A design that lands in two rooms

The anamorph that crosses a corner establishes the piecewise case: a design cast onto a floor and a wall is two homologies, continuous across the join and not smooth, with the scale jumping by a factor at the fold.

A corridor with a doorway in it is the same structure with a hole where the fold was, and the difference turns out to matter more than it sounds.

The object #

A corridor 2.4 metres wide and 2.6 high, four metres long, with a doorway 0.9 metres wide and two metres tall in its end wall, and a further wall three metres beyond that.

Seven planes: the floor, two side walls, three pieces of end wall around the opening, and the far wall in the next room. The design stands upright at the near end and the eye is 1.7 metres up, 2.4 metres in front of it — an ordinary standing reader looking down an ordinary corridor.

Cast the design and every one of those seven faces except the ceiling receives some of it.

The shares, and the depths #

The floor takes 56 per cent of the design, the two side walls 14 per cent each, the two pieces of end wall beside the doorway 6 per cent each, and the far wall 4 per cent.

The depths are what makes it a different object from a corner. Marks on the near floor are three metres from the eye. Marks on the end wall are 6.4 to 6.7. Marks on the far wall are 9.4 to 9.6.

So between two design points a hair apart — one landing on the end wall beside the doorway and one going through it — the distance from the eye jumps from 6.4 metres to 9.4, a factor of 1.47, with nothing in between.

What jumps and what does not #

The mark on the far wall is exactly where the ray put it, so nothing about the construction is unusual: it is the same anamorph, on a surface with a piece missing and a piece behind it.

What jumps is the scale. A design point’s neighbours land 1.47 times further apart on the far wall than they would have on the end wall, because a picture cast onto a surface further away is larger in proportion to the distance. So the picture painted in the next room has to be half again as large as the piece it continues, mark for mark, and a painter working from a scaled drawing would get it wrong by exactly that factor.

What does not jump is continuity in the picture. Every design point has a mark; the marks either side of the doorway’s edge are on different walls at different depths, and both are exactly where the design says. Seen from the design eye, the picture is seamless: the reader cannot tell which parts of it are in the next room, which is the whole trick.

This is the corner anamorph’s structure exactly — continuous and not smooth, with a scale jump at the join — with the difference that a corner’s two faces meet and a doorway’s two faces are three metres apart.

And the bisection that had to be abandoned #

The corridor is the object that broke a piece of machinery, and the break is instructive enough to be the rung’s methodological finding.

A design needs a band of heights whose rays meet the object. The stair essay finds that band by bisection, on an argument that is correct there: on a flight, a higher design ray is shallower and either clears the flight or does not, so hitting is monotone in the design’s height.

On a corridor it is not monotone. Going up the design, the rays hit the floor, then the end wall, then pass through the doorway onto the far wall, then hit the end wall above the doorway, then clear the corridor entirely. Hitting and missing are interleaved, and a bisection lands wherever its samples happened to fall.

The band here is therefore sampled — a hundred and sixty rays across the range, and the band is the outermost that land — which is slower and correct.

The general form of that error is worth stating in a place it can be found: a routine that assumes monotonicity gives an answer rather than a refusal when the assumption fails. It is the same shape as a bisection on a function with a corner in it, and the same shape as the plan reading that assumes the feet and the tips are coplanar — a construction that is exactly right under a condition nobody writes down.

The two numbers a designer can choose #

Two properties of the building decide everything about the far room’s share, and they pull in opposite directions.

How wide the doorway is. Widening the opening from half a metre to two metres takes the far wall’s share of the design from 2.5 per cent to 9.3 — and past a certain width the design starts escaping past the far wall’s edges, so the share stops climbing and the landed share begins to fall.

How far away the far wall is. Moving it from 1.5 metres beyond the doorway to eight takes the far wall’s share down, from 4.7 per cent to 2.6, and takes the scale jump up, from a factor of 1.23 to 2.24. A distant far wall receives less picture and has to paint it much larger.

So the far room is worth having when the doorway is wide and the wall behind it is close, which is the arrangement in which the two rooms read most nearly as one space. In a building where the doorway opens onto a large hall, the design’s continuation is a thin sliver painted at twice scale on a wall a reader can barely see through the opening — geometrically exact and practically pointless.

That trade is computable from a plan before anybody paints anything, which is the useful part: the design’s own share and the scale factor are two numbers a designer can read off the building’s dimensions.

And they are not two numbers. The scale jump is the ratio of the two depths,

$$\text{jump} \;=\; \frac{D_d + t}{D_d} \;=\; 1 + \frac{t}{D_d},$$

for a doorway at $D_d$ from the eye and a far wall $t$ beyond it. With the end wall at 6.4 metres, that gives 1.234 at $t = 1.5$, 1.469 at $t = 3$ and 2.25 at $t = 8$, against the 1.23, 1.47 and 2.24 measured — three points, one division.

The same factor governs the other column. A cone of rays leaving the doorway spreads in proportion to how far it travels, so the picture arriving at the far wall is the doorway’s own width times the jump — 1.3 metres at three, 2.0 at eight. Once that exceeds the far wall’s usable width the picture overruns it sideways, the surviving band is fixed by the wall rather than by the doorway, and the design’s share falls in inverse proportion to the jump.

Which is exactly what the sweep reports. Share times jump is constant: $4.7 \times 1.23 = 5.8$, $4.0 \times 1.47 = 5.9$, $2.6 \times 2.25 = 5.9$. Three arrangements spanning a factor of five in the far wall’s distance, and the product does not move.

So the two properties a designer was choosing between are one property with a conserved product, and what the constant means is a fixed width of picture — the far wall’s own width, measured back through the doorway into the design. Moving the far wall further away does not change how much wall there is; it changes how much design has to fit on it, and the two effects are reciprocal by construction.

The design rule that falls out is short. The far room’s contribution is worth having in proportion to $1/(1 + t/D_d)$, so it halves when the far wall is as far beyond the doorway as the doorway is from the eye, and it is negligible for any wall much beyond that. A doorway into a large hall gives $t$ of ten or twenty metres and a share of one or two per cent painted at three times scale, which is the “geometrically exact and practically pointless” case with the arithmetic attached.

Every face is still a homology, including the one through the hole #

The far wall is reached only by rays that pass through an opening in another face, which is a genuinely new configuration for this field — it is the first surface in the collection that a design reaches indirectly.

It changes nothing about the map. Four design points fitted to four marks on the far wall predict every other mark on it to $7\times10^{-15}$ metres, exactly as they do on a floor or a stair tread.

That is worth checking rather than assuming, because it was not obvious. The design region that lands on the far wall is the shadow of the doorway — a rectangle in the design, but only because the doorway is a rectangle — and a homology is a map between planes that does not care what shape its domain is or how the domain came to be that shape. Occlusion selects which design points land where; it does not bend the map.

So the count of homologies for this object is seven, and one of them is a homology through a hole.

What a designer gets and what they pay #

Two things follow that a person actually painting such a thing would want.

The far room is cheap picture and expensive paint. Four per cent of the design lands there, and it lands at half again the scale — so the area of wall that has to be painted for that four per cent is more than twice what the same four per cent would take on the end wall. A design that puts something important through the doorway is committing to a large piece of work in another room for a small piece of the picture.

And the far room is the part that fails first. The eye that reaches the most measures what happens as the eye rises: the corridor’s design-landing share falls from 100 per cent to 92, and every one of the lost rays is a ray that went through the doorway and over the far wall. So the part of the picture in the next room is both the smallest and the most sensitive to the reader standing in the wrong place — which is a poor combination and is an argument for keeping the design low enough that the doorway’s rays land well inside the far wall.

The stretch, which is where the corridor is worst #

One number in the corridor’s row is the worst in the whole set of objects, and it does not belong to the far wall.

The worst local stretch is 20, and it is on the floor — at the far end of the near floor, where the design’s shallowest rays graze along it and two design points a hair apart land twenty times further apart than the design says.

That is the general behaviour of a floor seen at a grazing angle, and it is why the plain floor has the worst stretch of any object in this set at 19: a flat floor viewed from a standing eye runs away toward the horizon, and the design’s last rays are nearly parallel to it.

So a corridor is a better object than a bare floor for a design, on every measure but one: the walls give the design somewhere to land at a sensible angle, and the floor’s grazing region is exactly the part the walls take over.

Why the reader cannot see the join #

The reason all of this works is worth stating once in plain terms, because it is the anamorph field’s founding fact and it does the whole of the work here.

From the design eye, the direction to a mark is the direction to the design point that made it. That is true whatever the mark’s distance, so a reader at that eye receives the intended picture exactly, with no information at all about which surface each part of it is on.

A corner anamorph exploits it across a fold; a vaulted ceiling across a curve; and a corridor with a doorway across a gap of three metres. In every case the surface is invisible from the one point the design was made for, and it is the only thing visible from anywhere else.

Two eyes give it away, which is an anamorph has one eye — stereopsis measures distance directly and reports the three-metre jump immediately. So a photograph of this corridor from the design eye is convincing and standing in it is not, which is the same asymmetry pavement paintings have, made larger by the fact that the depth discontinuity here is metres rather than centimetres.

What this adds to the anamorph field’s list of surfaces #

The field now has five kinds of receiving surface, and it is worth listing them because each was added to break something the previous one could not.

A plane, which is one homology and leaves the eye’s height free. A corner, which is two, continuous and not smooth. A vault, which is a curve and where the collineation stops existing altogether. A flight, which is eighteen and where most of them are unreachable. And now a corridor with a hole in it, which is seven, one of them reached indirectly, with a depth discontinuity in the middle of the picture.

The list has a direction to it. Each surface takes away one more of the properties a designer might have assumed: that the map is one map, that it is smooth, that it is projective at all, that every face can be used, and now that the picture’s parts are at comparable distances.

What survives all five is the construction itself — rays from an eye through a design, marks where they land — which is what the field’s first rung says an anamorph is and has needed no amendment since.

What it would take to see the join #

The picture is seamless from the design eye and the obvious question is how far a reader has to move before the three-metre jump becomes visible.

The answer is the anamorph field’s usual one and it is small. A wrong eye composes the intended picture with a central collineation on each face separately, and the departure grows linearly with the step — but here the two faces are at very different depths, so the same step produces a much larger departure on the near floor than on the far wall, and the two parts of the picture slide against each other.

That relative motion is what gives the join away, and it is the reason a corridor anamorph is harder to photograph convincingly than a corner one: at a step of a few centimetres the near and far parts have moved by different amounts, and the eye’s edge-matching machinery reads the resulting offset at the doorway’s rim immediately.

So the object’s tolerance is set by its deepest discontinuity rather than by its overall size, which is a rule of thumb worth having: an anamorph across surfaces at depths in a ratio of three is about three times less forgiving than one on a single surface at the nearer depth.

The short version #

A corridor with a doorway is seven homologies, one of them reached through a hole in another face. Four per cent of the design lands in the next room, at 9.4 metres against 6.4, so its scale jumps by half again across an edge the reader cannot see.

It is a corner anamorph with a gap where the fold was: continuous, not smooth, and with the two pieces three metres apart instead of touching.

And it is the object that broke the design band’s bisection, because on an object with a hole in it, whether a ray lands is not monotone in the design’s height.