Every essay — page 7
What survives Constructing a view Where to stand Surfaces that are not flat The other systems Light and mirrors Mirrors that are not cameras Measuring from one picture Through water and glass The real instrument The second eye What a pair is for Many pictures at once The eye that moves Systems that kept the measure What each system gave up What a machine computes The rectangle behind the lens The second projection Drawn confidently
Where to stand
A perspective picture is correct from exactly one point in the room. Every figure here computes where that point is, which turns the oldest complaint about wide-angle photographs into arithmetic.
The cylindrical mirror unbends it
Put a mirrored cylinder in the middle of the sheet and the light path from the eye to the paper bends once. The map that results is not a homography and not a projection in the plane sense at all — it varies its scale by a factor of seven across the design, which is why the smear is unreadable and why the mirror can put it back.
Standing in the wrong place
A picture read from twice the distance it is correct from depicts a scene twice as deep — and not one mark on the paper moves. The error is invisible in the picture, which is why it survives everywhere.
A floor anamorph is three numbers
An anamorph is described everywhere as a picture stretched until only one viewpoint can read it, which says what it looks like and nothing about what it is. Cast one onto a floor and fit a map to the marks, and the map turns out to be a planar homology — a line of fixed points, one point off it, and a single ratio. Those three numbers are not a description of the eye. They are the eye.
Where the anamorph still works
A picture correct from one point raises an obvious question that nobody answers with a number — how far may the eye move. The answer here is exact rather than approximate: a wrong viewpoint composes the intended picture with a central collineation whose axis is the line the picture stands on, so the error is zero along that line and grows linearly upward, and a step sideways costs precisely as much as the same step upward.
The marks name the place, not the height
Run the site's own round trip on an anamorph — hand the machinery the floor marks and ask for the eye back — and it returns the spot on the floor to eleven decimal places with nothing assumed at all. It does not return the height. What the marks fix is the product of the eye's height with the design's aspect ratio, and no amount of looking at the floor separates the two.
The ceiling that is not a plane
Paint the same design for the same eye onto a floor and onto a barrel vault, then fit the best possible homography to each set of marks. On the floor it misses by femtometres, because the map is a collineation and four marks determine every other. On the vault it misses by half a metre, and no choice of four marks helps — which is where the projective machinery this site has used since its first phase stops applying.
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.
Drawn confidently
The constructions taught in every book on the subject, measured. Some are fine. The point is that nobody knows which until the drawing is asked what solid it depicts.
The cube that is a box
The two-point cube every book teaches has a step it supplies no construction for. Place the two far edges symmetrically and the drawing depicts a cube for free; place them eight points apart — a difference invisible on the page — and it depicts a box 1.4 times shallower than it is wide.
Dividing depth by eye
Three methods for spacing a receding row, all of them taught, all of them wrong. Read back as distances, the best misplaces a post by 3.5 metres in a row that is supposed to be spaced 1.4. And the obvious way of checking them gives one of the three a perfect score.
The sixty-degree cone of vision
Every book says keep the subject inside a 60° cone. Measured, the marginal stretch the rule is nominally about is exactly zero from the station point — 1.000000000000 to 1, over 720 sampled points. The rule is a statement about the reader, and books do not obey it.