A picture that can be printed

The floors that unroll establishes which surfaces can be laid flat without stretching: the ones whose Gaussian curvature is zero, which is planes, cylinders, cones and the tangent developables, and nothing else. A sphere is not among them, which is why every world map is wrong and why counting cloud by counting pixels has to choose its surface before it counts anything.

This essay is that fact applied to a display, with one term held at zero so that it can be seen at all.

Holding the seat at zero #

The screen field’s difficulty has always been that its failures are entangled. A curved screen sends the wrong directions to a viewer, and the picture it was fed was also wrong, and separating the two takes care.

The matched arrangement removes one of them completely. Feed a cylinder its cylindrical picture and a dome its equirectangular one, and sit at each screen’s own centre of curvature, and both are exact — a tenth of a millionth of an arcminute, which is the arithmetic floor. Whatever is left after that is not about the seat and is not about the surface being the wrong one.

What is left is whether the picture can be made flat before it goes up.

The answer #

A cylinder unrolls. Its matched picture’s area scale varies by about a thousand-millionth across the whole surface, which is the arithmetic floor — so the picture is a flat sheet that can be printed, wrapped, and is correct when it arrives.

A dome does not. Its matched picture is stretched by a factor of 1.18 between the middle and the rim, and the stretch is there before anybody sits down. It is not a property of the seat, it is not removable by a correction at the viewer, and it is not a consequence of choosing the wrong surface — the surface is exactly the right one.

A flat panel is the control and comes back at exactly one.

Two failures with nothing to do with each other #

The point of holding the seat at zero is to show that the two things a curved screen can get wrong are genuinely independent, and here they are separated as far as they go.

A seat failure is angular. A viewer in the wrong place receives marks from the wrong directions, and the error is measured in arcminutes and is a property of where they are.

A development failure is areal. The picture is distorted on its way onto the surface, and the error is measured as a ratio and is a property of the surface’s shape.

A cylinder has the first and not the second: perfect development, and one seat. A dome has the second and not the first at its own centre: exact directions from the middle, and a picture that cannot be laid down without stretching.

That neither implies the other is the finding, and it is the reason this needed a separate rung. A field that only ever measured “how wrong is the picture” would have reported one number for two failures with different remedies.

What the eighteen per cent actually is #

A dome’s matched picture is authored equirectangularly, and equirectangular is the surface that stretches most at its own poles. The 1.18 measured here is the ratio between the largest and the smallest area scale over the dome’s surface, for the dome as this collection models it — an eight-metre span at a four-metre radius, which is a hemisphere’s worth of arc across and rather less up.

A full hemisphere would be worse and the number would be unbounded at the pole itself, which is where every equirectangular map’s stretch diverges. Domes are not full hemispheres for exactly that reason among others, and a planetarium’s content is authored knowing it.

The 1.18 is therefore a number about this dome rather than about domes, and its content is the sign rather than the magnitude: the stretch exists, is a property of the surface, and does not go to zero for any seat.

Why a printed screen and a projected one differ #

The distinction the measurement is about is not academic, because two ways of putting a picture on a curved surface differ exactly here.

A projected picture is drawn onto the surface by light, from a projector somewhere in the room. Whatever stretch the surface would impose is compensated by pre-distorting what the projector emits, and the render is distorted on purpose is where this collection measures that. Development does not arise: nothing is laid on the surface, so nothing has to be flat first.

A printed picture is made flat and then wrapped. It has to be a flat sheet, and if the surface is not developable there is no flat sheet that becomes it — the picture has to be stretched, or cut into gores, or accept the distortion.

So the developability question is a question about printed screens, and the printed curved screen is a real object: a painted cyclorama, a wrapped column, a curved backdrop, a printed panorama in a rotunda. All of those are cylinders, and this essay says why.

The area scale, and why it is the right measure #

Development is usually described as “can it be laid flat”, which is a yes-or-no question, and the measurement here reports a ratio. The reason is that the yes-or-no answer is not useful about a real surface.

Every surface can be laid flat somehow, with enough stretching. What matters is how much, and the honest measure is the area scale: how much a small patch of picture grows or shrinks on its way onto the surface, and how much that varies over the surface.

A developable surface has an area scale that is constant — one, if the picture is at the surface’s own scale — so its spread is exactly one and the yes-or-no answer follows. A non-developable one has a spread greater than one, and the amount is what decides whether it can be gored, whether the stretch can be absorbed by the material, or whether the picture has to be authored knowing it.

Reporting the ratio rather than the verdict is what lets a dome at 1.18 and a hemisphere at several be different answers, and it is the same reason the third column is area reports a scale rather than a yes.

What a material can absorb #

A practical bound, because the eighteen per cent has to be compared with something.

Printed vinyl stretches by a few per cent before it distorts visibly, and by ten before it tears. So a dome’s eighteen per cent is beyond what a single sheet absorbs, which is why domes are gored and cylinders are not — the difference is not that one is impossible, it is that one is outside the material’s own tolerance and the other is at zero.

That is worth stating because it turns a geometric statement into an engineering one with a threshold. A surface whose area spread is under a material’s tolerance can be printed in one piece regardless of its Gaussian curvature; a shallow dome, a gently doubly-curved panel or a cone with a small opening angle are all in that class.

The cylinder’s advantage is not that it is under the threshold. It is that its spread is exactly one, so the tolerance never enters and the answer does not depend on the material at all.

How many gores #

The threshold turns into a count, because the obstruction shrinks quadratically with the size of the piece. Gauss’s circumference formula puts the irreducible strain over a patch of radius $\rho$ at about $K\rho^{2}/6$, and a four-metre dome has $K = 1/R^{2} = 0.0625$ per square metre, so a single sheet reaching four metres of arc is stuck at $0.0625 \times 16/6 = 16.7$ per cent — which is the eighteen the measurement reports, arrived at from the curvature alone.

Cut it into $n$ pieces and $\rho$ falls by $n$, so the strain falls by $n^{2}$. Against a vinyl’s few per cent that is three gores; against a one per cent tolerance, five. Both are small numbers, and that is the useful form of the answer: a dome is not printable in one piece and is comfortably printable in a handful, and the handful is a square root rather than a proportion.

The same square root is why an atlas has the number of sheets it has, and why the cylinder’s exemption is worth so much. A cylinder needs one piece at any size; a dome needs $n \propto \sqrt{K}\,\rho$, which grows with the object. Doubling a dome’s radius leaves $K\rho^{2}$ unchanged only if the piece grows with it — so a bigger dome of the same shape needs the same number of gores, and a shallower one of the same span needs fewer.

Gores, and what they cost #

A dome can be printed if the picture is cut into gores — tapered strips, each narrow enough that its own stretch is small — and it is worth pricing that, because it is what is actually done.

The stretch within a gore falls with the square of its angular width, so halving the number of gores quadruples the stretch in each. Twelve gores over a hemisphere gives each one thirty degrees of azimuth and a stretch of a few per cent; twenty-four gives under one.

What gores cost is seams, and a seam on a picture surface is exactly the defect six flat pictures of everything measures on a cube map: a place where the surface’s smoothness fails and where the picture’s continuity has to be repaired by hand. A gored dome is a picture made of twenty-four correct pieces with twenty-three joins.

The trade is therefore stretch against seams, and it is the same trade a cartographer makes and for the same reason. A cylinder needs no gores at all, which is the whole of its advantage.

The cyclorama, which is the worked example #

There is one printed curved picture surface with a long history and it is worth naming, because it is a cylinder and the reason is this essay.

A cyclorama — a panorama painted on the inside of a cylindrical wall, viewed from a platform at the middle — is the matched arrangement, built. The wall is a cylinder; the viewing platform is at the axis, which is the centre of curvature; and the picture is authored so that its horizontal coordinate is the azimuth, which is what a painter standing at the axis and marking where things appear does automatically.

So a nineteenth-century panorama rotunda is a matched cylindrical display with its audience at the exact seat, and it is the only such thing anybody has built at scale. Its picture was painted in place rather than printed, which sidesteps development entirely; but a printed one would have worked, and a printed dome of the same size would not.

The arrangement’s own weakness is the one this row measured: it has one seat, and the platform is small. A rotunda’s viewing platform is a few metres across on a cylinder of twenty — the arrangement the seats a screen will accept measures the solid of, which is about two per cent off the axis — inside the region where the matched picture is still the better one, and only just.

Where the two exact cases part company #

It is worth putting the cylinder and the dome side by side one last time, because they are exact in the same sense and different in every other.

Both deliver every mark in exactly the direction it was drawn from, when the viewer is at the centre. Both have a single such seat and a linear penalty for leaving it. Both are fed a surface that is uniquely right and whose runner-up is measurably wrong.

They differ in one thing and it is the thing this essay measures. The cylinder’s exactness survives the picture being made somewhere else and brought, and the dome’s does not — because getting the dome’s picture onto the dome is itself a distortion, and the distortion is not the seat’s and cannot be corrected at the viewer.

That is the whole of what developability is worth to a display, and it is why the two exact cases have completely different industries around them: cylinders are printed and wrapped, and spheres are projected.

The developability of the other four surfaces #

Worth listing, since the row’s ranking has six surfaces in it and only two of them have been discussed here.

Rectilinear is a plane and lays flat trivially. Cylindrical unrolls exactly. Equirectangular is a chart of a sphere and lays flat trivially as a chart — it is already a rectangle — but the sphere it charts does not, which is the whole confusion the word invites and is the reason this essay measures the surface rather than the chart.

The two fisheyes and stereographic are all charts of a sphere too, and the same distinction applies: each is a flat picture, and none of them can be wrapped onto a sphere without stretching.

So the developability question has exactly two answers among the surfaces here — plane and cylinder yes, everything spherical no — and it does not discriminate among the four spherical charts at all. That is a different battery from the ranking in the surface a screen wants, and the two together say that the choice of spherical chart affects the directions and not the printing.

What is not developable and is not a sphere #

The two answers here are plane-or-cylinder and sphere, and it is worth noting the surfaces in between because a real screen is often one of them.

A cone is developable. Its Gaussian curvature is zero everywhere but the apex, it unrolls into a sector of a disc, and a conical screen — a slanted cylindrical backdrop, a tapered wrap — can be printed exactly.

A tangent developable — the surface swept by a curve’s own tangent lines — is developable too, and this collection has met it: the corners a floor cannot add uses one as the locus where a shadow acquires a corner. It is an unlikely screen and it is in the class.

A torus, a paraboloid, an ellipsoid are all doubly curved and none of them unrolls. A curved screen that bends in both directions is in this class however gently, and the only question is whether its area spread is under the material’s tolerance.

So the division is not between cylinders and everything else. It is between surfaces with zero Gaussian curvature and surfaces without, and the first class is larger than a designer might assume — which matters, because a screen shaped like a cone can be printed and a screen shaped like a very shallow dome cannot, however much more nearly flat the second looks.

What this adds to the field’s original result #

The floors that unroll measures developability on floors — surfaces an anamorph is cast onto — and finds the same division. What the screen case adds is the separation.

On a floor the two failures are entangled: a design cast onto a non-developable floor is distorted by the casting and by the surface together, and the essay has to argue that the second is present rather than measure it alone. Here the casting’s contribution is held at exactly zero by construction, so the surface’s contribution is the whole of what is measured.

That is worth having as a method rather than as a result. A term is best measured with every other term held at its own zero, and the matched arrangement is the first place in this collection where the seat’s term can be held at zero at all — which is a fair statement of what the whole row is for.

The short version #

A matched cylindrical picture can be printed flat and wrapped, and is correct when it arrives: its area scale varies by a thousand-millionth across the whole screen. A matched dome picture cannot; it is stretched by eighteen per cent between the middle and the rim, and the stretch is there before anybody sits down.

Both screens are fed the surface that is exactly right for them and both are exact at their own seats, so the seat’s term is held at zero and what is left is the surface’s alone. Two failures with different remedies, separated by holding one of them at nothing.