Machine Kinematics
solved, then drawn
Concept

Reflection — where it appears

The operation taking an angle to twice a mirror direction minus itself. Four bars perform it exactly, and from it come negation, doubling and addition — the whole of the angle arithmetic a linkage can do.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

the reflector, solved. A rhombus whose far vertex is held on a line through the pivot. One side is the input, the line is the mirror, and the other side comes out reflected in it — which is where negation, doubling and addition all come from. The relation it satisfies is (μ, θ) ↦ 2μ − θ, and across a sweep of 41 positions the worst departure from it is 1.3e-13 radians. Every joint here is the output of a Newton–Raphson solve on the bar lengths; nothing is placed by the formula the picture is about.

Four bars that add two angles

A rhombus on two links from one pivot points along the bisector of their angle, exactly, because a rhombus has equal sides. Hold its far vertex on a line and it reflects instead. From those two facts come negation, doubling and addition — and every whole-number combination of two angles a compiled machine needs.

computing · Compute What it costs to multiply an angle. Multiplying an angle by n is done by doubling and adding — 2k is one reflector applied to k, and 2k+1 is one addition of k's result and the original — so the cost follows the binary expansion of n and not n itself. Eight costs three doublings; seven costs six gadgets, twice as much, for a smaller number. The other line is what a chain of n−1 additions would cost. The difference matters because the highest multiple a curve of degree d needs is d, so the arithmetic in a compiled machine grows like d log d while the carrying grows like d⁴ — which is why the arithmetic is not what makes these machines large.

Doubling is cheaper than adding

Multiplying an angle by eight costs three gadgets and multiplying it by seven costs six. The cost of an integer multiple follows the binary expansion of the integer and not its size — which is why the arithmetic in a compiled machine grows like d log d while everything else grows like the fourth power.

computing · Compute Crank angle θ and crank angle −θ, joined. The coupler curve alone, with 12 chords, each joining the point the tracing point reaches at a crank angle θ between 0° and 180° to the point it reaches at −θ on the same assembly. Every chord is square to the dashed axis, to 7.5 × 10⁻¹⁶ in the cosine, and every midpoint lies on it. Measured at 48 pairs, the reflection of one point misses the other by at most 4.2 × 10⁻¹⁵. The crank's own angle is the pairing: the two places the curve crosses its axis are crank angles 0° and 180°, the only angles equal to their own negatives.

A symmetric curve from a lopsided machine

A four-bar with a crank of 1, a ground of 3 and a rocker of 2.5 has no symmetry anywhere in it. Make the rocker, the coupler and the arm from the rocker pin to the tracing point one length, and the curve it draws is its own mirror image to 4 × 10⁻¹⁵, about a line through the rocker pivot turned from the ground line by exactly half the coupler's angle at that pin.

curves · Coupler

Named alongside it

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

Assembly branchExact mechanismRhombusAngle bisectorAngle multiplierBinary expansionClosed formCoupler curveCrank-rockerDwellFrequency pairInscribed angle

All concepts