The screen that names the seat

The screen is a picture surface too settles what a curved display does to a picture, and the finding there is a difference in kind rather than in degree. A flat screen seen from any seat delivers a homography of the intended picture, so what reaches the reader is a correct picture of some scene — a differently shaped one. A curved screen delivers a map that is not a homography from any seat at all.

That is stated as a fact about the picture. It is also a fact about the reader, and the two are not the same statement.

The question turned round #

The screen field has been asking what happens to a picture on its way to a seat. Ask instead what the picture, once it has arrived, says about the seat it arrived at.

The question has an obstacle in it that has to be dealt with before anything can be measured. A reader does not receive a list of coordinates; they receive a set of directions, and those directions are read by an eye that is pointed somewhere in particular, or by a camera that framed the screen in some particular way. Neither of those is known. Both of them are, at most, a projective change of the picture.

So the honest question is not what does the reader see but what part of what the reader sees could not have been produced by their own framing. That is the residual against a homography, and it is the quantity every figure here is drawn on.

Put that way the flat screen’s answer is immediate and total. Its map from the intended picture to the delivered one is a homography; the observer’s own contribution is a homography; and a homography composed with a homography is a homography. There is nothing left over. A flat screen’s picture is consistent with every seat in the room, exactly.

Not “poorly determined”. Not “determined with a large uncertainty”. Not determined: the observation carries no information about the seat whatever, and no amount of care or resolution changes that.

And a curved screen does contain it #

Curve the panel and the composition stops closing. The delivered map is not a homography, the observer’s framing cannot make it one, and what is left over is a function of where the eye is and of nothing else.

The recovery is then the same shape as the anamorph field’s. Take a grid of picture points; compute what each candidate seat would receive; fit the four corners to the four corners with a homography, and ask the rest of the grid where it landed. That number is zero at the true seat and positive everywhere else, and a search over the room finds the zero.

On a 700 mm monitor curved at a metre — the ordinary desk object — four seats scattered through the room come back with a worst error of three and a half nanometres — three and a half parts in $10^{9}$ of the radius — at a cost of $2\times10^{-10}$ pixels. That is arithmetic rather than measurement, which is the point: the seat is not approximately in the picture, it is in the picture.

In units of the screen, and not in metres #

The recovery returns the seat as a multiple of the screen’s radius, and that is not a convention chosen for tidiness.

Scale the whole arrangement — the screen’s width, its radius, the reader’s distance, the reader’s offset along the sofa — by the same factor, and the set of directions the reader receives does not change at all. Done with a factor of four, so that every operation is the same operation on the same mantissa with a shifted exponent, the two pictures agree bit for bit. At a factor of 3.5, where the arithmetic itself has rounding in it, they agree to $2\times10^{-16}$.

So a screen 3.5 times larger, 3.5 times further away, in a room 3.5 times bigger, is not merely hard to tell apart from the original. It sends the identical picture.

This is the third time this collection has met that sentence, and the three arrivals are worth putting side by side because the mechanism is the same each time and the objects are not.

A mirror ball does not know its size: a ball 3.5 times bigger at 3.5 times the distance in a room 3.5 times bigger draws the bit-identical picture, and its outline gives r/d and refuses the size. The dome knows its offset in units of itself: a port’s picture depends on the ratio of its offset to its radius and on nothing else, to $9\times10^{-16}$ of a degree. And now a television.

The reason is a similarity group, and it has nothing to do with optics. Reflection is scale-free, refraction is scale-free, a pinhole is scale-free — and so is a cylinder. An arrangement built only from those has every observable a function of its dimensionless ratios, and a length can only enter where something breaks the similarity. The room breaks it. Knowing that the screen is 700 mm wide is what turns 1.2 radii into 780 millimetres, and that knowledge came from a tape measure rather than from the picture.

How sharply the seat is named #

A recovery that is exact on noiseless input is a statement about the geometry. What a reader wants to know is how far they may move before the picture stops being the one that was intended, and that is a different quantity: the width of the bowl rather than the depth of its minimum.

At one pixel of departure on a delivered picture 1,920 across, a curved desk monitor names its seat to a few millimetres sideways and about a centimetre along the sight line. A curved television at two and a half metres allows a few centimetres sideways and a quarter of a metre in depth. A cinema screen, whose curve is gentle beside its size, allows more.

Two things follow from that spread, and only one of them is comfortable.

The comfortable one is that the number scales with the object, as everything above says it must: the monitor and the cinema screen differ by a factor of twenty in both directions and the ratio between the two directions barely moves.

The uncomfortable one is that nobody sits still to a few millimetres. A reader at a desk breathes, leans, and reaches for a cup; a reader on a sofa moves further than the whole acceptable region between one scene and the next. So the delivered picture is, essentially always, not a projection of anything — and nobody notices, which is a fact about seeing rather than about projection and is exactly the boundary the wide-angle essay is careful about.

Every transverse direction costs the same #

The cost of a wrong seat turned out to have a property the first draft of this essay predicted incorrectly, and the correction is worth more than the prediction was.

The expectation was that a vertical cylinder would be forgiving of height: its generators are vertical, so moving the eye up and down leaves the horizontal geometry alone, and the acceptable region should be a slab. It is not. A step upward and a step along the sofa cost the same number to twelve digits, and so does a step at forty-five degrees to both: the anisotropy over twenty-four directions is $4\times10^{-14}$, which is nothing.

The reason is visible once the wrong prediction is discarded. A transverse step of any direction slides the eye against the screen’s own relief — the near part of the glass moves against the far part — and the part of the resulting map that is not a homography is set by how far the eye moved and by how much relief there is. Which way the step was taken decides only which way the relief slides.

A step along the sight line is different in kind rather than in size: it scales the relief instead of sliding it, and costs about a third as much.

That equality is not new to this collection either. Where the anamorph still works establishes that a step sideways and the same step upward cost a pavement anamorph precisely the same, and derives it from the central collineation a wrong eye composes the picture with. A curved television is not a pavement painting and its surface has a completely different symmetry, and the two directions agree there as well.

What the region looks like #

Collect the seats whose picture is within a stated tolerance and the result is a solid, which is the same object the room the eye may stand in measures for a pavement painting.

At one pixel the monitor’s region is about half a cubic centimetre — a slug of air a couple of centimetres long and seven millimetres across — pointing along the sight line. The television’s is a litre, at two and a half metres. And the volume goes as the cube of the tolerance allowed, because all three directions cost linearly in the distance moved: ten times the tolerance is a thousand times the room, which is the pavement anamorph’s own law arriving on an object that has nothing else in common with it.

The sharpness has a closed form, and it is three lengths #

The tolerances quoted above were measured by search. They also have an expression, and writing it down is worth the paragraph because it says which property of a screen decides them.

The whole effect lives in the glass’s relief. A cylinder of width $w$ and radius $R$ stands $w^{2}/8R$ nearer the eye at its edges than at its centre — 61 mm on the desk monitor. Move the eye sideways by $e$ and that relief slides against the picture: a screen point at $x$ picks up an extra image displacement $N(x/w)\cdot(x^{2}/2Rd)\cdot(e/x)$, of which the corner fit absorbs the constant and linear parts and leaves the quadratic. Evaluated at the screen edge, on a delivered picture $N$ pixels across at reading distance $d$:

$$\text{departure} \;\approx\; \frac{N\,e\,w}{8Rd}, \qquad\text{so}\qquad e_{\tau} \;\approx\; \frac{8\,\tau\,R\,d}{N\,w}.$$

Against the three screens measured: the desk monitor, 700 mm at 1000R read from 650 with 1,920 pixels, gives 3.9 mm — a few millimetres. A 1,230 mm television curved at 4000R and watched from 2.5 m gives 34 mm — a few centimetres. A twelve-metre cinema screen at 20,000R from fourteen gives 97 mm. All three are the numbers the search returned, from one expression with no fitting in it.

Read as a rule the expression is a little startling. Sharpness is proportional to the radius, so the flatter the screen the worse it names its seat, going to infinity exactly where the relief goes to zero and the recovery fails altogether — the two ends of this essay’s first finding turn out to be the same formula at its two limits. And every length appears as a ratio: $e/w = 8\tau (R/w)(d/w)/N$ has nothing dimensional left in it, which is the similarity argument above arriving a second time, as arithmetic rather than as an experiment.

The along-the-sight-line direction comes out of the same expansion. A step $\delta$ toward the screen scales the relief rather than sliding it, leaving a residual $\propto Nw^{2}\delta/Rd^{2}$, so the two costs stand in a ratio proportional to $w/d$ and to nothing else. Measured, the constant is about three tenths:

$$\frac{\text{cost along}}{\text{cost across}} \;\approx\; 0.3\,\frac{w}{d},$$

giving 0.32 for the monitor and 0.15 for the television against the 0.32 and 0.15 the search returns. The constant is not 1/2 because the two residuals are different shapes — the sliding one is quadratic across the picture and the scaling one cubic — and the corner fit absorbs more of one than of the other; the scaling is the part derived, and it is the part that transfers.

What it says is that the ratio depends only on the screen’s own half-width in units of its viewing distance — the angle it subtends, and not a fact about its curve at all. That is why the ratio barely moves between a monitor and a cinema screen while every other number in this essay moves by twenty: displays are chosen to subtend comparable angles whatever their size. A screen names its depth loosely for the same reason a photograph does — the depth direction is the one the picture is short in.

Multiply the three: the acceptable solid is an ellipsoid of semi-axes $e_{\tau}$, $e_{\tau}$ and $e_{\tau}d/0.3w$, so

$$V \;=\; \frac{4\pi}{3}\cdot\frac{d}{0.3\,w}\,e_{\tau}^{3} \;\propto\; \tau^{3},$$

which is the cube law the log plot shows, now with its constant. For the monitor that is 0.56 cm³ and for the television 1.0 litres — against the half a cubic centimetre and the litre the search found. The cube is not an empirical slope; it is three directions each linear in the tolerance, and the essay’s remark that ten times the tolerance buys a thousand times the room is the only thing three independent linear costs can mean.

Why this is the anamorph result and not a new one #

Stated in the field’s own vocabulary, the finding is short. A curved screen is an anamorph, and the content is the design.

An anamorph is a picture that is correct from one point, and the reason it names that point is that its marks lie on a surface which is not the picture plane — so a wrong eye composes the intended picture with something that is not a projective map of the picture, and the departure is readable. A flat pavement painting has a fold-free surface and gives back the place and not the height; a corrugated floor buys the height back, because curvature is exactly what breaks the projective family.

A curved television is a curved surface carrying a picture computed for a flat one. That is an anamorph nobody designed, made by a manufacturer’s decision about glass, and it names its eye for the same reason the pavement painting does.

What it is good for, honestly #

Three uses, in decreasing order of how much they are worth.

It settles what a correction can do. A display can pre-warp its content for one seat. Because the seat is in the picture, the correction is a statement about which seat, and correcting for one seat is not a favour to any other. That is measured in the essay on what one correction leaves an audience, and it is a consequence of this one rather than a separate finding.

It gives a curved screen a calibration nobody has to supply. A camera watching a curved display sees a picture whose departure from a homography names the point it was watched from — so the geometry is recoverable without a chart, a marker or a measurement of the room, up to the one scale the room has to give. Whether that is useful depends on whether anything is watching a screen, which is a question about products rather than about geometry.

And it tells a reader what their own eyes are being sent, which turns out to be the most interesting of the three, because a reader has two of them and they are at two different seats. What the two eyes are sent is that measurement.

What the measurement assumes #

Four things, and all four are stated rather than hidden.

The screen’s shape is known. A radius, a width, and that the surface is a cylinder. Without it the picture constrains the pair — shape and seat together — and one picture cannot separate them, which is the same one-view ambiguity two views give shape and no size is about.

The content is known. The recovery compares what arrived against what was intended, so it needs the intended picture. A viewer watching a film does have it, in the sense that the film is a projection of a scene and its straight lines are straight; a reader with a photograph of a screen and no idea what was on it does not.

The observation is noiseless here. Every number above is arithmetic. Real clicking on real pixels would put a floor under the recovery and the honest way to report that is the width of the cost bowl, which is what the sharpness figure is.

And the reader is a point. They are not; they are two points, at a fixed spacing, and the pair is a stronger instrument than either alone.

The short version #

A flat display, seen from anywhere, is a correct picture of something. That is why it does not name a seat: everything it could tell a reader about where they are has been absorbed into a scene that would look exactly the same.

A curved display, seen from anywhere, is a picture of nothing — and a picture of nothing is a much more informative object, because the particular nothing it is a picture of is a function of one point in the room.