The design that outruns the floor

The machinery that casts a design onto a floor refuses one thing: a design point level with the eye or above it. The ray through such a point runs parallel to the floor, or upward, and lands nowhere.

That refusal has a quantitative shadow, and the shadow is the practical constraint on the whole art form. As the top of the intended picture approaches eye level, the ray grazes, and the floor mark runs away.

The tangent #

Write f for the height of the design’s top as a fraction of the eye’s height. Then the floor depth the design needs is

d = z · f / (1 − f)

where z is how far the eye stands back from the ground line. It is a tangent in disguise: as f approaches one the denominator goes to zero and the depth goes to infinity.

The consequence is that the numbers do not behave like sizes.

Doubling the design’s height from forty per cent to eighty per cent multiplies the floor needed by six. The last hundredth of the way to eye level costs more floor than the whole of the first half.

The invariant that says it is a pole #

The formula can be checked from the drawing rather than taken on trust, and the check is the kind this collection prefers: a quantity that ought to be constant, computed at every point of the family.

Rearranged, d · (1 − f) / f = z — the depth needed, times the height still to go, divided by the height already reached, is the eye’s own distance from the picture. That should be the same number at every f.

Measured across a family running from a tenth of eye height to ninety-nine hundredths, it is the eye’s distance to four figures at every one of them.

A quantity that is constant over a family whose members differ by three orders of magnitude is not a coincidence, and it is a different kind of evidence from checking one case. A steep slope and a pole look alike over a narrow range; the invariant distinguishes them, because a slope would not have one.

Why an anamorph cannot contain a sky #

The refusal is not a technicality. It says something about what can be depicted.

Any picture of an outdoor scene has a horizon in it, and the horizon of the intended picture is at the intended eye’s own height — that is the horizon is at eye level, and it holds for the intended picture as for any other. So the horizon of the design is exactly the height the anamorph cannot reach, and everything above it — the sky, distant hills, the tops of buildings — has no mark on the floor at all.

A pavement anamorph is therefore restricted to what is below the horizon of the picture it is standing in for. It can show a hole in the ground, a chasm, a flight of steps going down, a creature standing on the pavement. It cannot show a landscape — unless the receiving surface curves upward, which is what a ceiling that is not a plane is for.

Every street painting anybody has seen obeys this, and it has always been described as a stylistic preference for holes and monsters. It is not a preference. It is the only thing available.

The stretch, which is the other half of the constraint #

Floor length is not the only cost. The far end of the design is stretched.

Two design points a centimetre apart near the bottom leave marks a few centimetres apart on the floor. The same two points near the top leave marks metres apart. The ratio — how many metres of floor per metre of design, at the far edge — is the last column of the table above, and it grows even faster than the length does.

At eighty per cent of eye height the far edge is stretched by a factor of forty-seven. At ninety-four per cent by four hundred and sixty-six.

The stretch has a closed form too, and it is the derivative of the same expression:

$$\text{stretch} \;=\; \frac{\mathrm{d}d}{\mathrm{d}y} \;=\; \frac{z}{h\,(1-f)^{2}} \;=\; \frac{(d+z)^{2}}{h\,z},$$

metres of floor per metre of design. Against the table: 5.4 at forty per cent, 9.6 at fifty-five, 21.5 at seventy and 48.5 at eighty, where the measured column gives 5, 9, 21 and 47 — the small shortfall being the difference between a derivative and the finite difference the table takes over its own top segment, which grows with the curvature and so with $f$.

The second form is the useful one. The stretch is the square of the mark’s total distance from the eye’s own ground position, over the eye’s height times its distance — the same expression the corner anamorph derives for its floor side, with the wall’s distance replaced by the mark’s. So the stretch grows as the square of the floor used: doubling the pavement quadruples the cost per unit of picture, which is why the last column of the table climbs faster than the one beside it and by exactly one power.

That has one obvious consequence and one that runs the other way from what would be guessed.

The obvious one is cost: a design stretched by a factor of five hundred at its far edge needs five hundred times as much paint per unit of picture there, and the painter is working on a region the size of a tennis court to produce a strip of picture a few centimetres tall.

The other is that the far end is forgiving, not fragile. A metre of floor contributes a millimetre of picture, so a slab join, a variation in the paint or a crack in the pavement is divided by the stretch on its way into the intended picture rather than multiplied — which is the same statement as the projection preserving angular size, since a floor irregularity far away subtends a small angle at the eye and arrives in the picture at that angle. Precision of execution is cheap at the far end and expensive near the ground line, which is the opposite of where a painter’s instinct puts the care.

What is fragile at the far end is the reader. Differentiating the mark’s depth with respect to the eye’s height gives $f$ times the stretch, so at eighty per cent of eye height a reader a centimetre taller than the design assumed needs the far marks thirty-nine centimetres further away. A design that reaches near the horizon is a design that has to be told the reader’s height to the centimetre — which is the room the eye may stand in and where the anamorph still works arriving as a design rule rather than as a measurement, and it is why the ambitious end of the table is not merely expensive but also the end that stops working when somebody stoops.

The two sensitivities are worth putting side by side, because they point in opposite directions and a designer has to trade them. Moving the eye back by a metre moves every mark in proportion — the whole design scales, and the picture is still a picture of the same thing seen from further away. Moving the eye up by a centimetre moves the near marks by centimetres and the far marks by tens of them, so the design shears rather than scaling and nothing about it survives. A pavement anamorph is robust to where along the sightline the reader stands and brittle about how tall they are, and the brittleness is concentrated entirely in the part of the design nearest the horizon.

A worked case, because the abstraction understates it #

Put a number on it for a real painting.

A street painter wants a design two metres tall — a figure standing, a doorway, a flight of steps. The viewer’s eye is 1.65 metres up. Two metres is more than eye height, so the design cannot be cast at all: the upper third of it is at or above eye level, and no floor anywhere carries it.

Reduce the ambition. A design 1.3 metres tall is 79 per cent of eye height. The eye stands 3.2 metres back, so the floor needed is 12.2 metres, and the far edge is stretched by about forty-four. The painter is working on a strip 12 metres long to produce a picture 1.3 metres tall, and the top ten centimetres of that picture occupy the last four and a half metres of the strip.

Reduce it again. A design 0.9 metres tall is 55 per cent of eye height, needs 3.9 metres of floor, and is stretched by nine at the far edge. That is a normal afternoon’s work on a normal pavement.

The three cases differ by a factor of two in the design and by a factor of infinity, then four, in the floor. Nothing about the intended picture’s appearance changes across them — all three are perfectly ordinary pictures — and the cost of making them is not remotely proportional to their size.

That disproportion is what makes the constraint feel arbitrary to somebody meeting it for the first time. It is not arbitrary; it is a pole, and a pole is what a fixed viewpoint looking along a plane always produces.

The same shape in three other places #

It is worth listing where else this collection has met a tangent of exactly this kind, because the family is larger than it looks and each member is usually explained locally.

A flat picture surface runs away. The half-width of a flat picture needed to cover a given angle grows as the tangent of that angle, so a rectilinear picture of a 180° field is infinitely wide. That is measured in the essays on picture surfaces that are not flat, and it is the same tangent with the eye and the surface exchanged.

A shadow under a low sun. The length of a shadow is the object’s height divided by the tangent of the sun’s altitude, so a sun approaching the horizon casts shadows that lengthen without bound. The refusal in the shadow machinery — a drawn tip above the horizon has no ground point — is the same refusal as the one here.

The vanishing point of a direction nearly parallel to the picture plane. As a ground direction turns to lie along the picture plane, its vanishing point leaves the paper as a pole, which is what makes the height-transfer construction undrawable about half the time.

All four are the same statement: a plane and a point of view generate a tangent, and the tangent is where the plane’s line at infinity is. Each one shows up as a practical limit in a different craft, and each is usually explained as a peculiarity of that craft.

What street painters actually do #

The constraint above is severe, and the practical responses to it are all instances of one move: put something other than a flat floor at the far end.

Run the design up a wall. Where the floor gives out, a vertical surface catches the rays that were about to escape. That is the corner anamorph, and it is a different construction, treated in the anamorph that crosses a corner.

Tilt the eye down. Designing for a viewer looking down from a balcony effectively lowers the horizon of the intended picture relative to the floor, and the design’s top can be well below the eye. Almost every very large street painting is designed to be seen from above for this reason and not for the reason usually given, which is that more people can see it.

Cut the design at its own horizon. Most anamorphs simply do not include anything near eye level: the intended picture is a hole, a pit or a staircase, all of which live in the bottom half of the frame.

Accept a shallow design. A picture whose top is forty per cent of eye height needs two metres of floor and is stretched by five — perfectly manageable, and it is the size of the classical table-top anamorphs.

The four together explain the entire visual vocabulary of the form, which is a pleasing amount of explanatory work for one tangent.

Where the marks stop being marks #

There is a limit before the pole that a painter meets first, and it comes from the pavement rather than from the geometry.

The rays reaching the far end of the design strike the floor at a grazing angle. At eighty per cent of eye height the last ray meets the pavement at about one and a half degrees. A pavement is not a plane at that scale: it is slabs with joints, a camber for drainage, kerbs, and a texture with a grain size of a few millimetres.

A ray grazing at one and a half degrees is displaced by forty times any bump it crosses. A five-millimetre step between slabs moves the mark by two hundred millimetres along the floor, which at that end of the design is a substantial fraction of a millimetre of picture — and the displacements are not random, because the slabs are laid in a pattern.

So the practical far limit of a pavement anamorph is set by the flatness of the pavement, and it arrives well before the geometric limit does. The same calculation on a polished floor indoors, or on a purpose-laid board, pushes it back by an order of magnitude, which is why the most convincing anamorphs are usually made in galleries.

This is the same accounting that the essays on curved receiving surfaces make in the other direction: a design cast onto a surface whose shape is not known comes back wrong in proportion to how wrong the shape is, and the constant of proportionality is set by the grazing angle. A pavement is a curved surface that nobody measured.

The eye’s distance is the only other knob #

The formula has two inputs: the fraction of eye height, and how far back the eye stands.

The second is a plain multiplier. Standing twice as far back doubles the floor needed at every design height, and does not change the shape of the curve at all — the pole is at the same place. So there is no distance at which the problem goes away.

What standing further back does change is the stretch, and it changes it favourably: the marks are spread over more floor but the rays hit the floor at a shallower angle only in absolute terms, not relative to the design. The stretch ratio at a given fraction of eye height is the same at every distance.

That combination — floor scales with distance, stretch does not — is why the very large paintings are more practical per unit of picture than the small ones, given that the floor is free. The limiting resource is paint and time, and those track the stretch rather than the length.

What this says about the map #

There is a way of stating the whole result that connects it to what an anamorph is.

The map from the design to the floor is a planar homology, and its centre sits at the eye’s height plus the eye’s distance from the picture. A homology sends the line at infinity of one plane to a finite line of the other and vice versa — and the finite line that goes to infinity is exactly the design’s horizon.

So “the design cannot reach eye level” and “a homology has a line that it sends to infinity” are the same sentence. The tangent in the formula is what that looks like from the inside.

That is a satisfying way to end up, because it means the constraint is not a fact about pavements. Every planar homology has such a line, so every anamorph on every plane has a line of its intended picture that it cannot carry, and where that line falls in the design is decided by the three numbers. What is peculiar to the pavement case is only that the line is the horizon, which is somewhere a picture usually wants to go.