The thread: Projected twice
A focal length is not an angle
Fifty millimetres means nothing until a rectangle is named behind it. The same lens is 39.6° across full frame, 26.6° across APS-C and 8.7° across a phone sensor — and the distance the resulting print is correct from depends on the ratio of the two, so two cameras matched on angle agree exactly whatever their formats.
The second projectionThe screen sets the distance
Every viewing distance quoted for a picture on a page is conditional on an assumed figure width. Replace the assumption with an actual chain — focal length, sensor width, display width — and the same 50 mm frame is correct from 9 cm on a phone, 83 cm on a monitor and 16.7 m in a cinema. Nobody is standing at any of them.
The second projectionA wide field on a small screen
A picture rendered at a hundred degrees and shown on a screen that subtends forty-nine is being read from two and a half times its own station distance, so the depicted space is two and a half times too deep. The stretch at the edges everybody complains about is correct; the complaint is really that nobody is sitting where it would be invisible.
The second projectionTwo pictures on one screen
A stereoscopic display puts a point where two sightlines cross, so the depicted depth is b·D/(b−d) and the disparity that reaches infinity is exactly the separation of the reader's eyes — 63 mm, at any screen distance whatever. The depth budget is set by the width of a head and by nothing about the scene.
The second projectionThe render is distorted on purpose
A headset renders a bent picture so its lens can straighten it, which is a lens's distortion polynomial run backwards, and the one case in which distortion is introduced deliberately. The round trip closes to a thousandth of a millionth of a pixel, and the price is that one rendered pixel becomes 0.493 delivered pixels at the edge of the field and one at the centre.
The second projectionTurning the cameras inwards
A stereo pair made by rotating two cameras toward a common point puts the same world point at different heights in the two pictures — up to thirty pixels here, on a frame of four hundred. Two eyes level with each other see every point at the same height, so a pair with vertical difference is a pair of pictures of no scene at all.
Light and mirrorsA shadow can be un-cast
A shadow looks like a lossy record — a flattened smear with the shape half thrown away. It is nothing of the kind. The map from the occluder's plane to the floor is a plane projectivity, so it has an inverse, and the outline comes back out of its own shadow exactly. What breaks it is not the light and not the shape: it is the floor not being flat.
Light and mirrorsTwo lamps and one map
A flat object lit by two lamps casts two shadows, and one is the other scaled about a point — ratio 1.2509 here, carrying every point of the first outline onto the second to 1e-15 m. No rotation and no shear is available to it, because a projection between two parallel planes has its axis at infinity. And the ratio is exactly 1 when the two lamps are at the same height, which makes a pair of shadows a measurement of the lamps.
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.
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.
Light and mirrorsThe 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.
Where to standA design that lands in two rooms
Cast a design down a corridor with a doorway in it and four per cent of the picture goes through the opening and lands on a wall three metres beyond, at 9.4 metres from the eye against the end wall's 6.4. The picture is continuous across the edge of the doorway and its scale jumps by half again, which is a corner anamorph's discontinuity with a gap in the middle of it instead of a fold.
Surfaces that are not flatShot on one surface, shown on another
Two picture surfaces are two charts of the same pencil of rays, so a reprojection between them is a change of coordinates and loses no geometry at all — bit-exact at all 408 sampled directions. What it costs lies elsewhere — 70 per cent of the source has nowhere to go, and the target wants ×5.49 the marks at its edge.
Surfaces that are not flatThe eye is a picture surface too
A retinal sphere behind an off-centre nodal point takes every measurement this collection puts to a lens or a screen, and answers all of them. It is the equal-area fisheye to within 110 micrometres of retina rather than the equidistant one everybody draws it as, and a flat picture at its own correct distance leaves the identical arc on it, to 1.4e-14 degrees.