Central collineation — where it appears
Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.
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.
What a flat map leaves alone
A projectivity of the plane is eight numbers in a matrix, and reading them tells a reader nothing. What it does is decided by its fixed points, and there are exactly three cases: three isolated fixed points, or a whole line of them with one point off it, or a whole line of them with the point fallen onto it. The middle case has five numbers instead of eight and every point slides along a line, and it is what most of the maps this site builds turn out to be.
Three constructions, one map
A shadow cast on a floor, an anamorph painted on one, and a reflection in a mirror were built in three different fields of this site, three phases apart, each with its own derivation and its own figure. Decomposed into their fixed points and lines, three of them are the same kind of map — a central collineation with a line of fixed points — and the fourth, a rectification, is not. That difference separates changing where a picture is seen from, from changing one picture into another.
Named alongside it
The objects these essays reach for when they reach for this one.
Characteristic ratioCollineationDemonstrationFixed pointPlanar homologyProjective mapAnamorphosisElationHomographyGround lineStation pointViewing distance