The thread: The viewer is in the geometry — page 3
A set cut for one eye
Build a colonnade four metres deep and cut every column so that its picture is the picture of one eighteen metres deep. The taper is forced rather than chosen — height and width both scale as the real depth over the intended one — and the match from the design eye is exact to sixteen decimal places. What gives it away is the second eye, and not by the ratio anybody would predict.
The second projectionThe evenness a curve buys
A curved screen is sold on evenness, and evenness turns out to be three quantities that disagree. On pixel pitch the curve wins from every seat; on the angle the glass is turned through it wins until three and a half metres along the sofa; on the plain distance from eye to glass — the reading the argument is usually made in — it gives up before half a metre.
Constructing a viewA picture with nothing straight in it
Every construction on this site is handed the horizon, and a photograph of a crowd, a hillside or a curved façade has no straight edge to give it. What such a picture does have is repetition — and three things of one height put the horizon exactly where the camera has it, from the picture alone. Two things do not, and three standing abreast do not either, and both refusals are the reader's ordinary situation.
Where to standThe 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.
The second projectionA projector in the viewer's eye
A projector paints a wall along its own rays, so from the projector's own position the wall's shape is invisible — exactly, on a cylinder, on a dome and on a plane alike. Move a hand's breadth away and the residual is pixels. The one place a projector can stand and ignore the shape of what it is throwing onto is the eye of the person watching.
Mirrors that are not camerasThe corner that answers every eye
Three mirrors at right angles compose into the point inversion, so a ray entering leaves antiparallel to itself whatever direction it arrived from — a picture surface with no correct viewpoint because every viewpoint is correct. Tilt one face half a degree and the worst returning ray is out by exactly one degree, twice the error; the best is out by a twelfth of it, which is why a tolerance quoted from one measured ray is a statement about that ray.
Where to standAn 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 second projectionMatching buys one seat
Feeding a curved screen its own picture surface makes the picture exact at one point and worse everywhere else than the flat picture it replaced. Forty centimetres along the sofa the two cross, and past that the matched picture is the worse of the pair — because a flat picture is mediocre everywhere and a matched one is perfect at a point and falls away from it faster.
The rectangle behind the lensA dolly zoom is a step and a zoom, and they meet at one depth
Step 1.5 m toward a subject 5 m away while shortening the lens to hold its size. Every mark moves along the line from the centre of the picture, to a ten-trillionth of a degree — outward if nearer than the subject, inward toward a limit if further, and not at all on the subject's own plane. The step and the zoom each move everything one way; the dolly zoom is where they cancel.
Mirrors that are not camerasTwo mirrors show fewer images than they make
Two mirrors at 55° generate seventy-one images of a point and an eye between them can reach six. The count the field teaches — three hundred and sixty over the angle, less one — is out by as much as sixty-six against the orbit and never by a whole image against what a viewer standing on the bisector actually sees. It is a correct rule about the eye, quoted as a rule about the mirrors.
The real instrumentFocusing moves the pivot past its best place
Focusing a fifty-millimetre lens to one metre carries its entrance pupil 2.63 millimetres forward of the camera body, and to half a metre 5.56 — which is more than the whole 5.53 that the pupil walks with field angle, so past a subject at 502 millimetres the focus decides where the pupil is. A panorama head aligned at infinity and used at a metre leaves 7.46 arcminutes along its seam; aligned at four metres it leaves 2.21, better than pivoting at the pupil at all.
Where to standThe 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 second projectionThe distance at which the eyes part
The two eyes' disagreement on a curved screen was measured at each screen's own sitting distance and reported as a null result. The sitting distance is a parameter and the chair moves — swept, the raw difference falls like the cube of it and the residual like the fourth power, and a viewer twenty-nine centimetres from a curved monitor crosses the fusion limit the null result was quoted against.
Where to standThe 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.
Light and mirrorsThe edge of a shadow is drawn on the object
The outline of a cast shadow is the image of a curve, and the curve is on the caster. It is not painted there: it slides when the lamp moves, it is not the outline the camera sees, and the two coincide only in the arrangement where no shadow is visible at all.
The second projectionA stereo picture is drawn for a level head
Every stereo pair is drawn for two eyes level with each other — a point's two images share a row and differ only across it. Tilt the head 10° in front of a television and that difference turns partly vertical, 14.46 arcminutes for anything drawn at infinity, and the two sightlines to a point stop meeting. At a desk monitor the same fifteen-arcminute limit arrives at 2.58°.
Where to standA 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.
What survivesTwo circles, one picture
A photographed circle leaves its own pose ambiguous, and not a little: two congruent circles in planes 23.6° apart draw the same conic to 1.1e-16 on normalised coefficients, both of them in front of the camera. On top of that the distance is free, so the family is two discrete poses each with one continuous parameter — and a plane one degree from either draws a conic 2.5e-4 away, which is what makes the agreement a measurement.
Where to standWhat an eye can paint
A flight of steps has eighteen faces and no eye reaches more than fifteen. Pointed at a cluster of blocks, a seating rake and a corridor with a doorway in it, the same measurement finds 8 of 21, 7 of 13 and 6 of 7 — and the plane, which offers its whole self to every eye, is the control that makes the law a law rather than a fact about stairs.
Where to standFacing the reader is not being reachable
A face turns toward the eye or it does not, and that is a dot product any reader can compute. Whether the eye’s rays actually land on it is a different question with a different answer — on a seating rake, three faces of ten that face the reader receive nothing, and they are 27 per cent of the facing area. On a corner the same test loses nothing at all, which is what makes the gap occlusion rather than arithmetic.
Surfaces that are not flatDrawn for the cylinder, shown on the cylinder
Four essays in this collection have named the arrangement where a picture's surface and a screen's surface are the same surface, and none has run it. It is exact — a tenth of a millionth of an arcminute — and the exactness is not the finding. Three separate things have to be true at once, and a curved television's own maker prints a sitting distance where the first of them is out by five degrees.
Where to standThe eye that reaches the most
A higher eye buys the faces occlusion was hiding and loses design off the far end of the object, so "the best eye" is not a question with an answer until somebody says which of the two they are paying for. On three objects the answer is as high as possible; on a corridor with a doorway in it the two quantities cross and the best height is two and a bit metres.
Where to standOne flight, two pictures
From the top of a descending flight every riser faces away, so the design lands entirely on treads. From the foot of the same flight the risers take 68 per cent of the design at a median stretch of 1.4, and the treads take the rest at a median of 2.6. Give each reader the faces the other cannot use and one staircase carries two pictures, with no face asked to hold both.
Constructing a viewThe slip that leaves no trace
A distance point put twenty-four pixels wrong moves the pavement by two and a half and leaves the reader's projective test reading exactly four thirds. The same slip on Alberti's section moves the drawing by the same amount and is caught, so the difference is not the size of the error — it is that one of them lands back on the set of correct drawings.