Machine Kinematics
solved, then drawn
Concept

Compiled machine — where it appears

A linkage built mechanically from a polynomial, whose tracing point stays on the curve the polynomial vanishes on. It is assembled from arm, multiplier, adder and transport gadgets, one set per term of the equation's expansion, and its size is a price the equation sets rather than the curve.

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

The machine compiled from a rectangular hyperbola. xy − 0.5, compiled: 20 bars and 20 joints, painted by what each part is for. The two-link arm at the pivot carries the tracing point; the reflectors and means build each term's angle; the rigid offsets fix the constant φₖ and the amplitude; the translators carry those directions out along the summing chain, whose last vertex is held on a line. That last constraint is the equation. Every joint drawn is the output of a Newton–Raphson solve on 36 equations, converged to 4.2e-16, and the polynomial at the tracing point is 2.2e-16.

A compiled machine and its own scale

A linkage compiled from a polynomial has bars whose lengths are the polynomial's coefficients and joints whose angles are its phases. Scale it and every coefficient scales — so the machine computes the same polynomial multiplied by a constant, which is a different polynomial with the same roots.

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

Named alongside it

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

Implicit equationAlgebraic curveCost modelFrequency pairSparsityTrigonometric polynomialTwo-link armApproximationCoefficientDegreeGadgetIdentifiable

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