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

Developable surface — where it appears

A curved surface that can be laid flat with nothing stretched, which is exactly a surface whose Gaussian curvature is zero everywhere. A cylinder and a cone are; a sphere and a saddle are not, and the difference decides whether a plate can be cut flat and bent or has to be pressed to shape.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

the eye the design is forthe cameracorrect from 14 cm, at 160 mm widerecovered to 2.5e-14 m

The wall under the paint

An anamorph's design is a set of rays from a known eye, fixed before any paint is applied — so the marks are a calibration target whose rays are known exactly and whose shape is not. One ordinary photograph supplies the second ray for each mark and the wall comes back at the arithmetic floor, on a dished floor, a ridged one, and a floor with a step in it. What it says between the marks is nothing, by the sagitta.

metrology · wallrecovery 020406011.5022.503candidate height for the eye, in metreswhat the best rectangle leaves over, in millimetresa flat floor: every height fitsthe eye that made the markscurvature costs the collineationand buys the parameter back

The height a flat floor cannot give

The marks of a floor anamorph name the eye's position on the floor exactly and say nothing about how high it was — every candidate height explains them perfectly, to one part in a thousand trillion. That is a fact about planes rather than about anamorphs. Ripple the floor by six centimetres and the family collapses: the true height explains the marks exactly and the nearest wrong one, five centimetres away, leaves two millimetres on a design 1.8 metres wide.

viewing · anamorphrecovery a flat plate · best view0a bent plate · best view0.1224a dished plate · best view0.2298a bent plate · along its generators0patches wrapping 29°dished / bent = 1.878

No view draws a curved plate true

The auxiliary view is descriptive geometry's answer to a foreshortened plane — turn until the plane is parallel to the paper and it draws at true shape. A bent plate has no such direction and a dished one has none twice over: the best view of the first is out by 1 − cos w and the best of the second by 1 − cos²w, worse by exactly 1 + cos w, because its normals need two parameters rather than one.

parallel · platepatch 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.

DemonstrationGaussian curvatureAnamorphosisDevelopmentDrawing systemOrthographicTrue scaleAmbiguityArea scaleAuxiliary viewAxis scaleCollineation

All concepts