Machine Kinematics
solved, then drawn
Concept

Translator — where it appears

A parallelogram that carries a direction from one point to another. It works only between two points a fixed distance apart, which is why a compiled machine spends most of itself moving angles around rather than computing them.

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

How much of a compiled machine is computing anything. Each machine's bars split two ways: the ones that build an angle — reflectors, means, rigid offsets, the arm — and the ones that carry a direction from where it was computed to where it is needed. On the smallest machines the arithmetic is nearly all of it. By the quintic the carrying is 80 per cent, and it goes on rising, because the arithmetic grows with the number of terms and the carrying grows with the number of pairs of them. That is the answer to why a universality construction is enormous, and it is not about the algebra being hard.

A parallelogram carries an angle, and only so far

A direction computed at the frame is no use where it is needed. A parallelogram carries one from point to point — but only between two points a fixed distance apart, and that single proviso is what makes a compiled machine quadratic in the number of terms and turns most of it into transport.

computing · Compute Every curve in the catalogue, compiled and counted. The bar count is not a formula: it is the number of bar constraints the compiled mechanism actually carries, asked of the object rather than predicted. A line costs five bars and a quintic four hundred and thirteen, which is the field's central number — exactness is paid for in size. The all-revolute column replaces the one prismatic pair that closes the chain with a Peaucellier cell, which costs seven bars whatever the curve, so it is more than half the machine on a line and a rounding error on a quintic. The last column is the polynomial evaluated at the traced point, worst over each machine's working arc.

Five bars for a line, four hundred for a quintic

Nine curves compiled and counted: a line at five bars, a circle at eleven, a lemniscate at fifty, a general quintic at four hundred and thirteen. The growth is a fourth power of the degree, and three quarters of the largest machine is not computing anything at all.

computing · Compute

Named alongside it

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

Summing chainAssembly branchCompiled linkageCost modelDegreeLinkageOverconstraintParallelogram linkageRigid bodyUniversalityWorking arc

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