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

Developable — where it appears

The property of a surface that can be laid flat without stretching, which holds exactly when its Gaussian curvature is zero everywhere. It is the property that decides whether an anamorph's design can be printed on a flat sheet and rolled on, which is a practical question with an exact answer.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

eye, on the axisa cone 29 cm across and 22 cm highthe design lies from 0.17 m to 0.47 m out

The cone that reads the floor

A conical mirror standing on a design turns it into a picture, and the map it performs is exact, one-to-one, and not a projectivity — a homography fitted to four of the marks returns those four to 7e-13 mm and puts the rest 2480 mm away, on a design 369 mm wide. The reason is that the cone turns the annulus inside out: the middle of the picture comes from the far edge of the floor and the rim from the near one.

mirrors · Conemirror the seatcentre of curvatureR = 1000 mm · seat at 650 mm51.4 px off a homography

The screen is a picture surface too

A curved television is one of the six named picture surfaces sitting in a living room, and from its own axis it delivers azimuth in proportion to the picture, exactly. What it is shown is a rectilinear picture from a sofa, and the difference is not a matter of degree — a flat screen from any seat shows a homography of the intended picture, so it is a correct picture of a transformed scene, and a curved one shows a map that is not a homography from any seat at all.

screen · Curvedscreen 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 025507510010203040how wrong the assumed curvature is, as a % of the true oneworst error in the recovered design (mm)0.91% buys 1 mma a ridged floor, k = 0.0644.0 mm at 41% out

How well the floor has to be known

“Supply the surface” is an instruction with no number in it, and an instruction with no number is a wish. Here is the number: the error in the recovered design is very nearly proportional to the error in the assumed curvature — the constant varies by 1.4% across a twentyfold range — so about nine parts in a thousand of the curvature buys one millimetre on a design 1800 mm wide, and 40% out costs 43.5 mm.

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

Named alongside it

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

HomographyGaussian curvatureRay tracingRectificationAnamorphosisArc lengthArea scaleConeInvertibilityIsometryleast squaresPicture plane

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