen left for infinity, none stalled, and eighteen arrived, but two of the eighteen arrived at the same configuration. On a general line no configuration is double, so two paths ending together means one of them jumped onto the other’s track on the way, and the configuration it should have reached was never found. A jump of that kind can only lower a count, never raise it. Eighteen on four lines, and seventeen with two paths merged on the fifth, is eighteen.

The same asymmetry protects the circularity. A merged path on an isotropic line would make the finite count too low and the circularity too high, and the ten isotropic runs agree on nine with every arriving path at a configuration of its own.

Both of those counts are counts of things that cannot be drawn. A complex line has no picture in the plane the curve is drawn in, and a configuration that arrived at a complex solution is not a position the machine can be put in. What can be drawn is the real shadow of the same arithmetic — a real line laid across the arm curve — and it is worth looking at, because it is where the surprise lives: the number of crossings a reader can point to changes as the line slides, and the number the count is about does not.

Slid across the curve, the line meets it in no real points at all near the origin, then in four, then in six, and the eighteen it is credited with never changes. The complex crossings are making up the difference, silently, and they are exactly the same kind of object as the nine at each circular point — with the one difference that these can be brought into view by moving the line, and those cannot be brought into view at all. The circular points sit at infinity in a direction the real plane does not have.

That is the whole reason this essay counts rather than draws. A degree is a statement about a curve over the complex numbers, and the moment a count is allowed to mean “how many crossings are visible” it stops being a property of the curve and becomes a property of where the line happened to be put.

Why half: z and z̄ #

The rule has a reason, and the reason says what would break it.

Write a point’s position as two complex numbers instead of two real ones: z = x + iy and z̄ = x − iy. For a real point z̄ is the conjugate of z, but as coordinates on the complex plane they are independent, and the isotropic lines are simply z = c and z̄ = c. In these coordinates a bar of length r between pins P and Q says

$$(z_P - z_Q)(\bar z_P - \bar z_Q) = r^2,$$

a z-difference times a z̄-difference. A point fixed to a body is a fixed complex combination of two of its pins, so its z is a combination of their z’s and its z̄ of their z̄’s. Nothing in a machine made of pins and bars ever mixes the two in any other way.

That structure decides how a point can reach infinity. For its position to grow without bound while every bar keeps its length, some product of a z-difference and a z̄-difference must stay fixed while one factor grows, and then the other factor must shrink. The point goes to infinity with z growing and z̄ staying finite, or the other way round. A point at infinity reached with z large and z̄ finite has x − iy negligible beside x and y, so y/x = −i, which is the direction of a circular point; with z̄ large and z finite, y/x = i, the other. So a pin-jointed curve meets the line at infinity only at the circular points.

The four-bar shows the mechanism in three lines. Put the crank’s pivot at the origin and the rocker’s at g on the real axis. The crank pin A satisfies $z_A \bar z_A = a^2$, so if $z_A$ grows then $\bar z_A$ shrinks towards nought. The rocker pin B satisfies $(z_B - g)(\bar z_B - g) = c^2$, so $z_B$ and $\bar z_B$ cannot both grow. If $z_B$ grows with $z_A$, then $\bar z_B$ tends to g, the coupler condition $(z_B - z_A)(\bar z_B - \bar z_A) = b^2$ has its second factor tending to g, and the first settles at $b^2/g$: both pins run off together with their z̄’s finite. If instead $z_B$ stays finite, the coupler’s first factor grows, its second must vanish, $\bar z_B$ tends to $\bar z_A$ and so to nought, and the rocker condition fixes $z_B$ at $g - c^2/g$. The remaining cases, with only B running off or with z̄ growing in place of z, end the same way with the roles exchanged: the coupler point, a fixed complex mixture of A and B, has one of z and z̄ growing and the other finite. No complex configuration of a four-bar carries its coupler point to infinity in any other direction.

The curve is real, and conjugation exchanges the two circular points, so it passes through them equally often. A curve of degree d meets the line at infinity d times in all, and if every one of those meetings is at a circular point, and the two are shared equally, each takes d/2.

A slide breaks the first step. A pin constrained to run along the x-axis has y = 0, which in these coordinates is z = z̄: its z and its z̄ are tied together and must grow together. A point can then reach infinity in a real direction, with z and z̄ both large, and the curve is free to meet the line at infinity somewhere other than at a circular point.

The two slider machines in the table show it. The elliptic trammel, a rod of length 3 with one end sliding on each axis and its tracing point 0.3 along, draws an ellipse: degree two, two finite meetings with each isotropic line, circularity nought. An ellipse that is not a circle does not pass through the circular points at all. The slider-crank, crank 1 and connecting rod 3 with the slide through the crank’s pivot, has a crank that draws a circle, circularity one as for any circle. Its connecting rod’s point, 0.4 along and 0.5 to the side, draws a quartic that meets each isotropic line in three finite points. It passes once through each circular point, not twice, and the other two of its four meetings with the line at infinity are elsewhere.

Written in z and z̄, the sextic fills a square #

The argument predicts more than a count. If the curve passes d/2 times through each circular point, its equation, rewritten in z and z̄, can have no power of z above d/2 and no power of z̄ above d/2. A line z̄ = c meets the curve where the equation, as a polynomial in z alone, has roots, and a line through a point of multiplicity d/2 has d/2 finite meetings, so the degree in z is d/2. For a sextic that means every term zʲz̄ᵏ has j ≤ 3 and k ≤ 3: a three-by-three square inside the triangle j + k ≤ 6 that a general sextic may use.

The four-bar’s sextic is available exactly, so this can be checked coefficient by coefficient.

Substituting x = (z + z̄)/2 and y = (z − z̄)/2i into the elimination’s sextic and collecting terms, no coefficient outside the square is larger than 2.2 × 10⁻¹⁸ of the largest, which is the arithmetic of the substitution and nothing else. Fourteen of the sixteen terms inside the square are present, and the corner $z^3\bar z^3$, which is $(x^2 + y^2)^3$, is 0.0055 of the largest. The two missing terms are $z^3$ and $\bar z^3$ alone.

The square says more than the leading form did, and the difference is the one that matters for what follows. The leading form $4b^2(x^2 + y^2)^3$ is the corner of the square, and it says the curve meets the line at infinity three times at each circular point. That would also be true of a curve passing once through a circular point and tangent there to the line at infinity. The empty cells outside the square, at lower total degrees such as $z^4\bar z$ and $z^5$, rule that out: they say the curve passes through each circular point three times as a point, with three branches. A triple point is what the isotropic slicing measures, since a general line through a point meets it once for each branch, and it is what the next section spends.

What the circular points cost in double points #

A curve of degree d has room for (d − 1)(d − 2)/2 double points. A curve that is the image of a configuration curve of genus g has that many less g, counted with multiplicity, and a point with k branches uses k(k − 1)/2 of them.

For the four-bar’s coupler sextic, degree 6 and genus one, the budget is ten less one, nine. Two triple points take three each, six in all, and three are left for the finite plane. A point the machine never reaches found exactly three finite double points on this curve by a construction that has nothing to do with infinity, and the agreement is the check that each circular point is an ordinary triple point spending exactly three. Of those three one is real, a place on the drawing where the curve crosses itself or stands isolated, and two are a complex conjugate pair that no drawing shows. Counted by the degree alone the curve would seem to be missing seven; counted with its circular points it is missing none.

For the Stephenson arm curve the same arithmetic is a prediction. Its degree is 18, so a curve of that degree with no singularities to spare would have room for a hundred and thirty-six double points. The arm’s configuration curve has genus seven, from the branch points counted in never three circuits, and the arm draws each of its points from a single configuration, so the curve in the plane has the same genus: a hundred and twenty-nine. The two circular points, nine branches each, take thirty-six each, seventy-two together. Fifty-seven are left for the finite plane.

More than half of the double points the genus allows, seventy-two of a hundred and twenty-nine, are therefore used at infinity, where no drawing shows them. Without the circularity, a count of the arm curve’s self-crossings and isolated points would be compared against a hundred and twenty-nine and would look far short of it however carefully it was made.

A sketch of the reason, and fifty-seven uncounted #

The reason is a sketch. The step from “each bar is a z-difference times a z̄-difference” to “the curve meets infinity only at the circular points” is argued for a point pushed to infinity, not proved for every complex branch of every machine. The measurement covers nine pin-jointed bodies of three machines at one set of lengths each.

The fifty-seven are not counted. They assume the two circular points are ordinary nine-fold points of the arm curve and every other singularity an ordinary double point. A tacnode or a triple point in the finite plane would use more than one each and leave fewer to find.

Nothing locates the slider-crank’s other two meetings at infinity. The quartic meets the line at infinity four times, twice at circular points, and the other two are not measured.

Circularity is measured at one coupler point per machine. A coupler point placed at a pin, or on a special line of a symmetric machine, can draw a curve of lower degree, and its circularity at that position is not surveyed.

What comes next: counting the arm curve’s fifty-seven #

Counting the arm curve’s fifty-seven. A double point is a place the tracing point reaches from two different configurations, which is a polynomial system in two copies of the machine with their tracing points set equal. Its path count is large, and the finite solutions with the two copies distinct are the double points. Whether it finds fifty-seven would test the genus, the ordinariness of the nine-fold points, and the arm curve’s birationality at once.

Where a slide sends a curve to infinity. The slider-crank’s quartic has two meetings with the line at infinity that are not at circular points. The direction of the slide is the natural candidate, and the paths leaving for infinity on a general line can be asked their direction of growth, as the degree essay did for the four-bar’s two. That would say whether a slide’s curve always meets infinity at the slide’s own direction, and with what multiplicity.

Real double points among the fifty-seven. Most of a high-degree curve’s double points are complex. How many of the arm curve’s are real crossings a drawing would show, how many are isolated real points, and how that count changes as the dwell linkage’s lengths are changed, is the question a designer would ask of a six-bar’s path.