Mechanical means of describing parabolas, the base, altitude,…

Only the input is driven; every other part of this model moves by physics (MuJoCo), and its ratios are measured. Run it live, change its parameters, or switch the friction off to see the output stop.

Modelled as

Brown's plan, the instrument lying on the drawing (no gravity): the parabola drawn on the paper with its dotted axis, the focus 8 cm below the directrix. The straight edge lies with its near side on the directrix; the square's stock bears against it (real contact, held up by a light spring as by the other hand) with its blade parallel with the axis, reaching 20 cm below the directrix. The hand sliding the square along the straight edge is the one driven joint. A thread (a chain of 70 links on real contact), as long as the blade's reach, is tied to a pin at the focus and to the blade's lower end and taken round the pencil, which rides along the blade's edge drawn up towards the straight edge by a light spring, so it is held where it is by nothing but the thread. A light readout from the focus, pinned at the pencil, records its distance from the focus and carries nothing.

Measured

As the square slides (about 5.5 cm either way) the pencil's distance from the focus follows its depth below the directrix with slope 1.061 (r² 0.9985): it stays as far from the focus as from the directrix, so it runs along the parabola. A point pencil would give exactly 1; for the thread wrapping this pencil (0.45 cm to the thread's middle) the generator computes 1.046 from the geometry, and the check holds the model to that within 0.03. The stock stays against the straight edge (within 0.01 mm); penetration stays under 0.5 mm.

Notes

The pencil is a 0.35 cm grooved peg so that the thread can be laid round it; the square traces the half of the curve on the stock's side only, as Brown's figure shows it; the thread along the blade shows a slight ripple of its links.

Film (MP4) · Still (PNG) · Spec (YAML) · Model (MJCF)