Machine Kinematics
solved, then drawn
Concept

Residual — where it appears

How far a candidate configuration is from satisfying the equations it is meant to satisfy, reported as a number rather than as a verdict. It is the quantity that separates a mechanism from a drawing on this site: a closure at 1e-15 is solved and one at 1e-3 is not, and the comparison between the two is what a claim of exactness rests on.

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

What a pattern that cannot fold leaves behind. The best a least-squares solve can do with the vertex closures, against the fold it is asked for. The Miura pattern closes at every angle, at the arithmetic's own floor — the line along the bottom is 10⁻¹⁵ and below. The same grid with its vertices moved by a tenth of a panel does not close at any angle at all: its residual starts at 5.9e-6 at the smallest fold and grows with it, and no seed and no number of iterations moves it. Both patterns have the same panels, the same creases, the same graph and the same developable vertices, and every one of those vertices folds perfectly well on its own.

Each one moves, and together they do not

Take the pattern a Miura sheet folds along and move every interior vertex by a tenth of a panel. Every vertex still folds on its own — each is a spherical four-bar with a freedom of its own — and the four of them together fold to no angle at all, with a residual that starts at six millionths and never falls.

networks · Network Which of these two tracks was made by the front wheel. The rear wheel of a bicycle is towed, so it points at the front wheel at every instant: the tangent to the rear track, extended forward by the wheelbase, lands on the front track. Done that way round the tangents land a mean of 0.13 mm off; done the other way round they land 461 mm off, a factor of 3625. It is a measurement rather than an eye for tracks, and it needs neither the wheelbase nor the direction of travel to be known in advance.

Which way did the bicycle go

Two tyre tracks in mud, and a question with a definite answer. The rear wheel is towed, so its tangent extended forward by the wheelbase must land on the front wheel's track — and it does, to 0.13 mm one way round and 447 mm the other. The test needs neither the wheelbase nor the direction of travel, and it returns both.

rolling · Rolling

Named alongside it

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

CompatibilityCrease patternDevelopabilityIdentificationInverse problemMisconceptionMobilityNetworkOff-trackingRedundant constraintRigid origamiRolling constraint

All concepts