The surface a screen wants

No picture surface keeps everything puts the six surfaces this collection implements on one battery — straightness, conformality, area — and finds that none of them scores on all three. It is a comparison made without a room in it.

Put a room in it. A screen has a shape and a seat, and the question of which surface it should be fed has an answer that the surfaces’ own properties do not give.

The comparison, and how it is made fair #

Each surface is authored onto the screen and viewed from the seat the screen’s curvature names, and the residual is the angle between the direction each mark was drawn to be seen in and the direction it arrives from.

The fairness is in the fitting. Each surface gets its own horizontal and vertical extents, fitted to minimise its worst angular error — because a comparison that gave them all the same numbers would be ranking them on scale rather than on shape, and every one of them can be scaled.

The answer #

Each screen’s own surface is exactly right on it, and nothing else is.

A flat panel wants rectilinear, which is the picture plane and is exact from anywhere on the axis. A cylinder wants cylindrical. A dome wants equirectangular.

The third is the one worth carrying. Equirectangular is the surface no picture surface keeps everything describes as preserving nothing at all — it bends straight lines, turns right angles and changes area — and it is exactly right for a sphere viewed from its own centre.

It preserves nothing about shape and it is precisely right about direction, and those are different questions. A viewer at the centre of a dome is asking only the second.

How much the choice is worth #

Here is the awkward result, and it is reported rather than buried.

On a curved television, the runner-up is equirectangular at 0.18 arcminutes. That is a fifth of what an eye resolves. The correct surface is exactly right and the next one is wrong by an amount nobody can see, so on this screen the choice is settled and practically free.

On a curved monitor the runner-up is 2.11 arcminutes, which is a mark visibly out of place. On a cinema screen it is 2.58. On a dome it is 50.8, which is nearly a degree.

So the choice’s value scales with how much of the sphere the screen covers, which is exactly what would be expected and is worth having as numbers: a screen subtending a few degrees can be fed almost anything and a screen subtending ninety cannot.

The runner-up’s gap is a cube #

Three of the four numbers in the section above come out of one expression, and it is worth deriving because it turns “the choice is worth more on a bigger screen” into a rate.

On a cylinder the runner-up is always equirectangular, and the two surfaces differ in exactly one thing: cylindrical puts a mark’s vertical coordinate at $\tan\psi$ for elevation $\psi$, and equirectangular puts it at $\psi$. Everything horizontal is identical. So the whole residual is the departure of $\tan\psi$ from a straight line over the screen’s own elevation range — and the fit is free to choose the best straight line, since it fits each surface’s vertical extent.

For $|\psi| \le \Psi$, $\tan\psi = \psi + \psi^{3}/3 + \dots$, and the best linear approximation to $\psi^{3}$ on a symmetric interval leaves a quarter of it. So the worst angular error is

$$\frac{\Psi^{3}}{12} \ \text{radians}.$$

Against the three cylindrical screens, using each one’s half-height over its own matched radius: a 1.23 m 16:9 television at 4 m gives $\Psi = 4.95°$ and 0.185 arcminutes; a 700 mm monitor at 1 m gives $11.15°$ and 2.11; a 12 m cinema screen at 16 m gives $11.92°$ and 2.58. The measured figures are 0.18, 2.11 and 2.58.

Three screens differing by a factor of ten in size, from one cubic term with nothing fitted.

The cube is the useful part. The value of choosing the right surface grows as the third power of how tall the screen is in angle, so the television’s near-indifference and the monitor’s ten-times-larger penalty are one relation rather than two facts: the monitor is 2.25 times the television in vertical half-angle, and $2.25^{3} = 11.4$. It also says the horizontal extent is irrelevant to this particular comparison, which is not obvious and is worth knowing — a very wide, short cylindrical screen can be fed an equirectangular picture with almost no penalty however far round it goes, because the two surfaces agree exactly in azimuth and the disagreement lives entirely in elevation.

And it explains why the dome is a different animal rather than a bigger screen. There the runner-up is not equirectangular, since equirectangular is the answer, so the gap is set by a different pair of surfaces and no term of this expansion applies. The 50.8 arcminutes is a comparison between two azimuthal rules and belongs to the fisheye family rather than to this one.

The ordering of the losers is informative #

The five surfaces that are not the answer come out in a consistent order, and the order says something about the six.

On every curved screen the ranking is: the matched one, then equirectangular, then the two fisheyes, then stereographic, then rectilinear. On the flat panel it is rectilinear, then stereographic, then the rest.

The pattern is that the surfaces closest in construction to the right one come nearest. Equirectangular and cylindrical share their horizontal law exactly and differ only in the vertical, so equirectangular is always second on a cylinder; the fisheyes are azimuthal rather than cylindrical — every fisheye is a different rule is where the two are separated, so they get the horizontal law wrong as well; and rectilinear is worst on any curved screen because it is the one surface with no azimuthal coordinate at all.

That is not a deep result and it is a useful sanity check on the fitting: a ranking that put stereographic second on a cylinder would mean the fit was doing something other than what it claims.

What the fit actually does #

The extents are found by coordinate descent — hold the vertical and search the horizontal by golden section, then the other way round, four times over — and the search’s own floor is worth stating.

The winner’s fitted residual comes out at about a hundred-thousandth of an arcminute rather than at the arithmetic floor, and that is the search’s resolution rather than the machine’s. The claim that the winner is exact rather than merely best is a separate one, made in drawn for the cylinder from an authoring derived in closed form instead of fitted.

Both are needed. The closed form says the right surface is exact; the search says no other surface can be scaled into being exact, which is a different statement and the one a ranking requires.

Why a fitted comparison and not a natural one #

A reader might object that each surface has a natural extent — the field of view of the content — and that fitting is a way of flattering the losers.

It is, deliberately. The question being asked is whether a surface’s shape suits the screen, and a surface authored at the wrong field is failing the second of the three conditions rather than the surface question. Fitting the extents removes the scale mismatch and leaves the shape mismatch, which is the quantity the ranking is about.

The alternative — every surface authored at the same nominal field — measures a mixture, and the mixture is dominated by the scale term because that term is exactly proportional and the shape terms are not. A ranking made that way would be a ranking of how close each surface’s natural field happens to be to the screen’s arc.

What a display is actually fed #

Worth setting the answer beside the practice, because the practice is not arbitrary.

A planetarium dome is fed equirectangular content — the format what a 360-degree photograph actually is establishes as the ordinary carrier for spherical pictures, which is the exact answer, and the reason usually given is that equirectangular is the format spherical content comes in. That is true and it is not the reason it works; it works because a sphere viewed from its centre delivers equirectangular coordinates exactly, and the coincidence is a real one rather than a convention.

A curved television is fed content authored for a flat panel, which is rectilinear and is the worst of the six on it — 1.79 arcminutes, which is above what an eye resolves and is nonetheless two orders of magnitude smaller than the error from sitting at the wrong distance. So the display’s surface mismatch is real, measurable and swamped.

A cinema screen is fed content authored rectilinearly and shown on a gentle cylinder, at 25.3 arcminutes — nearly half a degree, and the largest surface mismatch among the ordinary displays because a cinema screen subtends a wide field.

What the ranking would say about a screen nobody has built #

The four screens measured here are the ones people own, and the ranking’s structure suggests a fifth that would be worth building and is not.

A screen that is a section of a sphere rather than a cylinder — curved in both directions, like a dome but small and in front of one seat — wants equirectangular. Its runner-up would be cylindrical, at a price set by its vertical field cubed, which for a screen subtending twenty degrees vertically would be about three arcminutes.

That is a real difference and it is small, and it explains why doubly-curved displays are not made: the geometric benefit over a singly-curved one is a few arcminutes at the seat, and the manufacturing cost is that a sphere does not unroll, which a picture that can be printed prices separately and which is the argument that actually decides it.

So the ranking has a practical consequence in the negative: the screen shapes worth building are the developable ones, and among developable shapes the cylinder is the only one with a centre, so the cylinder is the whole of the interesting design space.

The runner-up gap as a design number #

The gap between the best surface and the second is a better number than either, and it is worth naming as a quantity in its own right.

It is what a display gains by knowing its own shape. On a curved television it is 0.18 arcminutes and the answer is that knowing its shape is worth nothing; on a curved monitor 2.11, on a cinema screen 2.58, on a dome 50.8.

Read the other way it is a tolerance: a screen may be fed any surface within its own gap and nobody will notice. That is the useful form for anybody deciding how much machinery to put into a rendering pipeline, and it says that a television needs none, a monitor needs a little, and a dome needs all of it.

The gap grows with the screen’s angular size, which it must, because at zero angular size every surface agrees — all of them are locally the tangent plane. So the gap is a measure of how much of the sphere the screen occupies, expressed in the currency of the error it causes.

The surfaces’ own virtues do not transfer #

The most useful thing this ranking does is separate two questions that the curved field has always kept together and that a reader is likely to run together.

Stereographic keeps every angle establishes that stereographic is the conformal surface and the only one. That is a statement about shapes drawn on it, and it is exactly true.

Stereographic is nonetheless fourth or fifth on every screen here, and last but one on a flat panel. Conformality is a virtue of a picture considered as a picture; it says nothing about whether a screen delivers the picture’s directions correctly, because that depends on the screen’s shape and not on the picture’s.

A surface’s properties are properties of the surface. Whether a screen wants it is a property of the screen. That distinction is the whole content of putting a room into the curved field’s comparison, and it is why the two rankings share no ordering at all.

The two coordinates each surface gets wrong #

It helps to say, for each loser, which of the two coordinates it is wrong in, because the ranking is otherwise a list of numbers without a mechanism.

On a cylindrical screen viewed from its centre, the delivered azimuth is exactly proportional to the horizontal picture coordinate and the delivered elevation’s tangent is exactly proportional to the vertical one.

Equirectangular gets the azimuth exactly right and the elevation’s law wrong — it uses the angle where the tangent is wanted. So its error is entirely vertical and cubic in the vertical field, which is why it is second everywhere and why its margin is so small on a shallow screen.

The two fisheyes are azimuthal: they measure a radius from the axis and an angle round it, so neither of their coordinates is the screen’s. Their errors are in both directions at once and they are consistently third and fourth.

Stereographic is azimuthal too, and its conformality buys it nothing here — it is fifth on every curved screen. A map that keeps every angle in the picture still has to deliver the picture’s directions, and those are different demands.

Rectilinear is worst on any curved screen because it has no azimuthal coordinate at all: it delivers tangents where azimuths are wanted, which is the largest possible mismatch of the horizontal law and grows fastest with the field.

That mechanism also predicts the order on a flat screen, and correctly: there the delivered coordinates are tangents in both directions, rectilinear is exact, stereographic is nearest because its radial law is closest to a tangent near the axis, and the cylindrical and equirectangular surfaces are further off because their horizontal law is an azimuth.

What happens off the matched seat #

The ranking above is taken at each screen’s matched seat, which is the only place any surface is exact. It is worth asking whether the ordering survives when the viewer is somewhere real.

It does not, in one respect that matters. At the sold sitting distance every surface is hundreds of arcminutes out, and the differences between them are tens — so the ordering is preserved and the gaps are swamped by a term none of the surfaces controls.

That has a blunt practical consequence. Choosing the right surface for a curved television is worth 1.6 arcminutes and sitting at the right distance is worth 319. A display engineer with a fixed budget of effort has an easy decision, and it is not the one this essay is about.

The exception is the dome, where the audience genuinely does sit near the centre of curvature, the seat term is small, and the surface term is the largest thing left. That is why the one display in the world that is fed its own matched surface is the one for which it matters.

Where this leaves the field’s own comparison #

No picture surface keeps everything ends by refusing to name a best surface, on the grounds that the three properties it measures cannot all be had and the choice depends on what a picture is for.

This essay names one, and the two are not in conflict — they are answering different questions and the difference is worth stating precisely.

The field’s comparison asks what a surface does to a scene: whether it bends straight lines, turns right angles, changes area. Those are properties of the map from directions to marks, and they hold wherever the picture is shown.

This asks what a screen does to a picture: whether the directions the viewer receives are the directions the picture was drawn from. That is a property of the map from marks to directions, which is the first map run backwards, and it depends on the screen.

So a surface can be excellent by the first battery and wrong by the second, and stereographic is exactly that: the conformal one, and fifth of six on every curved screen. The two comparisons share no ordering because they are inverse questions asked of different arrangements.

The short version #

Each screen has exactly one picture surface that is right on it, and it is the surface whose coordinates the screen’s own geometry delivers from its own centre: rectilinear on a plane, cylindrical on a cylinder, equirectangular on a dome.

How much the choice is worth depends entirely on how much of the sphere the screen covers. On a curved television the second-best surface is a fifth of an arcminute behind and nobody can see the difference; on a dome it is fifty, and everybody can.