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

Shear — where it appears

An affine map displacing every point along one fixed direction by an amount proportional to its distance from a fixed plane. It keeps parallels, midpoints and ratios along a line exactly and destroys angles and lengths, which is what an oblique drawing system does and what absorbing a second centre of projection turns out to do.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

horizoncentre of the picture0.58° of lean, invisiblethe horizontals are unaffected

The third point put where it looks right

Three-point perspective is taught as two vanishing points plus a third placed by judgement. The third is not free: two points and the centre of the picture fix it exactly. It survives being placed by eye because in the layout a book draws it belongs thousands of pixels off the paper, where 400 px of error costs less than a degree — and in a photograph taken looking up at a tower the same 400 px costs ten.

wrong · pointcount apart by 4.9e-13 pxcorrect from 17 cm, at 160 mm wideone eye again

The second eye is a shear

A picture drawn from two eyes is a picture drawn from one, of a different room. The map that puts the second eye away holds the picture plane still point by point and carries one centre onto the other, and the member of its family that matters turns out to be affine — a shear along the line joining the eyes, growing with depth, which is the same operation an oblique drawing performs.

conventions · twocentres 00.5001020406080the second eye, carried toward overhead (degrees)how far the rays miss their own best point (m)two axes 90° apartmiss up to 1.12 m

Two grounds, and what the second one costs

The miniature convention wants its floor drawn from overhead and its figures drawn from in front, and the two optical axes it asks for are exactly ninety degrees apart. Read as a picture with two centres rather than as a picture with none, the arrangement stops being a contradiction and becomes a quantity: the rays of the composite miss their own best point by more than a metre, and the absorbed reading is a floor that leans.

conventions · twogrounds apart by 6.4e-11 pxcorrect from 17 cm, at 160 mm wideone eye again

A parallel floor under a perspective room

Draw the floor without diminution and the people on it with it, and the picture has a centre at infinity glued to a centre in the room. The same map absorbs it — but there is no shear this time, and there cannot be: bringing a point in from infinity is not something an affine map does, so the room the picture is equally a picture of has its midpoints moved as well as its angles.

conventions · mixedcentre

Named alongside it

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

centre of projectionDemonstrationFree parameterAffine mapDiminutionDrawing systemHomologyPicture planeCamera calibrationConditioningFixed pointFocal length

All concepts