The fields
A field says what an essay is about. The order below is not alphabetical and is not a ranking: it is the order the argument builds, and each field assumes the ones above it. A projection destroys almost everything and keeps one quantity, so that comes first; a construction can then be checked against a projection rather than admired; and once the projection is exact it has a centre, which is the point the picture is correct from and the thing this site is named after.
Two of the fields are where the premise stops holding. Everything above Through water and glass assumes a picture is a projection through a centre, and a refracted picture is not — its rays, continued into the water, miss each other by millimetres rather than meeting at a point. A real lens fails the assumption a second way. Both are built as measured differences from the pinhole, with the dry number computed beside every wet one, which is why neither could have been written earlier: the whole content of both is a departure, and there has to be something exact to depart from.
Three of them raise the count. One picture fixes a ray, two fix a point, and many fix the track the camera moved along — up to seven numbers that no quantity of pictures supplies.
And the last three are the ones that are not perspective at all. A handscroll, a removed roof, a tilted ground plane, a composite of aspects, a construction whose lines spread with depth: systems that gave up the single station point and bought something exact with it. They are measured on what they preserve rather than on how far they fall short of a pinhole, which is the same method the parallel systems get and is the only one that says anything. The last field ends by turning the battery on perspective and pricing what it gave up in return.
Threads cut the other way: they say what an essay rhymes with, across fields.
What survives
A projection destroys length, angle, area and the ratio of lengths. One quantity comes through untouched, and almost everything checkable about a picture is checked with it.
Constructing a view
Horizon, vanishing points, measuring points. The classical construction drawn alongside the projection it is supposed to produce, so it can be checked rather than trusted.
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.
Surfaces that are not flat
A cylinder, a sphere, a fisheye. Every one of them is computed here rather than described, and every one is measured on the three things a picture surface can do to the world: bend its straight lines, turn its right angles, and change its scale. No surface escapes all three.
The other systems
Isometric, oblique, elevation. Not perspective done badly by people who had not worked it out yet, but a different answer to a different question — and the difference is measurable.
Light and mirrors
A shadow is a projection from the lamp; a reflection is the view from a camera on the far side of the mirror. Neither needs new machinery, and both are computed with the machinery already here.
Measuring from one picture
A photograph read backwards for the scene that made it: a height from a cross-ratio, a façade flattened by a homography, a plan of the ground. Every measurement is a ratio, because a single view has no size — and that is shown rather than said.
Through water and glass
A refracted picture is not a projection at all — its rays, continued into the water, miss each other by millimetres rather than meeting at a point. Everything the rest of this site rests on is measured here against the case where it fails, from the cross-ratio to Snell's window to a dome port that turns out to be an exact pinhole.
The real instrument
A lens is a departure from the pinhole, and the departure is the largest systematic error in every measurement made here. It bends straight lines, destroys the invariant, and can be recovered from nothing but the knowledge that some edges were straight — which is the round trip again, on a harder problem.
The second eye
One picture fixes a ray; two fix a point. Everything a pair of pictures determines about the eyes that made them — the image of one eye in the other's picture, the line a match must lie on, and the camera pose recovered from correspondences alone — with the recovery never shown a camera.
What a pair is for
Depth from two views is a reciprocal, so a fixed error in what is read maps to an interval that is not symmetric and eventually is not bounded. What a stereo pair can and cannot measure, computed rather than described, including the range past which one pixel reaches infinity.
Many pictures at once
Raise the count and the answer changes character. A sequence determines its own camera track and the scene together — up to seven numbers that no quantity of pictures supplies, counted here in the Jacobian and then walked along to show that they are free rather than merely small.
The eye that moves
A handscroll is drawn by an eye that travels, imaging one line at a time — orthographic along the roll and perspective across it, so a mile of river holds its scale while a single pavilion still recedes. Its rays miss their own best centre by metres, and the miss is exactly the length of track you unroll.
Systems that kept the measure
A removed roof, a ground plane tilted into the picture, a figure assembled from the aspect that identifies each part. Every one of them gives up the single station point and buys something exact with it — a floor a reader can measure with one ruler, a depth scale that does not bunch, a picture in which nothing is edge-on.
What each system gave up
The whole comparison on one battery: does it have a centre, does it keep true measure, does size fall with distance, is its depth range bounded, does it keep straight lines straight. No system answers yes to the first two at once, and the field ends by turning the table on perspective and pricing what it gave up to get its station point.
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.