Visual Plane
every picture states the point it is correct from
The second projection

A wide field on a small screen

A picture rendered at a hundred degrees and shown on a screen that subtends forty-nine is being read from two and a half times its own station distance, so the depicted space is two and a half times too deep. The stretch at the edges everybody complains about is correct; the complaint is really that nobody is sitting where it would be invisible.
17 min read 9 figures Projected twiceThe viewer is in the geometry

Worth reading first: The point you have to stand at · Wide angle is not distortion.

Rendered pictures are made at fields of view nobody photographs at. Ninety degrees horizontal is ordinary; a hundred is common; a hundred and twenty is a setting people choose deliberately. On a camera those would be 18 mm, 15 mm and 10 mm lenses on a full-frame body — specialist glass, used rarely, and known for producing pictures that look strange.

Rendered pictures at those angles also look strange, and the complaint is well documented: things at the edge of the frame are stretched, spheres become ellipses, weapons held at the side of the screen look enormous. The usual response is to call it a distortion and to look for a fix.

This site has already argued that it is not a distortion. Wide-angle is not distortion shows that the stretch is the correct consequence of projecting a large angle onto a flat plane, that it is exactly 1/cos⁡θ1/\cos\theta in the shape of a sphere, and that it vanishes completely when the picture is viewed from the point it is correct from.

What this essay adds is the number that says how far from that point anybody actually is, and it is not close.

A wide render read from a screen that subtends much lessA 27-inch monitor at 650 mm subtends 49.3°. A picture rendered at 100° is therefore being read from 2.60 times its own station distance, and the viewing field has already established what that does: depth is stretched by exactly that factor and nothing in the picture changes. The two routes to the number — from two angles, and from a focal length and a display width — agree to 1e-9.051050100150field of view the picture was rendered at — degreeshow many times the depicted depth is stretchedthe screen subtends 49.3°100° → depth ×2.6027-inch monitor at 650 mmsubtends 49.3°
Fig. 1 The depth exaggeration against rendered field of view for a 27-inch monitor at 650 mm, which subtends 49.3°. A picture rendered at 100° is read from 2.60 times its station distance, so the depicted space is 2.60 times too deep. The vertical guide marks where the render would have to be to match the screen.

The screen’s own field of view

A screen subtends an angle, and it is one line:

θs=2arctan⁡ ⁣(Ws2D)\theta_s = 2\arctan\!\left(\frac{W_s}{2 D}\right)

for a screen of width WsW_s at distance DD. A 27-inch monitor is 597 mm wide; at 650 mm it subtends 49.3°. A laptop subtends about 31°. A phone at arm’s length subtends 11°.

That angle is the one the reader’s eye is actually receiving. If the picture on the screen was rendered at the same angle, the reader is at the station point and everything is exactly right. If it was rendered wider, the reader is too far back by the ratio of the tangents:

df=tan⁡(θr/2)tan⁡(θs/2)\frac{d}{f} = \frac{\tan(\theta_r/2)}{\tan(\theta_s/2)}

At 100° rendered on a 49.3° screen that is 2.60.

Two routes, and the reason for checking they agree

This is the point at which a new field can accidentally re-derive something the site already had, and the fleet’s own standard for a repair pass names that as a trap: a phase that restates its own earlier derivation in new vocabulary has added nothing and looks like it has added a field.

So the factor is computed twice by different routes and the two are asserted to agree.

From two angles, as above: the ratio of the tangents of the half-angles.

From a focal length and a display width, which is the viewing field’s route, unchanged since it was built: the picture’s focal length in display millimetres, scaled by the ratio of the screen’s width to the display width, against the reader’s actual distance.

They agree to 10−910^{-9}, which is the tolerance a solver’s noise floor allows. The check is in this field’s gate and its value is entirely in the fact that the two computations share no code: one knows about angles and the other knows about lengths, and if the screen field were merely restating the viewing field the check would be comparing a function with itself.

A 60° picture, 160 mm wide, read from various distancesThe picture is correct from 13.9 cm. Read from an ordinary reading distance of 40 cm it depicts a scene 2.89× deeper than the one it was made from, and no mark on the page has moved.024620406080100how far the reader's eye is from the page (cm)how much deeper the depicted scene becomescorrect at 13.9 cm40 cm → × 2.89at 160 mm wide× 2.89 at 40 cm
Fig. 2 The viewing field’s route to the same number. Depth factor against the reader’s distance for a picture of a stated field of view — the quantity that field computes from a focal length and a display width. The screen field arrives at it from two angles instead, and the two are asserted to agree.

What 2.60 means in the picture

It means the depicted space is 2.60 times deeper than the space that was rendered.

A corridor modelled as ten metres long is seen as twenty-six. A room modelled square in plan is seen as a room two and a half times longer than it is wide. A character standing three metres away is seen as standing eight metres away, at a size that says three — which is a contradiction the visual system has to settle somehow, and settles by making them look small.

That last one is worth pausing on, because it is the common complaint about wide fields of view stated in a form that can be computed. The complaint is usually “everything looks far away and the world feels large”. The measurement is that the depicted distance to everything is 2.6 times the modelled distance, while the drawn size is unchanged — so every object is depicted at 2.6 times its distance and its own size, which is to say at 2.6 times its real size, at the wrong place.

The same picture, read from 40 cm instead of 19 cmNot one mark has moved: the reconstruction re-projects onto the drawing to 6e-14 px. What has changed is the solid the drawing depicts — a cube at 18.8 cm, and 2.12× as deep as it is wide from 40 cm.the picture — identical at every viewing distanceplan: the true cube, and the solid depicteddepth × 2.12, width × 1.00correct from 18.8 cm at 160 mm wideread from 40 cm — depth × 2.12
Fig. 3 The stretch as a picture. The same drawing read from further than its station point depicts a deeper room, and the re-projection is exact — every stretched point lands on the mark already there. Nothing in the picture can reveal the error, which is why the complaint about wide fields of view is always about a feeling rather than about a discrepancy.

Why the setting is chosen anyway

The wide field of view is not a mistake and it is worth being clear about what it buys, because the argument is not that everybody should render at 49°.

It shows more of the scene, which for anything interactive is the whole point: peripheral awareness, seeing what is beside the viewer, not being surprised. A 49° window on a world is a letterbox.

It matches the eye’s own field better in one sense. Human vision spans about 200° horizontally with both eyes. A 49° screen is a small window into a world; a 100° render at least contains the angular content a person standing there would have.

And that second argument is where the confusion lives, because it is half right in a way that is easy to miss. The render does contain the right angular content. What it cannot do is deliver it, because delivery depends on the screen’s subtense, and the screen’s subtense is 49° whatever the render says. Putting 100° of content into a 49° window compresses the angles by the ratio of the tangents — which is the same 2.60, wearing its other hat.

So the choice is a genuine trade rather than an error: more of the world, at the price of a picture nobody is standing in the right place to read. Both halves are computable and the trade is between two quantities that cannot be maximised together.

How wide the picture gets as the field of view opensOn a flat plane the picture's half-width is tan(θ/2): it multiplies by 6.6 between 120° and 170° and is unbounded at 180°. On a cylinder it multiplies by 1.42 over the same range and keeps going past 180° without incident.0246850100150field of view across the picture (degrees)half-width of the picture, in focal lengthsplanecylinderstereographicequidistantcut off at eight focal lengthsthe plane crosses it at 166°
Fig. 4 The ceiling on the trade. A flat picture’s half-width goes as the tangent of the half-angle and is unbounded at 180°, so the strategy of rendering wider has a hard end. A 170° flat picture is eleven times wider than a 90° one at the same centre scale, and its edges are unusable at any viewing distance.

The angle at which the two meet

There is a rendered field of view for every screen at which the reader is exactly at the station point, and it is worth writing down because it is not a number anybody uses.

For a 27-inch monitor at 650 mm it is 49.3°. For a laptop it is 31.5°. For a phone at arm’s length it is 11.1°.

Every one of those is narrower than any camera lens a photographer would call normal, and much narrower than any rendered field of view anybody chooses. Which means the mismatch is not a matter of a poor default — there is no setting in common use that puts a reader at the station point of a screen, and there could not be, because the setting that would is a picture so narrow nobody would accept it.

That is the honest form of this essay’s finding. The complaint about wide fields of view is real, its usual explanation is wrong, and its correct explanation implies that the only fix is unacceptable for reasons that have nothing to do with geometry.

A wide render read from a screen that subtends much lessA 27-inch monitor at 650 mm subtends 49.3°. A picture rendered at 50° is therefore being read from 1.02 times its own station distance, and the viewing field has already established what that does: depth is stretched by exactly that factor and nothing in the picture changes. The two routes to the number — from two angles, and from a focal length and a display width — agree to 1e-9.051050100150field of view the picture was rendered at — degreeshow many times the depicted depth is stretchedthe screen subtends 49.3°50° → depth ×1.0227-inch monitor at 650 mmsubtends 49.3°
Fig. 5 The setting that would be right, drawn on the same curve. At 50° rendered on a 49.3° screen the depth factor is 1.01 and the reader is at the station point — and a 50° window is a picture in which a person can see almost nothing beside them. The geometry’s own answer is one nobody will take.

The two fixes that are not fixes

Two responses to the wide-field-of-view complaint are common enough to be worth pricing, because both change something real and neither addresses the mismatch.

Rendering onto a curved surface. The stretch at the edges of a flat picture is the flat plane’s doing — a cylinder or a sphere spreads the angles evenly and the edges do not run away. The curved field measures exactly what each surface costs, and the cost is straight lines: a cylindrical projection bows every horizontal that is not at eye level.

That is a genuine trade and it does not touch this essay’s number. The depth factor comes from the screen’s subtense against the render’s, and rendering onto a cylinder and then showing the result on a flat screen 49° wide leaves the reader exactly where they were. The stretch has been redistributed and the station point has not moved.

Narrowing the field of view for the middle of the picture only. Several variants exist — render the centre at a narrower angle and the periphery at a wider one, blend between them. This produces a picture that is not a projection of anything from anywhere, which is a real cost and is usually judged worth paying because the alternative is worse.

Both are reasonable engineering. What neither does is put the reader at a station point, because the reader’s position is fixed by furniture and the screen’s subtense is fixed by its size — and those two quantities, between them, decide the whole thing.

The same 120° view, projected onto a plane and onto a cylinderOn the plane every straight line stays straight (1e-13 px of bend) and the edges stretch; on the cylinder the stretch is even and straight lines bow by up to 449 px.flat picture plane — straight lines stay straightcylindrical picture surface — even stretch, bowed linesone scene, two picture surfacesneither is the distorted one
Fig. 6 The first response, priced by the field that owns it. A flat picture’s edges run away at a wide angle and a curved surface spreads them evenly, at the cost of bowing every straight line. It is a real trade and it changes what the picture looks like; it does not change where the picture is correct from, which is what this essay’s number is about.

The rule that was already about this

The wrong field measured a piece of received advice and found that it was right for a reason nobody states. The sixty-degree cone of vision is taught as a rule about drawing — keep the subject inside a 60° cone — and it is not a rule about the drawing at all. A 60° picture is correct from 0.87 of its own width; a 90° one from 0.50. The rule is a hedge against a reader who will stand further back than that, which most readers do.

This essay is that rule met from the other direction, with the reader’s actual position measured rather than assumed. And the two agree: a 49° screen wants a picture of about 49°, the taught cone says keep it under 60°, and the reason in both cases is the same one — a reader at a comfortable distance is a reader at the station point of a fairly narrow picture.

Which makes the taught rule better than it looks. It was formulated for paintings, by people with no way to compute anything in this essay, and it lands within about ten degrees of the answer the arithmetic gives for a modern screen. That is not a coincidence: both are constrained by the same fact about where people sit relative to things they look at.

What the sixty-degree cone is a rule aboutA 60° picture is correct from 0.866 of its own width — 13.9 cm at 160 mm wide. A reader at an ordinary 40 cm is 2.89× too far back and sees a scene 2.89× too deep. The rule cannot fix that; it only makes the error smaller by making the pictures narrower.0510255075100125field of view of the picture (degrees)how much deeper the scene looks, read from 50 cmno error60° → × 3.61160 mm wide, read from 50 cm60° is correct from 13.9 cm
Fig. 7 The taught rule with its subject restored. The cone of vision is a constraint on the reader’s distance disguised as a constraint on the picture, and reading it correctly turns a matter of taste into arithmetic. A screen is a picture whose reader’s distance is fixed by furniture, so the same constraint binds harder.

The one display that can meet its own station point

A headset is the exception, and it is worth stating why, because the reason is structural rather than a matter of tuning.

Every display in the table of the previous essay has a subtense set by two independent quantities — its width and how far away it is — neither of which the picture controls. A headset controls both. The optics fix the apparent size and distance of the panel, so the subtense is a property of the device, known in advance, and constant.

So a headset can be rendered at exactly its own subtense, and is. Which makes it the only display in ordinary use whose pictures are read from their own station points, and the only one where the whole of this essay’s argument has nothing to bite on.

That is a genuinely unusual position for a medium to be in and it is worth noticing what it costs to get there: the viewer’s head has to be clamped to the display. Brunelleschi’s peephole achieved the same thing by clamping the viewer’s eye to the panel, and the two are the same solution five hundred years apart — enforce the station point by making every other position physically unavailable.

Which suggests the general statement. A picture’s station point can be stated, or it can be enforced, and enforcing it has always required restraining the viewer. Everything else — a print, a page, a monitor, a cinema screen — states it and hopes.

Brunelleschi's panel, and where the eye had to beThe Baptistery is about 25.6 m across and the door it was painted from about 53 m away, so it subtends 27.2°. On a 290 mm panel that it fills 100% of, the picture is correct from 60 cm — which is a mirror at arm's length, or half of one, depending on the single number nobody knows.the piazza, in planthe Baptistery, 25.6 m53 m27.2°the cathedral doorthe panel, and the eye it needsthe Baptistery fills 100%the eye is 60 cm back — off this sheet290 mm panel · 27.2° of Baptisterycorrect from 60 cm · 27° across
Fig. 8 The first device to enforce a station point rather than state one. The panel, the drilled hole and the mirror fix where the eye must be, and the viewer has no freedom at all. A headset is the same solution with the restraint moved from the eye to the head, and it is the only display in common use that is correct from where it is looked at.

The one number a rendered picture could print

A closing practical thought, since this site’s whole convention is that a figure states the distance it is correct from.

A rendered picture knows its own field of view exactly — it is a parameter, not a measurement — so it could state its station distance in picture widths, once, at the moment it is made. A 100° render is correct from 0.42 of its own width; a 60° one from 0.87.

Nothing does this. A rendered frame carries no record of the angle it was made at, so a viewer wanting the number has to be told it separately or work it out from a setting menu. Which is the same gap the site’s own strip exists to close, and it is a one-line gap: the quantity is known, it is dimensionless, and printing it would let anybody compute their own depth factor from a tape measure.

That is not a proposal so much as an observation about what is missing. Of all the pictures in the world, rendered ones are the only category where the station point is known exactly by the thing that made them, and it is discarded.

What is not claimed

Two boundaries, because this essay is closer to the perception literature than most on this site and the site has been explicit since its foundation that it computes the geometry of pictures and takes no position on how one is seen.

Nothing here says a wide field of view looks bad. It says the depicted depth is stretched by a computable factor. Whether a viewer notices, minds, adapts, or prefers it is a question about people, and this site’s machinery has no standing to answer it.

And nothing here says a viewer is doing arithmetic. The depth factor is a statement about what the picture is a projection of from where the reader is. Whether the visual system reconstructs that projection, or uses other cues, or does something else entirely, is not a geometric question — and there is good reason to think it does not, since pictorial depth cues compete and viewers at the wrong distance mostly do not notice.

What the geometry supplies is the number, and a number is a better thing to argue about than an impression.

Seven identical spheres across a 84° frameThe outer sphere images 27% wider than the central one. That is what a correct rectilinear projection does, and it vanishes if the picture is viewed from 9 cm.54 px69 px84° across27% wider at the edge
Fig. 9 And the artefact everybody names, computed rather than deplored. Spheres across a wide frame are stretched by exactly 1/cos θ, which is correct — it is what a sphere at that angle does when projected onto a plane. The stretch is invisible from the station point and unavoidable from anywhere else, and both halves of that are this essay’s subject.

Where this goes next

54 px69 px84° across27% wider at the edge Wide angle is not distortion A wide lens stretches shapes at the edge of the frame by exactly 1/cos θ — 3% at 28° across, 41% at 90°. Every bit of that is what a correct rectilinear projection must do, and every bit of it disappears if the picture is viewed from the point it was made for. Nobody views it from there. the picture — identical at every viewing distanceplan: the true cube, and the solid depicteddepth × 2.12, width × 1.00correct from 18.8 cm at 160 mm wideread from 40 cm — depth × 2.12 Standing in the wrong place A picture read from twice the distance it is correct from depicts a scene twice as deep — and not one mark on the paper moves. The error is invisible in the picture, which is why it survives everywhere. drawn: 1.309 : 1subtended at the station point: 1.000000000 : 1the faint circle is the mean radius, for comparisonthe drawn centre sits 3.42 px from the axis's own mark The sixty-degree cone of vision Every book says keep the subject inside a 60° cone. Measured, the marginal stretch the rule is nominally about is exactly zero from the station point — 1.000000000000 to 1, over 720 sampled points. The rule is a statement about the reader, and books do not obey it.

Shares its objects with

Essays that name at least two of the same things, and that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

cone of visionDemonstrationDepth compressionEdge stretchfield of viewMarginal distortionStation pointSubtended angleViewing distanceViewing position