Homology — where it appears
Named by 14 essays across 5 fields — each of them below, with the objects they name alongside it.
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.
The curvature a shadow reports
A flat floor makes a shadow a homology, so four marks predict the rest exactly and a curved floor mispredicts them by millimetres. Turn that round and the mispredict is a measurement of the floor. It recovers a dished floor's own curvature to a per cent — and returns 0.482 for a floor made of two planes, whose curvature is zero everywhere, with a residual of ten microns.
Measured down from the waterline
Whatever stands so far above the water, draw its reflection the same distance below. Through a vertical picture plane that is not an approximation — it is the reflection, to the arithmetic floor. Tilt the camera twenty-two degrees and it is eleven pixels out. Draw a gull, which touches the water nowhere, and guessing its waterline point wrong by two metres of depth costs fifty.
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.
A mirror that is not parallel to the wall
Carry the depth in front of the glass an equal depth behind it, square to the wall. That is exact for a mirror hung parallel to the wall and 1.26 metres — 107 pixels — out for one turned 20°. Two invariants survive the turn instead, and one of the two nearly did not survive being tested, because it had been written in a form that could not fail.
The room the eye may stand in
An anamorph is correct from one point, and one point is not a thing a person can occupy. Fix a tolerance on the picture and the set of eye positions that meet it is a solid — for a design 1.8 m wide and a ten-millimetre tolerance it is 36 mm long, 14 mm across and 31 cubic centimetres altogether, a spindle pointing along the line of sight. Ten times the tolerance is a thousand times the room.
An anamorph has one eye
From the design point exactly — a camera's single eye — the floor marks give the design back to sixteen decimal places. A head carries two eyes 63 mm apart, and neither of them is the design point. The difference between the disparity the floor gives and the disparity an upright board would give runs to 47 arcminutes, against a stereoacuity of a few tens of arcseconds. This is why pavement paintings are photographed.
The design that outruns the floor
A pavement anamorph of a design forty per cent of eye height needs two metres of floor. Eighty per cent needs thirteen. Ninety-four per cent needs fifty, ninety-nine per cent needs three hundred and seventeen, and the sky needs an infinite one — because the ray through a design point level with the eye never comes down. Depth times the height still to go, divided by the height already reached, is the eye's own distance at every point of the family.
The anamorph that crosses a corner
Cast one design onto a floor and the wall at the end of it, from one eye. Each plane gets a collineation of its own; the two agree on the line where the planes meet, exactly, because a point of that line is a point of both. What they do not agree about is scale — the design runs 7.8 times its own size along the floor and 2.0 times up the wall, and the jump at the join is a factor of 3.9.
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.
A stair does not use all its faces
A corner anamorph is two homologies and a flight of nine steps is eighteen, which is arithmetic and is the least of it. What a flight has that a corner does not is that which faces exist and which faces can be painted are different questions. From the top of a descending flight not one riser is reachable at any eye height, so half the planes are unpaintable by construction — and from the bottom of an ascending one, 58 per cent of the picture lands on risers that are 36 per cent of the surface.
The residual has a shape
A flat-floor map mispredicts a shadow by millimetres on any floor that is not flat, and the number everybody quotes is the worst one. Tune a dish, a ridge and a step until all three mispredict by exactly 25.0 millimetres and the scalar can no longer tell them apart — by construction. The signed residual around the ring still can. The second harmonic of it reads 0.09%, 2.79% and 19.37%, a factor of two hundred across three floors the headline number calls identical.
The floor is a choice of coordinates
Four rungs of this field have measured what a curved floor costs a shadow reading, in millimetres. It costs nothing. A shadow mark sits on the ray from the lamp through the occluder's edge, and where along that ray the floor caught it is a fact about the floor alone — so un-casting in rays returns the occluder to three parts in ten thousand of a millionth of a millimetre on a plane, a dish, a ridge and a step alike, while the same shadow read in the plan is wrong by up to 216 millimetres.
The lamp and the floor cannot both be recovered
Every member of a one-parameter family of lamp-and-floor pairs draws the identical photograph to a fraction of a thousandth of a pixel — one member swings the lamp 90 cm and tilts the floor 4.3° and the picture does not move at all. A tape measure or a post of known height each pin the true member uniquely, and so does a wall known in advance to be square to the floor, whose angle to the recovered floor peaks at exactly 90° at μ = 1.000 and nowhere else; a receiver merely known to be parallel drifts thirteen times more weakly.
Named alongside it
The objects these essays reach for when they reach for this one.
AnamorphosisGround planePicture planeResidualConditioningDevelopable surfaceReceiving surfaceShadow projectioncentre of projectionDemonstrationGaussian curvatureleast squares