Planar homology — where it appears
Named by 14 essays across 4 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.
The marks name the place, not the height
Run the camera-recovery round trip on an anamorph — hand it 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 floor that is not a plane
A shadow on a flat floor is a homology, so four marks determine the whole map and the rest of the outline comes back exactly. Dish the floor and the same four marks mispredict the rest by 5.67 mm; ridge it and 9.07 mm; put a step in it — two planes, each of them exactly a homology — and 74.95 mm. The receiver's shape is what breaks the projective description, and it breaks it worst where the surface is flattest.
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 every projective construction made for a floor stops applying.
A shadow across an edge
A straight rod's shadow crossing the crease between floor and wall is two straight pieces, each dead straight to 1e-15 m, meeting at 35.08°. The corner is a fact about the room and not about the rod. Fit the floor's map from four marks and apply it across the whole shadow and the part on the wall comes back up to 78.9 cm from the object — the wrong map, applied confidently.
The bay repeated by a straightedge
Draw one bay, then repeat it by diagonals alone — no measurement, no scale, no arithmetic — and after twelve bays the constructed corners sit 1e-12 px from the corners the camera projects. It is exact because the operation being iterated is a homology of the picture, not because the draughtsman was careful, and that separates it from every construction in this site's `wrong` field.
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.
Two 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.
Three constructions, one map
A shadow cast on a floor, an anamorph painted on one, and a reflection in a mirror are usually treated as three different subjects, 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.
What 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.
A 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.
One 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.
The lamp and the floor cannot both be recovered
Every member of a one-parameter family of lamp-and-floor pairs draws the identical photograph to a fraction of a thousandth of a pixel — one member swings the lamp 90 cm and tilts the floor 4.3° and the picture does not move at all. A tape measure or a post of known height each pin the true member uniquely, and so does a wall known in advance to be square to the floor, whose angle to the recovered floor peaks at exactly 90° at μ = 1.000 and nowhere else; a receiver merely known to be parallel drifts thirteen times more weakly.
Named alongside it
The objects these essays reach for when they reach for this one.
Projective mapDemonstrationAnamorphosisCentral collineationReceiving surfaceCharacteristic ratioHomographyCollineationFixed pointShadow projectionViewing positionPicture surface