Machine Kinematics
solved, then drawn
Concept

Circular points — where it appears

The two complex points at infinity, (1 : i : 0) and (1 : −i : 0), through which every circle in the plane passes. A line through one, x + iy = c or x − iy = c, meets a curve in its degree less the number of times the curve passes through that point.

Named by 10 essays across 2 fields — each of them below, with the objects they name alongside it.

Two circles, four answers, two of them nowhere. A four-bar with its crank held at 52° is two circles: the coupler pin is 3.5 from the crank pin and 3 from the far ground pivot. Two quadratics in two unknowns, so Bézout's number is four — and the tracker finds two. The other two paths run off to infinity, and they do so for every pair of circles ever drawn: two circles meet the line at infinity in the same two points, and those are what the fourth and third answers are.

Two circles, four answers

A four-bar with its crank held still is two circles, and two circles meet twice. Bézout's theorem says four. The two missing answers are not a rounding error and are not special to these link lengths — they are the same two points for every pair of circles ever drawn, and they are the beginning of a way of counting that has so far been done by hand.

algebra · Algebra How many times each curve passes through the circular points. For each body of five machines, the degree of the curve a point on it draws, from a random complex line, and the number of finite points where that curve meets a line through the circular point I, x + iy = c, and one through J, x − iy = c. The difference is how many times the curve passes through that circular point. Four-bar, crank: degree 2, 1 and 1 finite, circularity 1; four-bar, coupler: degree 6, 3 and 3 finite, circularity 3; four-bar, rocker: degree 2, 1 and 1 finite, circularity 1; Watt six-bar, coupler: degree 6, 3 and 3 finite, circularity 3; Watt six-bar, second coupler: degree 6, 3 and 3 finite, circularity 3; Watt six-bar, output: degree 2, 1 and 1 finite, circularity 1; Stephenson six-bar, coupler: degree 6, 3 and 3 finite, circularity 3; Stephenson six-bar, arm: degree 18, 9 and 9 finite, circularity 9; Stephenson six-bar, output: degree 2, 1 and 1 finite, circularity 1; slider-crank, crank: degree 2, 1 and 1 finite, circularity 1; slider-crank, coupler: degree 4, 3 and 3 finite, circularity 1; elliptic trammel, rod: degree 2, 2 and 2 finite, circularity 0. Every body of the three machines built from pins alone has circularity exactly half its degree; the slider-crank's coupler curve and the trammel's ellipse do not.

Nine times through each circular point

A line through a circular point meets a curve at that point once for every time the curve passes through it, so counting its finite meetings measures how often, with no equation. The Stephenson six-bar's arm curve, degree eighteen, meets such lines in nine finite points: it passes nine times through each circular point. Every curve drawn by a body of a machine built from pins alone does this at exactly half its degree; a slide breaks it.

algebra · Algebra The closed path of a geared five-bar's pin at four gear ratios. A five-bar with two cranks of length 1 on pivots 3 apart, couplers 3 and 2.5, and a gear on each crank, drawn over one whole cycle at ratios 1 : 1, which closes after 1 input turn; 2 : 1, which closes after 1 input turn; 3 : 2, which closes after 2 input turns; 5 : 3, which closes after 3 input turns. Both assembly branches of the coupler pin are drawn, one in each colour. The curves they make together have degree 6, 10, 16, 26.

Every rational gear ratio has a degree

Mesh a gear on each crank of a five-bar and the pin where its couplers meet draws a closed curve at every rational ratio. Its degree is 6 at 1 to 1, 16 at 3 to 2, 68 at 13 to 8 and 178 at 34 to 21: four times the larger term of the ratio plus twice the smaller, read off the span of one polynomial in one variable. Along the golden ratio's convergents the degree grows by the golden ratio at each step, and at the golden ratio itself no fit finds any.

algebra · Algebra The same five-bar with its gears meshed outside and inside. A geared five-bar drawn over a whole cycle at four ratios, with its two cranks turning against each other in the top row and together in the bottom one — an external mesh and an internal one, which is the same machine with the ratio's sign changed. Both assembly branches are drawn. The curves have the same degree in both rows — 1 : 1 at 6, 2 : 1 at 10, 3 : 2 at 16, 5 : 3 at 26 — and they are not the same curves: the lower ones are the maximally circular members of their degree and the upper ones are not, which is a difference nothing about the drawing shows.

The mesh inside keeps the half

Mesh a geared five-bar's two gears inside each other instead of side by side and the cranks turn together rather than against each other. The curve its pin draws has exactly the same degree at every rational ratio — and it passes through each circular point half that degree, which is as often as any curve can, where the counter-rotating machine manages between a third and two fifths. The sign of the ratio is the whole difference.

algebra · Algebra Six coupler points of one slider-crank: three at other distances from the crank pin, three exactly a rod away. A slider-crank with a crank of 1 and a rod of 3 on a slide through the crank's pivot, drawn at one position with the closed curves six points of its rod trace over both assemblies. A point is given as (u, v) in the rod's own units — u along the rod from the crank pin, v across it. On the left (0.4, 0.5), (−0.3, 0.2), (1.2, −0.6): their distances from the crank pin are 0.64, 0.36, 1.34 rods. On the right (0.6, 0.8), (0, 1), (−0.8, 0.6), each exactly one rod from the crank pin, as far as the slider pin is. All six are quartics, all six are bounded — no real branch runs off the page — and nothing in the drawing tells the two panels apart. The difference is at infinity.

Where a slide puts the rest of the degree

A slider-crank's connecting rod draws a quartic that passes once through each circular point, which leaves two of its four meetings with the line at infinity unaccounted for. They are not along the slide. A point u along the rod and v across it sends them to the complex slopes [2v ± i(1 − u² − v²)] / [(1 + u)² + v²], whatever the crank, rod or offset — confirmed by slicing and by the fitted equation — and they are real only for points exactly a rod's length from the crank pin, where they merge into one direction at half the point's angle.

algebra · Algebra Four poses: four slider circles through one body point, and its four images on one line. Left, the moving body's own frame: for each of the four ways of choosing three of the four prescribed poses, the circle of body points whose three images are in a line. All four pass through one point, (1.006, 0.560), the slider point. The first two also share the body point that sits on the pole P₁₂, where two images coincide; the other two circles do not pass through it. Right, the fixed frame: the body in its four poses, and the slider point's four images, which lie on one line to 3e-16 of its length. A block on the body at that point, running in a straight guide along that line, carries the body through all four poses. There is no other.

The slide a fourth pose leaves

Three prescribed poses leave a whole circle of body points that want a slide. A fourth leaves exactly one, found where two of those circles meet a second time. It had been predicted as one or three, because a real cubic meets the line at infinity in one real point or three. But both of Burmester's cubics pass through the two circular points, so only one real point is left. Four hundred random four-pose problems each have one slide and one slot pin, both found with compasses.

synthesis · Synthesis The one real double point a four-bar owes is the one that leaves when its rocker becomes a slide. Left: a four-bar with a crank of 1 pivoted at the origin, a coupler of 3 and a rocker of 5 pivoted at (3, −5), so that the rocker's circle touches the line y = 0 where a slider-crank's slide would run. The coupler point is (0.4, 0.5) in the coupler's own units, and the curve it draws over both assemblies is shown with the circle through the three cognate pivots, on which all three finite double points lie. Two of them are a complex pair near the crank; the third, the only real one, is an isolated point at (0.92, −5.32) that no configuration reaches. Right: the same picture divided by the rocker length for rockers of 4, 10, 20, 50, 200, 100,000, with the drawn rocker's point ringed. The machine shrinks to the origin, the pivot circle settles onto a fixed circle, and the isolated point settles onto one point of it, (−0.484, −0.580), marked with a cross: the value d(|w|² − 1)/(w̄ − 1) for w = 0.4 + 0.5i and d the direction from the slide to the rocker's pivot.

The double point a slide cannot keep

A four-bar's coupler curve owes three finite double points and a slider-crank's owes two, so a four-bar whose rocker grows into a slide has to lose one. Parity decides which: the one that leaves is always real, and always an isolated point no configuration reaches. It runs off along d(|w|² − 1)/(w̄ − 1) times the rocker's length, a law that knows nothing of the crank, the rod, the offset or where the rocker touches the slide. On the circle one coupler length from the crank pin the law is zero, and all three double points leave together like the cube root of the rocker.

algebra · Algebra A slider-crank whose crank becomes a slide loses both its double points. Left: a slider-crank with a crank of 5 pivoted a crank's length before the point where its crank pin crosses the slide, a rod of 3 and the slide along y = 0, drawn with the slides' eventual crossing at the origin. The rod point (0.3, 0.2) in the rod's own units draws the curve shown, and its two double points are the dots: isolated points, at (−7.90, 2.47) and (−2.10, −2.47). Right: the same double points divided by the crank, with the crank's pivot at the origin and the crossing at (1, 0), for cranks of 5, 10, 30, 100, 300, 1000. The pair settles symmetrically onto the crosses at ±(0.595, −0.505) — the law ± i(|w| − e^(i·arg w))/√(1 − |w|²) for w = 0.3 + 0.2i — and leaves in proportion to the crank.

The trammel takes both double points

Grow a slider-crank's crank without bound and its crank pin's circle flattens into a second slide: the machine becomes the elliptic trammel, whose curves are conics with no double points at all. Both of the slider-crank's double points have to go, and parity cannot say how. They go together, symmetric through the crank's pivot, at ±i(|w| − e^(i·arg w))/√(1 − |w|²) crank lengths — a law that knows nothing of the rod or the slide's offset. Inside the one-rod circle they are two real isolated points, outside it a complex pair. And on the circle with the rod as its diameter, where the trammel draws straight lines, one of them does not leave at all: it settles on the crossing of the two slides.

algebra · Algebra Taken in one step, the limit is a family of laws. A four-bar with its crank a pivoted a crank's length to the left of the point where its two pins' circles cross, and its rocker c pivoted a rocker's length below it, both growing with c = λa, so that near that point it becomes the elliptic trammel. The coupler point (0.3, 0.2) in the rod's units has three double points, all leaving in proportion to the machine; plotted divided by a + c, with the crossing at the origin. Faint dots: the real ones at 49 ratios from a thousandth to a thousand. Large dots: at λ = 1, at (-0.110, -0.161), (-0.827, -0.176), (-0.059, -0.692), all three real. At either end one point settles where the one-step limit taken first would put it — the four-bar's law with the rocker grown first, its mirror with the crank grown first — and the other two collapse onto the crossing at this scale. In between, all three are spread.

A four-bar becomes a trammel along a family of laws

A four-bar becomes the elliptic trammel when its crank and its rocker both grow without bound, their pivots receding at right angles, and its three double points must all leave. Taken one link at a time the limit sends one point off along the four-bar's law and then the other two along the trammel's. Taken in one step it does something neither order sees: all three leave together, in proportion to the machine, at positions set by the coupler point and by λ, the rocker's length over the crank's. The two one-step orders are the family's two ends, λ → ∞ and λ → 0, and at equal rates the coupler point (0.3, 0.2) has its three double points up to 0.86 crank lengths from where either order puts them.

algebra · Algebra Every boundary touches the circle on the rod. The coupler's plane in the rod's units, crank pin at (0, 0) and rocker pin at (1, 0). Shaded: the coupler points whose three departing double points are all real at a ratio of rocker to crank of 1, the region where R = λ⁴A + λ²B + C is negative; 8.1% of a window from −1.4 to 2.4 along the rod and ±1.5 across it. Dotted: the circles one rod from each pin, the boundary at the family's two ends. Dashed: the circle with the rod as its diameter. Every boundary in the family touches that circle and never crosses it; at this ratio it touches at (0.50, ±0.50), the points where −B/2A = λ². Inside it all three are real at every ratio.

Three discs decide the real double points

A four-bar growing into the elliptic trammel loses three double points, and how many of them are real depends on the coupler point and on the ratio λ of rocker to crank. The boundary of the region where all three are real is one polynomial, R = λ⁴A + λ²B + C, quadratic in λ² and short when written in the squared distances from the two pins. Its sign is the count at every one of 23,100 cells tried. As a quadratic in λ² its discriminant is the circle with the rod as diameter times a figure of eight, so that circle is touched by every boundary and crossed by none, and a coupler point's whole history along the family is settled by three discs.

algebra · Algebra

Named alongside it

The objects these essays reach for when they reach for this one.

Double pointSlider-crankSolutions at infinityDegreeAlgebraic curveElliptic trammelGenusWitness setCoupler curveGear ratioGeared five-barAssembly branch

All concepts