Arc length — where it appears
Named by 2 essays across one field — each of them below, with the objects they name alongside it.
Also named here as isometry — the same set of essays touches all of them, so they are one junction rather than several.
Undoing a picture made on a curve
Three rounds of work here have measured what a curved receiving surface costs going forward — 5.67 mm on a dish, 9.07 mm on a ridge, 529.4 mm on a vault. None of them asked whether the design can be got back. It can, exactly, and the price is stated precisely: you have to supply the surface. Told the floor, the recovery returns a design to 1.1e-12 mm; told nothing, and fitting the four marks every rectification tool fits, it is 111 mm out.
The floors that unroll
A ridged floor curves visibly and can be laid flat without stretching anything — 7.4e-9 of strain across the patch. A dished floor curves less and cannot be laid flat by any means whatever. The difference is one number, Gaussian curvature, and it is the number Gauss proved no bending can change: 0 for the ridge, 0.0144 per square metre for the dish, and no cleverness in the flattening touches it.
Named alongside it
The objects these essays reach for when they reach for this one.
DevelopableGaussian curvatureIsometryPicture planeRectificationAnamorphosisArea scaleConeConformalCylinderHomographyInvertibility