Machine Kinematics
solved, then drawn
Concept

Two-link arm — where it appears

Two equal links from one fixed pivot, whose tip reaches every point of a disc of twice their length. It is what a compiled machine is built on: the tip's coordinates are two-term Laurent polynomials in the links' angles, so any polynomial in them is a finite sum of cosines.

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

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 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

Named alongside it

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

Algebraic curveFrequency pairTrigonometric polynomialConjugate pairCost modelDegreeExact arithmeticImplicit equationLaurent polynomialSymmetryWorking arc

All concepts