Three conditions, and three prices

Drawn for the cylinder, shown on the cylinder establishes that the arrangement is exact and names three things that have to hold for it. This is what each of them is worth.

The method throughout is to break one and hold the other two, so that what is measured is one condition’s price rather than a mixture — and so that the three can be shown to be three conditions rather than one condition stated three ways.

The seat #

The picture is exact from the screen’s centre of curvature and from nowhere else, and the curve of error against distance has three features worth separating.

It is linear. A tenth of the radius too close costs sixty-six arcminutes; a twentieth costs thirty-two, which is half. A well-set optimum is usually quadratic, so a small departure costs the square of itself and there is a margin. Here there is none: the price is proportional to the mistake all the way down.

It is asymmetric. A tenth too far costs fifty-five arcminutes against the sixty-six for a tenth too close, so coming forward is about a fifth more expensive than going back. That is the opposite of what a viewer is usually told about a large screen.

And it saturates in one direction only. At a hundred radii the error is six hundred arcminutes and approaching a bound, because a picture seen from very far away approaches an orthographic view of the screen, which is a fixed thing. Going forward it grows without limit, because a viewer far enough in has the screen wrapping round them.

The horizontal scale #

The second condition is that the picture’s own horizontal extent is the screen’s arc, and it is the cleanest of the three.

A picture authored at a wider field is a correct cylindrical picture of a world scaled by the ratio, laid on a screen that subtends less of it. Its price is exactly proportional and exactly symmetric: ten per cent either way costs fifty-three arcminutes, and fifty per cent costs two hundred and sixty-four, which is five times fifty-three to the digit.

That exactness is worth pausing on. It means the horizontal condition has no optimum in the ordinary sense — it has a zero, and the error is the absolute value of the mismatch times a constant. There is no second-order term at all, because the two maps differ by a scaling of the azimuth and a scaling of the azimuth is linear in the scaling.

It also means the condition is the easiest of the three to satisfy in practice: it is a number in the rendering pipeline rather than a fact about furniture, and a display that knows its own arc can be fed the right field.

The vertical law #

The third is the subtle one, and it is the one a reader is most likely to have got wrong.

A cylindrical picture surface divides a mark’s height by its horizontal distance from the axis. An equirectangular surface divides by nothing and takes the elevation as an angle. The two agree at the middle of the picture and part company toward its top and bottom, by the difference between an angle and its tangent — which is third order in the angle.

So the price depends entirely on the picture’s vertical field. A sixteen-by-nine panel subtends about five degrees above its own middle from its centre of curvature, and authoring it equirectangularly instead of cylindrically costs 0.28 arcminutes — a third of what an eye resolves, which is to say nothing.

A dome subtends twenty-nine, and authoring it cylindrically instead of equirectangularly costs eighty-four arcminutes, which is a degree and a half.

That the same condition is negligible on one screen and dominant on another is the reason it has to be measured separately rather than folded into “use the right surface”. On a television it genuinely does not matter which of the two vertical laws is used; on a dome it is most of the error.

An arcminute, and what it is worth #

Every number here is in arcminutes, and it is worth saying what one is before the numbers are read.

An arcminute is a sixtieth of a degree. It is roughly the finest detail a good eye resolves at ordinary contrast, and it is the angle a millimetre subtends at three and a half metres. So an error of one arcminute is at the threshold of being seen at all, and an error of sixty — one degree — is a mark visibly out of place: on a curved television at its sold distance it is a mark two centimetres from where it should be.

That makes the three prices readable. The vertical law’s third of an arcminute is invisible. The horizontal scale’s fifty-three for a ten per cent mismatch is a mark almost a centimetre out. The seat’s three hundred and nineteen at the sold distance is five degrees, which is ten moon-widths.

Quoting a picture surface’s error in arcminutes rather than in pixels or in a residual against a homography is what makes those comparisons possible, and it is the change of currency drawn for the cylinder had to make before any of this could be measured.

Why the horizontal price is exactly proportional #

The symmetry and the exact proportionality of the scale condition are unusual enough to be worth deriving, because they are what makes it the one condition a display can fix confidently.

A picture authored at azimuth extent U and shown on a screen of half-arc A delivers, at picture coordinate u, a direction of azimuth uA where the picture intended uU. The angular error is therefore u(A − U) — linear in the mismatch, linear in the position across the picture, and worst at the edge where u is one.

There is no higher-order term anywhere, because both maps are linear in u. So the worst error is exactly |A − U|, the price is exactly proportional, and it is exactly symmetric because the absolute value is.

That also says where the error lives: entirely at the picture’s edges, and not at all in its middle. A scale mismatch is invisible at the centre of the screen and grows steadily outward, which is a signature quite unlike the seat’s — a seat error is worst at the edges too but has a different profile, and the two could be separated by a viewer who could measure at three points across the picture.

The vertical law’s cube #

The third condition’s dependence on the vertical field is the one number in this essay that a reader could not have guessed, so its arithmetic is worth setting out.

A cylindrical surface’s vertical coordinate is the tangent of the elevation; an equirectangular one’s is the elevation itself. Their difference, for a small elevation θ, is θ³/3 — the first term of the tangent’s expansion beyond θ.

So the error is cubic in the vertical field, for a field small enough that the first term is the whole story. A curved television’s picture reaches 4.9° above its own middle from the centre of curvature, and a dome’s 29.4°: a ratio of 5.94, whose cube is 209. The measured prices are 0.28 arcminutes and 83.8, a ratio of 297.

That is the check, and the gap between 209 and 297 is the check working rather than failing. The cubic law is the first term of an expansion, and at twenty-nine degrees the next term is contributing about a third — which is exactly what the difference between the two ratios says. A condition claimed to be cubic whose two prices agreed to the digit over a sixfold field would mean the expansion had been assumed rather than measured.

The two intermediate screens confirm it. A curved monitor reaches 11.1° and pays 3.25 arcminutes; a cinema screen reaches 11.9° and pays 3.98. Divide each price by the cube of its own field and the three shallow screens give 0.0024, 0.0024 and 0.0024 — the same constant to two figures — while the dome gives 0.0033, which is the next term arriving where the expansion says it should.

Why three and not one #

A reader might reasonably suspect that the three are the same condition seen from three sides — that all of them are “use the right picture surface” — and the measurement is what settles it.

They fail independently. The seat can be wrong with the scale and the law right; the scale can be wrong with the seat and the law right; and the law can be wrong with the other two right. Each of those three arrangements is constructed here and each produces a nonzero residual with the other two at their exact values, which is what independence means.

They also fail with different shapes. The seat’s price is linear and asymmetric and bounded on one side; the scale’s is linear and symmetric and unbounded; the law’s is a third-order function of the vertical field, so it is negligible on a shallow screen and dominant on a deep one.

Three failures, three shapes, three prices. A single condition would have one.

A fourth condition that is not one #

There is an obvious candidate for a fourth condition and it turns out to be absorbed by the others, which is worth showing because it is the sort of thing that would otherwise sit as an unstated assumption.

The picture has to be laid on the screen the right way round and the right way up. That is a real requirement and it is not a fourth condition, because a picture laid on backwards is a picture whose horizontal scale is negative — which the second condition already covers, at a price of twice the full field.

And the picture’s centre has to be the screen’s centre. A picture shifted along the screen delivers every mark at an azimuth offset by a constant, which is a rigid rotation of the whole set of directions. A viewer who turns their head absorbs it exactly, and a viewer who does not sees a picture that is correct and pointed somewhere else.

That second one is worth its paragraph. A rotation of the whole picture is not an error in the picture; it is an error in where the viewer is looking, and the two are genuinely different things. Every measurement here is of the residual after the best rigid alignment, for the same reason the screen field’s own measurements are of the residual after the best homography: a difference the viewer can absorb by moving is not a difference the picture carries.

Straightening does not move the eye makes the same distinction in the construction field, about a keystone correction, and it is the same argument: a transformation the reader can undo for free is not part of the error.

What “no margin” means for a real display #

The linearity is the practical finding and it is worth spelling out, because it contradicts an intuition that most optimisation problems support.

A quadratic minimum forgives small mistakes: being one per cent off costs a hundredth of what being ten per cent off costs. A linear one does not: one per cent off costs a tenth of what ten per cent costs. So a display arrangement that is nearly matched is nearly as wrong as its nearness, and there is no band around the optimum inside which the answer is effectively exact.

Numerically, on a curved television: a viewer one per cent off the matched seat — four centimetres — is already about six arcminutes out, which is six times what an eye resolves. There is no seat within a metre of the matched one at which the picture is imperceptibly wrong.

That is a strong statement and it is what the linearity implies, so it is worth double-checking against the shape of the curve rather than trusting the slope. The figure’s own points at a twentieth and a tenth of the radius bear it out: thirty-two and sixty-six arcminutes, a ratio of 2.1 against the 2 exact linearity gives and the 4 a quadratic would.

The condition a display can actually fix #

Of the three, exactly one is under a manufacturer’s control and it is the second.

The seat is furniture. A television matched at four metres is a television for a four-metre room, which the screen sets the distance prices in the ordinary flat case, and no rendering choice changes that.

The vertical law is a rendering choice and on a television it is worth a third of an arcminute, which is below the point of caring.

The horizontal scale is a rendering choice and it is worth fifty-three arcminutes for ten per cent. A display that knows its own arc and is fed content authored for that arc gets it exactly right; one fed content authored for a flat panel of the same width is authored at the chord’s field rather than the arc’s — a wide field on a small screen is the same mismatch on a flat panel, which on a curved television is a mismatch of a few per cent and a few tens of arcminutes.

So the single available improvement is the one nobody talks about, and the two that are talked about — the curvature and the seating distance — are the two that cannot be changed by anything in the signal. The evenness a curve buys prices the curvature itself on a different battery, and reaches the same kind of answer.

Where the numbers come from #

Each price is the worst angular error over a grid across the whole picture, which is the right summary and is worth defending against the alternative.

A mean would be smaller and would say less. A picture whose middle is right and whose edges are two degrees out is not a picture that is nearly right — the eye goes to the part that is wrong, and a viewer reading a straight line across the screen sees the whole of the error at once. So the worst is reported, and the mean is available beside it for anybody wanting the other reading.

The grid is thirteen by thirteen across the picture, which is the same density the screen that names the seat uses for its own residual, which is dense enough that the worst point is found rather than approached: the error is smooth in both directions, so a finer grid moves the answer by less than the last digit quoted.

What breaks first in a real room #

Setting the three side by side at plausible mistakes gives an ordering, and it is not the one a reader would guess.

A viewer sitting a metre and a half in from the matched seat — which is every curved television — is out by three hundred and nineteen arcminutes. Content authored for a flat screen of the same width is out by a few tens. The vertical law is out by a third of one.

So the seat dominates by an order of magnitude over the scale and by three over the law, and the whole of a real display’s error is furniture. That is a slightly deflating conclusion for a row about picture surfaces, and it is the honest one: the surface question is exactly settled and worth almost nothing, and the question worth everything is where the sofa is.

The seats a screen will accept is the other end of that argument, and one picture and three people is what happens when there is more than one sofa.

The short version #

The three conditions fail independently, with three different shapes, at three very different prices. The seat is linear, asymmetric and bounded going back; the horizontal scale is linear, symmetric and exactly proportional; the vertical law is third order in the picture’s vertical field, so it costs a third of an arcminute on a television and eighty-four on a dome.

None of them is quadratic about its own optimum, so none of them forgives a small mistake. And in any real room the seat’s term is ten times the scale’s and three hundred times the law’s, which makes the surface question exactly settled and practically minor.