The ceiling that is not a plane

This site has spent six phases treating pictures as projective objects. Lines go to lines, cross-ratios survive, four correspondences determine a homography, and the whole apparatus of vanishing points, harmonic conjugates and rectification follows. It has been applied to shadows, to mirrors, to photographs of photographs, to the recovery of a camera and — one rung down — to an anamorph on a floor, where the map turned out to be a homology and three numbers rebuilt every mark.

None of that is a way of thinking about pictures. It is a description of what happens on a plane, and it stops.

This essay measures where.

The construction, unchanged #

A vault anamorph is made exactly the way a floor anamorph is made, and the machinery is deliberately the same.

The design is specified as what a visitor is meant to see: a set of points on a notional plane one unit in front of the eye, looking up at a stated pitch. That is how a ceiling design is actually given — the architecture that is supposed to continue upward, the figures that are supposed to be floating there — rather than as marks on a surface.

Each design point becomes a direction. Each direction becomes a ray from the eye. The mark goes where the ray meets the vault, and the vault here is a half-cylinder of radius $R$ whose axis runs horizontally at the springing height, painted on the inside.

That is the whole of it. From the design eye the construction is exact by definition: the direction to the painted mark is the direction the design asked for, so the reconstruction returns the design, and it does so to 1.8e-15 over fifty-four points. There is nothing approximate about a vault anamorph and nothing hand-made about this one.

The measurement #

The question is not whether the construction works. It is whether the map from the design to the marks is a projectivity, and that has an exact test.

Fit a homography to four of the correspondences and measure how far it misses on the rest. Four points can always be matched exactly by a homography, so a residual quoted over the four it was fitted on is a statement about the solver; the correspondences that matter are the fifth onward. The four are taken from the design’s own corners, because four nearly collinear points give a matrix that satisfies its own four and misses everything else.

On the floor, over thirty-six correspondences, the worst miss is 1.4e-15 m. That is the double-precision floor. It is not that a homography approximates the floor anamorph well; it is that the floor anamorph is a homography, and four marks determine all the others exactly.

On the vault, at the same design and the same eye, measured in the surface’s own developed coordinates — the arc length round it and the distance along it, which is the surface a painter actually works on — the worst miss is 529.4 mm.

The ratio between the two is 4e+14.

The units the miss has to be quoted in #

There is a trap in that measurement and this essay fell into it while it was being written.

Quoted in millimetres the vault’s miss barely moves with the radius: 476.8 mm at a 2.4 m vault, 529.4 mm at 4 m, 458.7 mm at 6 m, 480.4 mm at 8 m. Read as a residual that is what it says — the vault’s departure from being a plane does not depend on how flat it is — and that is plainly false.

The reason is that a barrel vault’s developed surface scales with its radius. The same design cast onto a larger vault spreads over more surface, so a miss of about the same number of millimetres is a smaller miss. Quoted as a fraction of the marks’ own extent the numbers behave: 10.3%, 7.7%, 5.1%, 4.3%, falling monotonically as the vault flattens.

That is the version worth carrying, and the version that was written first is the site’s own habit failing for a moment — a residual is a ratio, and quoting one in absolute units is quoting the scale of the drawing rather than the quality of the fit.

Why no choice of four helps #

There is an objection worth answering, because it is the first one anybody raises: perhaps a better four would fit.

It would not, and the reason is structural rather than numerical. A collineation maps lines to lines. Take four points of the design lying on one straight line in the intended picture; their rays lie in one plane through the eye; that plane cuts the cylinder in an ellipse, so the four marks lie on a curve rather than a line. A map that sent a line to a curve cannot be a collineation, whichever four correspondences are used to try to build one.

It follows that the cross-ratio is gone too. Four collinear points of the design have a cross-ratio, and the four marks are not collinear, so the quantity is not merely changed — it has no value on the marks at all. Everything this site builds on the projective invariant is unavailable on a curved receiving surface, and unavailable in the strong sense that the objects the invariant is defined on are not there.

The vault has to be given a reach #

A ray that leaves the eye going downward never meets a vault overhead, and a ray along the vault’s own axis never meets it either. Both cases return nothing rather than a mark, and both are checked in the site’s gate rather than assumed.

That refusal matters more here than in the flat case. A design wider than the vault’s own reach has points with no mark, and a routine that clamped them to the rim would paint a picture nobody asked for and report success. In the figure above, twenty-seven of eighty-one rays miss on the widest design tried — a third of it — and the machinery records the misses and keeps a gap in the list rather than shortening it.

This is the same discipline the site records for a shadow: a figure that counts what it can draw is not counting what happened. Rays that miss the vault are not rays that landed somewhere convenient.

Two surfaces, one question, two fields #

That last sentence is worth stopping on, because it identifies two things this site has kept apart for four phases.

The curved field asks what a picture surface does to a scene: a fisheye, a cylinder, a stereographic sphere, each taking directions and producing marks. This essay asks what a receiving surface does to a design: a floor, a vault, a dome, each taking rays and producing marks.

They are the same map with the arrow reversed, and the finding transfers in both directions.

From the curved field: only the plane keeps every straight line straight, by Beltrami’s theorem, and every other surface bends the image of some line. From here: only a flat receiving surface admits a collineation. Those are the same statement, because a collineation is exactly a map that takes lines to lines, and the theorem says the plane is the only surface for which that holds.

So the boundary this essay measures is not a new one. It is a theorem the site already had, arriving in the field where it decides whether a construction is possible rather than whether a photograph looks bent.

The dome, and the one case that is exact #

A hemispherical dome is worth a paragraph because it is where the site’s own machinery already has an exact answer in a neighbouring field, and it is not the answer this essay is about.

The refraction field found that a spherical port centred on the entrance pupil bends nothing at all — every ray meets it at normal incidence, so Snell’s law has nothing to act on, and the departure is $7 \times 10^{-15}$ degrees. The dome here is the same sphere in a different role: a ray from the eye at the centre meets it at normal incidence, and the mark lands at exactly the angular position the design asked for.

That exactness is about directions and not about the map. The design is still cast onto a curved surface, four collinear design points still land on a circle rather than a line, and no homography fits any better than it does on the vault. A dome centred on the eye is the most convenient receiving surface there is — the paint is spread perfectly evenly, every direction is treated alike — and it is no more projective than any other.

So the two exactness results sit side by side without touching. One says the sphere adds no bending; the other says curvature removes the collineation. The first is about a ray’s direction and the second about a family of rays’ incidences, and a surface can be perfect at one and useless at the other.

What the paint costs #

There is a second consequence of the curvature and it is the one a painter feels rather than measures.

A vault is at grazing incidence to the eye’s rays near its springing, and nearly normal to them overhead. The area of surface a fixed patch of the design covers therefore runs away toward the springing, in the same way a flat picture’s area scale runs away at its edges — the curved field’s $\sec^3$ behaviour, arriving here as square metres of plaster per square degree of picture.

That is why the designs actually painted on vaults put their detail overhead and their sky at the springing, and it is a geometric constraint rather than a compositional preference. A figure placed low on a vault is drawn many times larger than one placed high, from paint that is many times more oblique to the eye, and the ratio is computable from the vault’s radius and the eye’s position before anything is drawn.

What survives, and what a painter uses instead #

Something has to survive, because vault anamorphs were painted and they worked.

What survives is the construction itself. The ray from the eye to the mark is still a ray, the direction is still the direction the design asked for, and the reconstruction from the design eye is still exact. Nothing about the making of a vault anamorph is harder than the flat case; one ray-casting routine handles both, with the surface as an argument.

What does not survive is every shortcut. A painter working on a floor can lay out four marks, run a straightedge, and get every other mark from incidences alone — the diagonal construction, the harmonic conjugate, the repeated bay. On a vault none of those work: four marks determine nothing, the diagonals of a rectangle’s image do not cross at the image of its centre, and every point has to be cast individually.

Which is exactly what the historical method was. The scaffolding grid, the string from a fixed point, the cartoon pricked and pounced along a ray — those are ways of casting each point rather than constructing it, and the geometry says they had to be.

The number worth carrying #

Femtometres against half a metre, at the same design and the same eye, is not a difference of degree.

It is worth stating in the form that makes it a boundary rather than a comparison. A map that takes four marks and gives every other one exists if and only if the receiving surface is a plane. Above that boundary the map is five numbers and can be constructed with a straightedge; below it there is no map, and every mark has to be cast.

Everything the projective half of this site does — the vanishing points, the cross-ratio, the harmonic construction, the rectification, the fixed-structure classification of one rung down — lives above it.

The short version #

A vault anamorph is built exactly as a floor anamorph is built, and from its own eye it is exact.

It is not a projectivity. The best homography fitted to a floor anamorph’s marks misses by 1.4e-15 m; fitted to a vault’s, at the same design and the same eye, it misses by 529.4 mm, and no choice of four correspondences repairs it — because four collinear design points cast marks on an ellipse, and a map that takes a line to a curve is not a collineation.

The plane is the only receiving surface that admits one, which is Beltrami’s theorem arriving in a field where it decides what a painter can construct rather than what a photograph looks like. Above that boundary a picture is five numbers. Below it, every mark has to be cast.