The anamorph that crosses a corner

A design cast onto a floor runs away as its top approaches eye level: the design that outruns the floor puts the numbers on it, and they are severe. The obvious repair is to stop the floor and put a wall there.

That is not a patch. It is a different object, and it is worth working out what kind.

Two maps, one design #

Each plane on its own is the case this collection already understands. A design cast from an eye onto a plane is a planar homology — a line of fixed points, one centre off it, one ratio — and those three numbers are the eye, which is what a floor anamorph is three numbers establishes.

So the corner anamorph is two homologies. The design is divided by one line: the design points whose rays reach the floor before the wall, and the ones that do not.

Which line? The one whose ray passes through the join. With the eye 1.65 metres up and 3.2 metres back and the wall 3.2 metres out, that is the design point at 0.825 metres — exactly half the eye’s height, which is not a coincidence and falls out of the arithmetic when the wall stands as far from the picture as the eye does.

Below that line the design goes on the floor. Above it, on the wall.

Where the join line falls in the design #

The half-of-eye-height answer above is a special case, and the general one is worth having because it is what a painter needs before starting.

The ray from the eye through a design point at height y lands on the floor at a depth of z·y / (h − y), where h is the eye’s height and z its distance from the picture. Setting that equal to the wall’s distance w and solving for y gives

y = h · w / (z + w)

So the join line sits at the fraction w / (z + w) of the eye’s height. With the wall as far out as the eye stands back, that fraction is a half — which is the case drawn above. With the wall twice as far out it is two thirds; with the wall at half the eye’s distance it is a third.

Two things fall out of that immediately.

The join never reaches eye level. The fraction approaches one only as the wall goes to infinity, which is the pure floor case. So a corner anamorph always has a piece of design left over above the join, and that piece goes on the wall — and the wall has no such limit, because the rays meet it more and more squarely as they go up.

A wall converts the pole into a finite problem. The whole difficulty of the floor case is that the last few per cent of design height costs unbounded floor. A wall at any finite distance removes it, in the way a fold in the floor removes a different one: everything above the join is on the wall, at a stretch that is bounded and in fact improves with height.

That is the strongest thing that can be said for the corner construction and it is worth saying plainly. It is not a convenience. It is the difference between a design that can include its own horizon and one that cannot.

They agree on the join #

The first thing to check is continuity, and it is exact for a reason that is almost too simple to state.

A point of the join is a point of the floor and a point of the wall. The ray from the eye through the design point that aims at the join lands there, and it lands there whichever surface is thought of as catching it. So the two maps produce the same mark, and the design is unbroken across the corner.

That is not an argument about limits or about the maps being well behaved. It is the observation that a ray hits what it hits.

Measured, the design point aimed at the join lands on the join to within arithmetic noise of the plane it is measured against, which is the check rather than the claim.

And they disagree about scale #

Continuous is not smooth, and here the derivative jumps.

Take two design points a hair apart, just below the join line, and measure how far apart their marks are on the floor. Then take two design points a hair apart just above it, and measure how far apart their marks are on the wall.

7.755 on the floor, 2.000 on the wall. The design is stretched by nearly eight times along the floor at the seam and by exactly two up the wall, and the ratio between them is 3.877.

The reason is the angle the rays make with each surface. A ray about to reach the join is grazing the floor — it has come almost all the way down and is running nearly flat — so a small change in its direction moves its floor intersection a long way. The same ray meets the wall nearly head on, so the same change in direction moves its wall intersection much less.

The jump is therefore the ratio of two grazing angles, and it can be computed from the geometry without casting anything. Both stretches differentiate in one line.

On the floor the mark sits at depth $Z = zy/(h-y)$, so $\mathrm{d}Z/\mathrm{d}y = zh/(h-y)^{2}$, and at the join, where $h - y = hz/(z+w)$, that is

$$\text{floor stretch} \;=\; \frac{(z+w)^{2}}{h\,z}.$$

On the wall the ray continues past the picture plane, so a design point at $y$ lands at $h + (y-h)(z+w)/z$ and the stretch is $(z+w)/z$, with no $h$ and no $y$ in it — constant all the way up the wall, which is the precise form of the claim that the wall holds its scale. Dividing,

$$\text{jump} \;=\; \frac{z+w}{h}.$$

For the arrangement drawn — eye 1.65 m up and 3.2 m back, wall 3.2 m out — those are 7.758, 2.000 and 3.879, against the 7.755, 2.000 and 3.877 measured.

Three readings, and the third is the one worth carrying.

The wall’s stretch does not depend on where the design point is, only on how much further the wall is than the picture. So the wall is not merely better than the floor at the seam; it is uniformly as good everywhere above it, which is why the design can run up it indefinitely.

The jump is bigger for a reader standing further back. It carries $z + w$ over $h$, so a corner anamorph made for a viewer across a hall has a seam four or five times sharper than one made for a viewer standing close, on the same wall.

And it cannot be removed. Bringing the wall right up to the picture plane, $w \to 0$, leaves a jump of $z/h$ — the eye’s distance measured in eye-heights, which is 1.94 for this arrangement and larger for any reader further away than their own height. There is no wall position that makes the map smooth, because the floor is grazed and the wall is not, and the two can be brought together only by removing the floor entirely. That is the reach a flat floor has and the pole it runs into meeting the one repair available, and the repair costs a seam whose size is fixed before the wall is placed.

What the jump means for the painting #

The scale jump is the whole practical content of the construction, and it cuts both ways.

It is what makes the corner worth having. The floor at the seam is stretching the design by nearly eight, and it would go on getting worse — the next few centimetres of design would need metres more floor. The wall takes over at a stretch of two and holds it: the wall is nearly perpendicular to the rays, so it is an efficient receiving surface, and a design can run up it for a long way without the stretch growing much.

It is also a visible seam in the work. The painter crosses the join and the required scale changes abruptly by a factor of four. Nothing about the design changes there; nothing about the intended picture marks that line. So a corner anamorph has a line running through it, at a place decided by the room rather than by the picture, where the brush strokes change size fourfold.

The mark of a badly made one is that this line is visible in the reconstruction, because the painter smoothed the transition — which is exactly the wrong thing to do, since the correct map is not smooth.

Where the join should go #

The wall’s distance is usually not a free choice — it is where the wall is — but when it can be chosen, the arithmetic says something.

Moving the wall closer moves the join lower down the design and reduces the jump, because the rays reaching the join have not grazed as far. Moving it further does the opposite: more of the design goes on the floor, at increasing stretch, and the jump at the seam grows.

In the limit of a very distant wall the floor case is recovered, with all the trouble that implies. In the limit of a wall at the ground line the whole design goes on the wall, and the anamorph degenerates into an ordinary picture painted on a wall — no smear at all, correct from everywhere.

So the family runs from “an ordinary picture” at one end to “an unusable smear” at the other, with the interesting cases in between, and the parameter is the wall’s distance measured against the eye’s.

That is a tidier description of the design space than the usual one, which treats floor anamorphs and wall paintings as different kinds of thing.

The classical instances #

The construction is not new and its best-known examples are architectural rather than painted on pavements.

A room painted so that its ceiling appears to continue upward, with the illusion running across the cornice, is a corner anamorph with the join at the top of the wall. The cornice is exactly the line where the derivative jumps, and the painters who worked this way put the join at an architectural feature deliberately — a moulding, a string course, the edge of a vault — so that the fourfold change of scale is hidden by something the eye already expects to be a boundary.

That is a good piece of craft and it has a geometric reading. The seam is unavoidable and its position is decided by the room; the choice available is whether it coincides with a real edge or falls across an open expanse. Putting it on an edge costs nothing and hides the one visible artefact of the method.

The other classical instance runs the other way: a design on a floor that continues up a stair. Each tread and riser is a plane, so the map is piecewise projective with many pieces, and the joins fall on the stair’s own edges — which is where the eye expects a discontinuity anyway, for exactly the same reason.

Both are the same observation. Put the seam where the architecture already has one, and the fact that the map is continuous but not smooth stops being visible.

More than two planes #

Nothing in the construction cares that there are two.

A design cast into the corner of a room meets a floor and two walls; a design cast into a stairwell meets a dozen surfaces. Each plane gets a homology, adjacent planes agree on their common line, and the whole map is piecewise projective — continuous everywhere, smooth nowhere along the edges.

That is the same structure as the cube map among picture surfaces: six planes, each one a perfectly good perspective picture, joined along edges where the derivative jumps but the picture does not break. The essays on picture surfaces measure the kink at a cube map’s seams and find it exactly the same kind of thing.

Which is worth noting because it means the corner anamorph is not exotic. It is a cube map with the roles of the eye and the surface exchanged: instead of an eye at the centre of a box projecting outward, an eye outside a box projecting in.

What is not available #

Two things a reader might expect are not on offer, and both are informative.

There is no single collineation. Four marks do not determine the rest. The map is a collineation on each piece, so four marks on the floor determine the floor part and say nothing about the wall part; a homography fitted to marks from both planes fits neither. That is the same failure the vault produces in the ceiling that is not a plane, arrived at by a different route: there the surface curves, here it is flat everywhere and bends along a line.

There is no unrolling that helps. A floor and a wall meeting at a right angle can be flattened onto paper — the two planes unroll about their common edge with no strain at all, since a dihedral is developable. So the design can be printed flat and folded into the corner, which is a real practical advantage over a vault. What that does not buy is a projective description: the flattened picture is two homographies with a fold between them, and the fold is where the scale changes fourfold.

Being developable and being projectively describable are different properties, and this collection has confused them before. A dihedral has the first and not the second.

Reading a corner anamorph backwards #

The recovery is worth a paragraph, because it behaves differently from the single-plane case in a way that is useful.

Handed the marks on a single plane, the recovery returns the eye’s position on the floor exactly and does not return its height — only the product of the height with the design’s aspect ratio. That is the ambiguity measured in the marks name the place, not the height, and it is a fact about planes.

Handed marks on two planes, the ambiguity is gone. Raising the eye and stretching the design is a family that works for one plane; it does not work for two at once, because the two planes’ homologies respond differently to the change. The join line moves, the split of the design between the surfaces changes, and no rescaling of the design fixes both.

So a corner anamorph names its own eye completely — position and height — from the marks alone, with nothing assumed about the design. Two planes are enough where one is not, and the mechanism is the same one that the height a flat floor cannot give finds in a rippled floor: something that breaks the plane’s own family of solutions.

The two are worth setting side by side, because they are the same repair applied differently. A ripple in the floor breaks the family with curvature. A corner breaks it with a second plane. Either will do, and the second is a great deal easier to arrange.

The general statement #

An anamorph is a map from a design to a receiving surface, determined by one eye. What the receiving surface contributes is the whole of the difference between the cases this collection has measured:

• A plane gives a collineation: three numbers, four marks determine every other, straightedge constructions apply.
• A dihedral gives two collineations sharing an axis: continuous, not smooth, still flat, still printable, and no four marks determine it.
• A developable curved surface — a vault — gives neither: no collineation anywhere, and half a metre of error from the best homography that can be fitted, but still printable.
• A doubly curved surface — a dome — gives no flat design at all, so there is nothing to print.

The four are a ladder, and each rung gives up one property. The corner is the rung that is usually skipped, and it is the one that gives up the least: it costs the projective description and keeps everything physical.