A wider screen keeps its seat under one rule

A wide field on a small screen measured how far a rendered picture stands from its own viewing position. A picture rendered at 100° across on a 27-inch monitor at 650 mm, which subtends 49°, is read from 2.6 times its station distance, so the space it depicts is 2.6 times too deep. All of that took the screen’s shape as given. Wide screens have arrived since, in 21:9 and 32:9, and a renderer handed a wider screen has a decision to make that the single-screen account never meets.

The decision has two common forms, and the people who play on these screens have strong views about them. The first keeps the vertical field and lets the horizontal grow: more of the world appears at the sides. It is usually called hor+. The second keeps the horizontal field and lets the vertical shrink, so that a wider screen shows less above and below. That one is called vert−, and among players it is close to an insult, since a person who paid more for a wider monitor sees less of the game. The argument is conducted entirely in terms of how much of the world shows. It can also be asked what each rule does to the picture’s station point, and the answer turns the discussion around rather more than it should.

One seat, read off either axis #

A flat picture has one station point. It is on the axis through the picture’s centre, at the distance from which the screen subtends the field the picture was rendered at. That distance can be read off either axis, since both give the same place:

$$s = \frac{h/2}{\tan(v/2)} = \frac{w/2}{\tan(u/2)}.$$

The second form is the one that the wider field on a small screen used. The first is the one that matters here, because it has no width in it. Take three screens that are close to the same height, as modern monitors of these shapes are: a 27-inch 16:9, a 34-inch 21:9 and a 49-inch 32:9, all near 336 mm tall.

Under hor+ the vertical field does not change when the screen widens. It comes from the renderer’s reference setting, and at the common 90° across at 16:9 it is 59°. So the height is the same, the vertical field is the same, and so the station is the same: about 298 mm at each of these three screens. The picture on the wider screen has more on it, but it is the same picture on its central part, drawn from the same point in space.

That is the important property of hor+. It is often described as the rule that gives a person more, and that is true. What it does geometrically is to hold the seat still. The picture’s central region is identical across all three screens, which is a large part of why it feels natural to a player moving between them: objects are the same size on the glass, at the same place, with more added at the sides.

Under vert−, it is the horizontal field that holds. Then the second form of the formula applies and the station point moves back in direct proportion to the width. At 90° across, it stands at half the width from the glass: 299 mm on the 16:9 screen, 399 mm on the 21:9, 597 mm on the 32:9.

So it is possible to state what the two rules really are, in a way the usual comparison does not. Hor+ keeps the station and changes what is shown. Vert− keeps what is shown across the width and moves the station back. They are different answers to a different pair of questions from the one usually argued.

What it does at one seat #

Now put a person in a chair. The chair does not move when the monitor is replaced: on all three screens the eye is at about 650 mm, since that is where a desk puts it.

Hor+ gives a factor of 2.17 on the 16:9 screen, 2.19 on the 21:9 and 2.18 on the 32:9. Each is the same mistake to within what a monitor’s height varies by. Going from a 27-inch to a 49-inch monitor, a person keeps exactly the depth error they had before, as wide-angle is not distortion would describe it: each picture is correct from a place well in front of the person’s nose.

Vert− at 90° across gives 2.17, then 1.63, then 1.09. On the 32:9 screen the chair is within nine per cent of the correct place. Without intending it, a renderer following the despised rule has arrived close to the geometry that everybody who has thought about the matter says should be the goal. It has done so by showing the same 90° of horizon, spread across a much wider piece of glass, viewed from a distance at which the glass really does subtend about that angle.

This is not an argument in favour of vert−. The player’s complaint, which the next section takes up, is real. But the complaint and the geometry have parted company in an interesting way, in that the rule people want is the one that keeps the error, and the rule they reject is the one that repairs it.

What the rule pays for #

The complaint is about the world in the picture, and it can be counted.

Going from 16:9 to 32:9 under hor+, the scene grows from 1.42 to 1.82 steradians, an increase of 28 per cent. Under vert− it falls to 0.77, a loss of 46 per cent. These are large numbers, and they are the player’s point. A competitive player who can see 28 per cent more of the world from one screen than another has a clear advantage, and a person sold a wider monitor who then sees less above and below has a fair grievance. So this is a real trade between two legitimate goals: seeing more, and seeing it right. The usual discussion hears only the first of those, because the second has no champion. It does not show up in a screenshot.

The extra solid angle under hor+ has a character of its own. It comes at the edges, where a flat picture spends the most glass on each degree. On the 32:9 screen under hor+, the outermost ten degrees of the 127° field takes up a large share of the width. An object in that band is drawn at about twice its true proportion across. A viewer at the station sees it correctly, because the ray from the edge arrives from far off to the side. A viewer at 650 mm sees it from a much narrower angle than it depicts.

At the edge of the 32:9 screen, hor+ depicts 63.4° while the seat sees 42.6°, a difference of 20.9°. Vert− depicts 45.0°, and the difference is 2.4°. That is the wide monitor’s whole visual complaint, in one number. A person looking to the side on a very wide hor+ picture sees things that seem to swell and swing as they cross the frame, and the reason is that the outer third of the picture is built for an eye at 298 mm. The evenness a curve buys took a different route to the same stretch of glass: curving the screen puts its edges closer and turns them toward the eye, which improves the angle at which the glass is seen, but it does not touch this difference, because the picture is still flat in the renderer.

The field the seat itself wants #

Set the renderer’s field from the seat, and the difference is zero everywhere by construction.

That field is 49.4° across on the 16:9 screen, 63.0° on the 21:9 and 85.1° on the 32:9, with about 29° vertical on all three. Look at the shape of that. The vertical is constant, and the horizontal grows with the width. That is hor+ exactly. It is the correct rule for a wider screen at one seat. Its only fault is its starting value: 59° vertical at the reference, where this desk wants 29°. So the ideal is not a compromise between the two rules. It is the first one, with a much smaller field.

It helps to state the whole situation that way. A seat at a fixed distance has a correct vertical field, set by the screen’s height. Hor+ preserves whatever vertical field it was given as the width changes, and so it is the correct form for changing a screen’s width. It just inherits a setting that is about twice too large from the habit of 90° across. Vert− takes a large field and keeps it fixed as the screen widens, and it comes closer only because a very wide screen eventually grows to fit that large field at the seat’s distance. One of these rules is right for the wrong reason, and the other is wrong about the setting and right about the form. The screen at which vert− matches can be found by setting the station to the seat: at 90° across, a width of 1,300 mm, which at this height is an aspect of 3.9 to 1 and is not quite the sort of thing that sits on a desk.

Nobody plays at a matched field, for the reasons a wide field on a small screen gave: at 29° up a person sees very little of the world, and the sensation is of looking through a letterbox, not of being in a place. So what the player’s rule of thumb has really settled, in choosing 90° and then hor+, is that they would rather be shown much more of the world at a depth exaggeration of two than be shown it correctly with a small field. It is a defensible choice, and it has the merit of being stated rather than assumed once it is named this way.

A slider labelled in degrees #

There is a practical trap in all of this that catches almost everybody who sets a field in a menu. A number of degrees on a slider is a statement about one axis at one aspect ratio, and programs disagree about which. Three conventions are common. Some give the true horizontal field at the current screen’s aspect. Some give the vertical field. And a long-standing one gives the horizontal field the picture would have had if the screen were 4:3, whatever shape the screen actually is.

Those conventions differ a good deal. A slider reading 90 under the 4:3 convention means a vertical field of 73.7°, which at 16:9 is 106.3° across and at 32:9 is 138.8°. Its station on a 336 mm screen is 224 mm, so from the same chair the depth factor is 2.9, where the same number under the true-horizontal convention at 16:9 would give 2.2. A player who moves between two programs at “90” has moved the picture’s station by a third, without knowing it, and the two settings are not comparable in any way the numbers suggest. A published recommendation of a best field, given without its convention, says almost nothing.

The station point is a much more useful thing to quote, since it needs no convention at all. It is a distance, in millimetres, on the axis of the screen. It can be measured with a ruler against the chair, and its ratio to the chair’s distance is the depth factor, directly. A renderer could print it next to the slider with one line of arithmetic, and anyone who has followed the argument this far has reason to wish it would. The one number a rendered picture could print made a similar plea about the seat, and the case is stronger once screens of several shapes are in use, since an angle no longer travels between them and a distance does.

Where each rule is the right one #

Different uses want different things from this, and a few of them are clean.

Competitive play wants the most world, and hor+ delivers that. Its fixed station also means a player’s eye for sizes and distances transfers between monitors, which is a real advantage.

Cinema frames a picture as a director composed it, and the composition’s width is what is fixed. Vert− is the form a film takes, with letterboxing on a narrower screen and a matching reduction of the field. In film, nobody complains that a wider print has less vertical field, since it is the same shot.

Simulators and multi-monitor rigs are one case where this is treated correctly as a matter of course. A racing or flight simulator can be given the screen’s size and the seat’s distance. It then computes the matched field and often an off-axis projection for each panel, in the manner of a tile as an off-centre frustum. Speed and gaps judged from a matched picture are more accurate, and people who drive simulators seriously set it up that way, however narrow it looks.

Interfaces and anything to be read want vert− or an equivalent, with text anchored to the centre, since on a very wide hor+ picture text placed at the edge is not within comfortable reach of the eye.

When the seat moves, or the screens differ in height #

The screens have nearly one height, as these three do. A rule stated in terms of the vertical field is simplest in that case. Two monitors with one width and different heights behave oppositely, and vert− is then the rule that keeps the station.

The seat is held at 650 mm. People move back from a very wide monitor, a little, though with one of these on a desk they usually cannot go far. Moving back increases the factor under hor+ and pushes vert− past its match. Viewed from 800 mm, the 32:9 hor+ picture is 2.7 times too deep.

The eye is on the axis, and on a screen a metre wide nobody stays there. A person looks at the side of a 32:9 monitor by turning the head, and the eye may well be 100 or 200 mm from the central axis by then. That changes each figure here slightly, and it is what the curved screen is partly responding to.

These are pure rectilinear renderings, and a cylindrical projection, which some renderers now offer on wide screens, is a third rule that changes the outer picture rather than the seat. Drawn for the cylinder measured that case.

Still open: what a person’s head does at the edge #

Everything here treats a fixed eye, and on a screen subtending 85° at the seat that is untrue, since a viewer looks at the edge by turning the head. A turned eye is displaced from the axis, by an amount that depends on where the neck is, and so it stands further from the station point laterally as well as in depth. Standing in the wrong place measures the effect for a picture viewed off to one side, with a fixed gaze.

The measurement that settles what the turn does adds a pivot behind the eye and a gaze that follows a mark across the screen, and works out the mismatch between what is depicted and what is seen at the eye’s position for that gaze, for both rules. It asks whether a head turn makes the outer third of a hor+ picture better or worse than the figures above; whether it matters which side of the axis the player sits on; and, since a person who turns the head to see the edge is looking at it from a different seat, whether that seat is closer to the station than the fixed eye’s seat is.

The rule that gives a wider screen more world is the rule that keeps its depth error unchanged. The rule that gives it less world moves the seat back toward the chair. Neither is wrong; only one of those facts usually gets mentioned.