Visual Plane
every picture states the point it is correct from
Concept

Isometry — where it appears

A map between surfaces preserving every distance measured along them, which exists between a plane and another surface only when that surface is developable. That is the exact condition for a design to be printable flat and applied to a surface without stretching.

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

111 mm outcorrect from 21 cm, at 160 mm wide1e-12 mm with the floor · 111 mm without

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.

curved · curvedrectify the floor, in placeunrolled — 2.611 m of surfacea ridged floor, k = 0.062.6 m of plan is 2.611 m of floor

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.

curved · developable 00.2500.5000.7501-1-0.50000.5001across the plate, from one edge to the otherdrawn length of a unit of surfacealong the generatorsround the benda plate bent through 57°one ruler in one direction

The drawing and the development

A bent plate gets two flat pictures on the same sheet and each is exact in what the other loses. The parallel drawing keeps the generators at one scale and stretches the arc over a factor; the development keeps every length on the surface and keeps nothing of the shape in space. Neither is the plate and the pair of them is.

parallel · platepatch

Named alongside it

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

Gaussian curvatureArc lengthDevelopablePicture planeRectificationAnamorphosisArea scaleAxis scaleConeConformalCylinderDemonstration

All concepts