Machine Kinematics
solved, then drawn
Concept

Algebraic curve — where it appears

The set of points in the plane where a polynomial in two coordinates vanishes. It is a curve given as an equation rather than as a list of positions, which is what makes it possible to demand that a mechanism be right along the whole of it rather than at sampled places.

Named by 16 essays across 3 fields — each of them below, with the objects they name alongside it.

What the machine draws, against where the polynomial vanishes. Two objects, found two ways. The thin line is the set where x^4 + 2x^2y^2 + y^4 − 1.2x^2 + 1.2y^2 is zero, walked over a grid with no mechanism involved. The marks are where the compiled machine's tracing point went, one per converged solve, over the 147 positions of its working arc. The machine's constraint set never mentions the polynomial, so evaluating it at each traced point is an independent check: the worst value over the whole arc is 3.7e-13. The arc is 1.30 radians of the driving angle and not the whole turn, and past that arc it draws something else.

A demand that is an equation

Every field of this collection is handed its demand geometrically — three positions, a sampled path, a ratio at each angle — and hands back a mechanism that is right at those places and approximately right between them. This one is handed a polynomial, and the mechanism that comes back satisfies it everywhere it moves.

computing · Compute Two routes to the same number, along one sweep. The polynomial x^4 + 2x^2y^2 + y^4 − 1.2x^2 + 1.2y^2 evaluated at the tip of the arm as the second angle goes right round, by two routes that share nothing: once by putting the tip's coordinates into the polynomial, and once by adding up 5 cosines of whole-number combinations of the two angles. The two curves are drawn on top of each other and the worst gap between them is 8.9e-15. Where the line crosses zero is where the machine may stand: 2 crossings at this α, which is how many assemblies the compiled machine has at this driving angle.

Every curve is a sum of cosines

Put the two angles of a two-link arm into any polynomial in x and y and what comes out is a constant plus a finite sum of cosines of whole-number combinations of them. Nine curves, three hundred random angle pairs each, and the two routes agree to 1.8 × 10⁻¹⁴.

computing · Compute What each curve costs, in cosines. Every polynomial in the two arm angles is a constant plus a sum of terms A cos(mα + nβ + φ) with whole-number m and n, and the number of those terms is what a machine has to build. The count is not the degree and not the monomial count: a circle costs one term, a general line two, and a lemniscate — degree four — costs five, fewer than the cubic above it, because its symmetry cancels frequency pairs the cubic keeps. Each row's expansion was checked against a direct evaluation of its own polynomial at random angles, worst disagreement 1.8e-14.

A circle costs one term

A line expands to two cosines, a circle to one, a general conic to six. A lemniscate is degree four and costs five; a general cubic is degree three and costs eight. What a curve costs is not its degree — it is how many frequency pairs its own symmetry fails to cancel.

computing · Compute Three linkages, one polynomial. Roberts's theorem says three different four-bars draw the same coupler curve. Here each one is traced, and each trace is fitted separately for the sextic that vanishes on it — on a common normalisation, or the comparison would be between three polynomials in three coordinate systems. The twenty-eight coefficients agree across all three to 5.0e-7. The three traces are drawn on top of one another and the curve is the same object each time; the test shares nothing with the construction that produced the cognates, which is why it is a test.

Three linkages, one equation

Roberts's theorem says three different four-bars draw the same coupler curve. Fitted separately for the sextic that vanishes on each trace, on a common normalisation, the twenty-eight coefficients agree across all three to 5 × 10⁻⁷ — a test of the theorem that shares nothing with the construction the cognates came from.

curves · Coupler What degree a coupler curve actually is. A four-bar's coupler point traced at 602 solved positions, then asked which polynomials vanish on those points. For each degree the design matrix's smallest singular direction is taken and its worst residual plotted. Degrees four and five leave nothing near zero; degree six drops by ten decades and degree seven buys nothing further. The textbook sentence — a coupler curve is a sextic — comes back as a measurement, and the decision is made by the gap between the smallest singular value and the next, which here is a factor of 2.8e+3 — at 15, 21, 28 and 36 monomials respectively.

The equation a four-bar satisfies

Every textbook says a coupler curve is a sextic. Traced at six hundred solved positions and fitted at degrees four through seven, the answer comes back as a measurement: nothing vanishes below six, degree six drops by ten decades, and degree seven buys nothing — with the decision made by a gap of 2.8 × 10³ rather than by a residual.

algebra · Algebra A line across the four-bar's coupler curve. The curve traced by a point on the four-bar's coupler, over every real configuration, with the machine drawn faintly at one of them. A real line crosses it at 4 marked points. Written as polynomials, the machine and the line have 8 paths to track; 6 arrive, 2 leave for infinity, and the arrivals draw 6 distinct points — 4 real and 2 complex. A random complex line gives 6 as well, which is the curve's degree.

A degree counted on a line

A curve of degree six meets a general line in six points, and that sentence is a way to measure the degree with no equation in it. Written as polynomials, a four-bar and a random complex line have eight paths to track; six arrive on every line tried, the other two run off towards the circular points, and a sum of the six stays straight to fifteen figures only when none is missing.

algebra · Algebra Every body of the Stephenson six-bar, sliced by one line. One random complex line, and a point on each body of the Stephenson six-bar required to lie on it. Every system has Bézout number 32. crank: 8 configurations drawing 2 distinct points, each 4 times, so degree 2; ternary coupler: 12 configurations drawing 6 distinct points, each 2 times, so degree 6; rocker: 8 configurations drawing 2 distinct points, each 4 times, so degree 2; arm: 18 configurations drawing 18 distinct points, each 1 time, so degree 18; output: 12 configurations drawing 2 distinct points, each 6 times, so degree 2. The configurations are what the tracker counts; the points are what the curve has.

The curve nobody eliminates

A point on the arm of the site's dwell six-bar draws a curve whose equation nobody writes down and no fit can find: at degree eighteen a fit needs a hundred and ninety coefficients, and its singular values have no gap to decide by. Sliced by a random line, the same machine has thirty-two paths to track and eighteen arrive, on every line tried, all eighteen distinct and all on one curve.

algebra · Algebra Bars against terms, over the whole catalogue. One mark per compiled machine. The bar count rises much faster than the term count, and the reason is the summing chain: term k has to have its direction carried to the k−1th vertex of the chain, one parallelogram per hop, so the carrying costs a translator for every pair of terms. Nine curves, from five bars to four hundred and thirteen, on a term count that goes from one to eighteen.

What universality is worth

The linkage exists, it is four hundred and thirteen bars, and it draws ten degrees of its curve. All three are true and only the first is in the theorem — which is the ordinary shape of a result about what exists, and the reason it was worth building one to find out.

computing · Compute 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 A rectangular hyperbola, compiled from a multiple of its equation. The machine compiled from p · (1 + x² + y²) for a rectangular hyperbola, with the translators — the parallelograms that carry a direction from where it is produced to where it is needed — in their own colour. The factor 1 + x² + y² is at least one at every real point, so every equation here vanishes on exactly the same curve. The machines do not agree: 20 bars at p, 75 bars at p · (1 + x² + y²), 144 bars at p · (1 + x² + y²)². This one solves 29 positions over an arc of 0.508 radians, and at every one of them the original polynomial reads 4.93e-14.

The price is on the equation

A line costs five bars. The same line, written as its own equation multiplied by a factor that is never zero, costs fifty — and the machine compiled from the longer equation draws the same line just as exactly. Every cost this field quotes belongs to a polynomial and not to a curve, and the cheapest equation of a given curve is a quantity nobody here has.

computing · Compute Two circles, one term. The curve r² = 1.44 together with r² = 2.56, whose squared radii sum to 4.00 — four times the square of the arm's link length. Their product equation expands to 1 term, which is what either circle costs on its own, so the second circle is free. The machine compiled from it has 17 bars against 11 for the single circle, runs over 5.200 radians against 1.560, and stays on the outer component throughout: its radius varies by 4.15e-13 over 240 solved positions. A mechanism moves continuously and the two circles are disjoint, so no assembly of it reaches both.

Two circles for the price of one

Search every multiple of a curve's equation by a polynomial of degree two and the cheapest is the curve's own equation, on four curves and by exhaustion. On the circle a second multiplier ties — and what it describes is two concentric circles, whose squared radii sum to four times the arm's link length squared, at exactly the cost of one.

computing · Compute The cheapest curve of each degree. The curve of degree 6 whose equation expands into the fewest cosines of the arm's two angles: cos 3θ = 0.3: 3 concentric circles about the pivot. It costs 1 term — cos(3α − 3β + …) — and the machine compiled from it has 27 bars. The dashed circle is the arm's reach. The dots are the machine's tracing point over 140 solved positions; it draws one component and stays on it. At every even degree the cheapest curve costs one term and at every odd degree two, whatever the degree.

The cheapest curves at each degree

Search every equation of degree up to six, not the multiples of one curve, for the fewest cosines a compiled machine has to build, and the answer does not depend on the degree at all: one term at every even degree, two at every odd one, each reached by exactly one set of frequencies. The cheap curves are concentric circles placed where Chebyshev's polynomials put them — k of them for the price of one — and, at odd degree, a line through the pivot with circles that come free. One symmetry of the arm forbids anything cheaper: its two links can be swapped without moving the tip.

computing · Compute A quintic of eighteen terms, and the nearest curve that costs a few. The quintic x^5 + 0.4x^2y − y = 0, whose equation expands into 18 cosines of the arm's two angles, inside the arm's reach (dashed). Drawn over it, the curve of the same degree with at most 4 terms that comes nearest to it inside the reach, found by trying every set of terms of that size: it keeps 0,1|1,0, 1,-2|2,-1 and is at most 2.77e-2 from the quintic and the quintic at most that from it — 1.4% of the reach. The slider steps through the budgets at which the nearest curve improves.

Four terms of eighteen draw the quintic

A quintic whose equation costs a compiled machine eighteen terms is drawn to 1.4% of the arm's reach by four of them and to 0.02% by ten, found by trying every set. The four are not its four largest. They are a line through the pivot bent by the square of the radius, the family the cheapest curves come from, and a curve close to that family is nearly free. An ellipse, a parabola, a lemniscate and a folium are not close to it, and each costs its whole equation to come within four per cent.

computing · Compute 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 Moving the pivot adds terms everywhere but one locus. The folium x³ + y³ − 1.5xy = 0, drawn in its own coordinates, and the square of pivot positions from −1 to 1 searched at steps of 0.05 (shaded). With the arm's pivot at the origin, where the catalogue puts it, the equation expands into 11 cosines of the arm's two angles. Moved anywhere in the square off the line x + y = 0 (heavy), it expands into 12; at (0.5, 0.3), for instance, 12. 41 of the 1681 pivots tried cost less than 12, all on that locus, and none costs less than the catalogue's 11.

Moving the pivot only adds terms

A compiled machine's cost is the number of cosines its curve's equation expands into, and moving the arm's pivot gives the same curve a different equation. So an expensive curve might only be expensive because of where the arm was put. Searched over 1,681 pivots, it never is. Moving the pivot adds the equation's own derivatives at the pivot as new terms, and they cancel only on one locus per curve — a centre, or a line — on which the catalogue's pivot already sits. Anywhere else the ellipse costs 6 instead of 4, the folium 12 instead of 11, the lemniscate 9 instead of 5, the parabola its 6 everywhere, and the quintic that four terms nearly draw costs 30 instead of 18.

computing · Compute 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.

Implicit equationDegreeFrequency pairCost modelCoupler curveTrigonometric polynomialTwo-link armSparsityCircular pointsDouble pointSingular valueSymmetry

All concepts