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 5 essays across one field — 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

Named alongside it

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

DegreeDouble pointWitness setGear ratioGeared five-barGenusSlider-crankSolutions at infinityAlgebraic curveAssembly branchBezoutComplex solution

All concepts