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

The screen is a picture surface too

A curved television is one of the six named picture surfaces sitting in a living room, and from its own axis it delivers azimuth in proportion to the picture, exactly. What it is shown is a rectilinear picture from a sofa, and the difference is not a matter of degree — a flat screen from any seat shows a homography of the intended picture, so it is a correct picture of a transformed scene, and a curved one shows a map that is not a homography from any seat at all.
24 min read 20 figures The surface is a choiceThe viewer is in the geometryNot a projection at all

Worth reading first: A wide field on a small screen · The point you have to stand at · When the picture surface is not flat.

Everything the screen field has said so far treats a display as a rectangle of a stated width at a stated distance. That is all the arithmetic needed: the chain from sensor to screen cares how wide the picture is and not what shape it is, and the answer — that almost nobody is standing where the picture says to — comes out of two lengths and a division.

A screen that curves breaks that, and it breaks it in a way this site already has a whole field for. A curved display is a picture surface: the pixels are equally spaced along the glass, so the arc length across the screen is the picture’s width, and a viewer on the axis receives azimuth in proportion to the picture coordinate. That is the cylindrical picture surface, in a living room.

A curved screen, from above, with the seat and the axis markedA 700 mm screen bent to a radius of 1000 mm, seen from above, wrapping through 40.1° of arc. The lower mark is the seat, 650 mm from the middle of the glass; the upper one is the centre of curvature. From the centre of curvature the screen delivers azimuth proportional to the picture coordinate, to 2.8e-17 radians, so it is exactly a cylindrical picture surface — and from anywhere else it is not, by 0.216 radians at six tenths of the radius. Four picture points fix a homography from the seat and the fifth misses it by 51.4 px of a 1,920-pixel picture, which is what the next figures are about.the seatcentre of curvatureR = 1000 mm · seat at 1521 mm22.0 px off a homography
Fig. 1 A curved monitor from above, with the seat and the centre of curvature marked. From the centre of curvature the screen delivers azimuth in proportion to the picture coordinate; from the seat it does not.

From the axis it is exact, and only from there

Put the eye at the centre of curvature and every screen point is at the same distance and at an azimuth proportional to its position along the glass. That is the cylindrical map’s definition, and the agreement is at the arithmetic floor — not a good approximation, the same function.

Move the eye off that point, along the axis, and it stops. The azimuth is no longer proportional to anything; at six tenths of the radius the departure is degrees rather than parts per million.

A curved screen, from above, with the seat and the axis markedA 700 mm screen bent to a radius of 1000 mm, seen from above, wrapping through 40.1° of arc. The lower mark is the seat, 1000 mm from the middle of the glass; the upper one is the centre of curvature. From the centre of curvature the screen delivers azimuth proportional to the picture coordinate, to 2.8e-17 radians, so it is exactly a cylindrical picture surface — and from anywhere else it is not, by 0.216 radians at six tenths of the radius. Four picture points fix a homography from the seat and the fifth misses it by 33.4 px of a 1,920-pixel picture, which is what the next figures are about.the seatcentre of curvatureR = 1000 mm · seat at 1000 mm33.4 px off a homography
Fig. 2 The seat moved to the centre of curvature. The rays from the eye to equally spaced picture points are now equally spaced in angle, which is what makes the arrangement a cylindrical projection and nothing else does.

So there is one exact arrangement and it has two conditions: the eye is on the axis, and the picture was drawn for a cylinder. Neither is what happens. A television at four metres of radius is watched from two and a half, and it is shown a rectilinear picture, which is what every camera and every renderer produces.

The same 110° 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 544 px.flat picture plane — straight lines stay straightcylindrical picture surface — even stretch, bowed linesone scene, two picture surfacesneither is the distorted one
Fig. 3 The two surfaces, from the curved field, drawn at the same angular width. A curved screen is asking the reader to accept the right-hand one, and it is being sent the left-hand one.

Two families stay straight, and it is the cylinder’s own two

A drawn line on a curved screen is seen straight when the eye and the line are coplanar. Two families manage it and they are not chosen for convenience.

A horizontal line lies in a horizontal plane, and a viewer at the same height is in that plane. It is seen straight, exactly.

A vertical line is one of the cylinder’s own generators, which is a straight line in space. It is seen straight, exactly.

Everything between them is a curve on the cylinder and is not coplanar with anything.

Two families stay straight and everything between them bendsThe departure from straightness a 700 mm screen of radius 1000 mm puts into a drawn line, against the line's own angle on the picture, seen from 650 mm. A horizontal comes back at 0.0e+0° and a vertical at 3.5e-31°, which are arithmetic zeros rather than small numbers: a horizontal lies in a plane through the eye and a vertical is one of the cylinder's own generators. Between them the bend rises to 0.0485° at 33°. This is the same straight family lib/surfaces.js measures on a cylindrical picture, and a curved television is that surface with a sofa in front of it.00.0020.004020406080the drawn line's own angle on the picture (degrees)seen bend (degrees)0.0044°R = 3300 mm, seen from 650 mmtwo exact zeros
Fig. 4 The departure from straightness against the drawn line’s own angle on the picture, from horizontal to vertical. Two exact zeros, and a maximum well before the middle.

Those are the same two families the curved field measures on a cylindrical picture, arriving here without being sent for. A curved television is not similar to a cylindrical picture surface in the way a metaphor is similar; it is one, so its straight family is the cylinder’s straight family, and a reader who knows the field already knows which lines will bend.

The maximum is not in the middle either, which is worth a moment. A drawn line at fifteen degrees off horizontal bends more than one at forty-five, because the bend depends on how much of the line’s length is running round the curve and on how far the two ends are from the eye’s own plane, and those two peak at different angles.

The lines each surface leaves aloneA curved picture surface does not bend everything. Each panel draws the family of world lines the surface images as straight lines: two-dimensional for the plane, and a one-parameter family for every other surface here — running through the picture's centre on an azimuthal surface, and parallel on a cylindrical one.planeevery line—cylinderone parameter · parallelno meeting point1 of 2 keep a curvethe signature has three values, not eight
Fig. 5 The general form, from the curved field: every surface keeps some family of lines straight, and which family it is, is the surface’s signature.

Where to sit, and the answer is not in the room

For a picture drawn for a cylinder there is an exact seat. For an ordinary picture there is a best one, found by search rather than by formula, and the search is over how far back the viewer sits.

The best seat for an ordinary picture is not in the roomHow far a curved television's picture is from being a projection of anything, against where the reader sits: four picture points fix a homography and the worst of the other forty-five is plotted, in pixels of a 1,920-pixel picture, for a 1,230 mm screen bent to 4000 mm. The minimum is at 6.76 m and leaves 3.8 px; the distance the picture's own field of view asks for is 1.69 m and leaves 15.1 px; the sofa, at 2.6 m, leaves 9.8 px. There is no seat at which the number is zero, because a rectilinear picture on a cylinder is not a picture from anywhere — the best seat is the one that curves least, which is the one furthest away.0102030246how far back the seat is (metres)distance from a homography (px)the sofa: 5.3 pxR = 7500 mmnever zero
Fig. 6 How far a curved television’s picture is from being a projection of anything, against where the reader sits. The minimum is not at the distance the picture’s field of view asks for, and it is not on any sofa.

The search is over the seat’s distance and the cost it minimises is the residual against a homography, which is the measurement the next section is about. It is worth noticing that the search returns a minimum rather than a zero, and bestSeatFor asserts exactly that: the best cost has to be positive. A search that reported zero would mean the map had become projective somewhere, and it never does.

The best seat is nearly seven metres from a television a metre and a quarter wide. That is not a recommendation; it is the plot’s way of saying the obvious thing, which is that the further back a viewer sits the less the curve does, and the limit of the best seat is the seat at which the screen might as well be flat.

Three candidate seats are worth naming because a reader will have heard the first two argued for.

The centre of curvature is the one the marketing describes: every part of the screen at the same distance. It is the exact seat for a cylindrical picture and it is four metres back from a domestic television, which is further than most rooms.

The station point is the one the picture asks for: the distance at which the screen subtends the field of view the picture was rendered at. It is the shortest of the three and the worst of the three, because the curve does most at close range.

And the sofa, which is where the viewer is, and is in between.

A wide render read from a screen that subtends much lessA 27-inch monitor at 650 mm subtends 49.3°. A picture rendered at 65° is therefore being read from 1.39 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°65° → depth ×1.3927-inch monitor at 650 mmsubtends 49.3°
Fig. 7 The flat version of the same question, from earlier in this field: the picture’s own station point against where anybody actually sits. On a flat screen sitting wrong stretches depth and moves no mark; here it does something else.

The difference that is not a matter of degree

Everything above is a quantity, and a reader could reasonably conclude that a curved screen is a flat screen with a small extra error on it. It is not, and the distinction is this essay’s headline.

Sit off to the side of a flat screen and the picture reaches the eye through a projective transformation — a plane seen from a different centre is a homography of itself, which is the keystone and is exactly the same statement as the picture plane being a choice. Four picture points fix that homography and every other point lands where it predicts, at the arithmetic floor, from every seat in the room including one a metre and a half off to the side where the distortion is violent.

So the off-axis viewer of a flat screen is seeing a correct picture of a differently-shaped scene. That is this site’s recurring answer and it holds here without qualification.

A flat screen is a homography from every seat and a curved one from noneFour seats, two screens of the same width, and one measurement: four picture points fix a homography and the worst of the other forty-five is reported, in pixels of a 1,920-pixel picture. Every flat row is an arithmetic zero, including the seat 1.5 m off to the side where the picture is a violent keystone — a keystone is a homography, so the picture is still a correct picture of a scene that has been transformed. Every curved row is pixels: 9.8, 42.9, 27.0, 28.9. That is not a worse version of the same thing; it is the other kind of thing.flat · seat 10curved · seat 19.8flat · seat 20curved · seat 242.9flat · seat 30curved · seat 327.0flat · seat 40curved · seat 411.61,230 mm wide, flat and at R = 4,000 mmflat: exact at every seat
Fig. 8 Four seats, two screens of the same width, one measurement: four picture points fix a homography and the worst of the other forty-five is reported. Every flat row is an arithmetic zero and every curved row is pixels.

Sit anywhere in front of a curved screen and the map is not a homography. A cylinder is not a plane, and no centre of projection turns one into the other; the fifth point misses by pixels, from every seat, including the axis.

That is not a worse version of the flat case. It is the other kind of thing: the flat screen’s error is a change of scene, and the curved screen’s is not a scene at all.

The picture plane tilted 14°Pointing the camera up tilts the picture plane with it, and three things happen at once: the verticals converge — 3.59° between the outer two — the horizon drops 213 px below the middle of the frame, and the vertical vanishing point arrives at 3425 px from the principal point. They are one fact: the product of those two offsets is f².correct from 20 cm, at 160 mm wideverticals converge 3.59° · horizon 213 px off centre
Fig. 9 The result the flat case leans on. One eye and two picture planes give a homography of the picture for any scene — so a plane seen from the wrong place is a picture of a room that has been transformed, and every mark is accounted for.

And no correction fixes it for two seats

The obvious repair is to pre-warp: apply a fixed map to the content so that one chosen seat sees the intended picture. Projectors do this and panels can.

On a flat screen the correction is a homography, and composing two homographies gives a homography — so correcting for the near seat leaves the far seat seeing a projective transformation of the picture, which is still a correct picture of a scene. The second viewer loses nothing that a viewer of an uncorrected flat screen had.

On a curved screen the correction is not a homography and neither is what the second seat is left with. Correcting for one seat more than doubles the residual at the other.

Correcting for one seat takes nothing away, because there was nothing thereFour numbers, in pixels of a 1,920-pixel picture, for two seats 1.5 m apart. A flat screen shows a homography of the picture to both of them, at the arithmetic floor, and still does after being pre-warped for the first — because composing two projective maps gives a projective map, so the second viewer keeps a correct picture of a differently-shaped scene. On the curved screen the second seat is 20.7 px from a picture before the correction and 47.1 px after it. The correction more than doubles the error and the finding is not that it damaged something: the second seat never had a picture to damage.flat · uncorrected, at seat B0flat · corrected for seat A0curved · uncorrected, at seat B20.7curved · corrected for seat A47.1two seats, one screenflat: exact · curved: 47.1 px
Fig. 10 Four numbers for two seats. The flat rows are arithmetic zeros before and after the correction; the curved rows are pixels before and more pixels after.

The finding is not that the correction damages the second viewer, and stating it that way would be a small mistake worth avoiding. assertTheCurvedScreenHasNoSharedAudience checks the third thing explicitly: the curved screen’s uncorrected picture is not a homography for the second seat either. There was nothing there to damage. What the correction does is take the picture from wrong-for-everybody to right-for-one-and-worse-for-the-rest, and calling that a loss requires forgetting what the starting point was.

The render is bent so the lens can straighten itThe pale grid is what the eye is meant to receive. The dark one is what the renderer actually draws — the same grid pushed through the inverse of the lens — so that the lens's own distortion undoes it. Every node returns to within 4.6e-13 px of where it started. This is the lens field's polynomial run the other way round, and it is the one place on this site where distortion is introduced on purpose.k₁ = -0.15 · the round trip closes to 4.6e-13 pxpale: what the eye receives · dark: what the renderer drawsthe inner 86% of a 72° frame, where the inverse is exact
Fig. 11 The correction on a flat surface, from the screen field: a pre-warp that costs resolution and buys a rectangle. On a plane it is a projective map and composes; on a curve it does neither.

The measurement, and why it is four points and not a fit

The instrument used throughout is one this site keeps reaching for and it is worth stating plainly, because a different one would have measured a different thing.

Four correspondences determine a homography — eight degrees of freedom, two equations each — so four picture points and their perceived positions fix the map exactly, with nothing left over. Then the other forty-five points of the grid are asked where they landed. On a flat screen they land where the four predict, at the arithmetic floor. On a curved one they do not.

The alternative is a least-squares fit over all forty-nine, and it is worse for a reason that has nothing to do with accuracy. A fit returns a compromise, and its residual is a mixture of two things — that the map is not a homography, and that the fit split the difference between the points. Four-and-predict separates them: the residual at the four fitted points is zero by construction, and everything else is the map’s own departure.

That is exactly the instrument the shadow on a curved floor uses, and it was written there for the same reason. A reader who has met it once will recognise the shape of the answer: fitted points exact, predicted points out by the surface.

What the receiving surface costs a four-point fitThe same lamp and the same occluder, cast onto four surfaces. Four marks of the shadow are matched to four known points of the occluder, the map they determine exactly is built, and the other sixty-eight points are predicted by it. On the plane the prediction is right to 3e-13 mm; on the others it is not, and the fit is still exact at the four everywhere — 2e-13 mm — which is what makes the rest a prediction rather than a residual.a flat floor3e-13 mma homology — four points determine ita dished floor5.67 mmnot a homologya ridged floor9.07 mmnot a homologya floor with a step74.95 mmnot a homologyworst error of the predicted pointsexact at the four fitted, on every surface
Fig. 12 The same instrument in the light field. Four marks fit a map and the rest of the outline is asked where it landed — exact on a plane, millimetres out on anything else.

The dome, where the surface does something a cylinder does not

A cylinder unrolls. Cut it along a generator and lay it flat and nothing stretches — which is the developable surfaces result and is why the pixels can be equally spaced along the glass in the first place, with no distortion introduced before anybody looks.

A sphere does not. An image laid on a dome is stretched before a viewer exists, by an amount that is a property of the surface, and the stretch between the middle and the rim of a hemispherical dome is a fifth again.

Gaussian curvature decides whether a floor can be unrolled at allA surface can be laid flat without stretching exactly when its Gaussian curvature is zero everywhere — Gauss's theorem, which says that quantity survives any bending. Three of these four floors have none, including the ridge, which curves visibly. The dish has 0.0144 per square metre, and no cleverness in the flattening removes it.floorGaussian curvature, worst over the patchcan it be unrolled?a flat floor0yes, exactlya ridged floor0yes, exactlya floor with a step0yes, exactlya dished floor0.0144by nothing whateverfour floors, k = 0.06three at zero, one at 0.0144 m⁻²
Fig. 13 The distinction, from the curved field: a surface that flattens without stretching and one that does not. A cylindrical screen is the first and a dome is the second, and the difference happens before the viewer.

So a dome has two problems where a cylinder has one. It is not a homography from any seat, like the cylinder; and its content has been stretched non-uniformly to get onto it, which no cylinder does. Reporting both as “distortion” hides that one of them is a viewing problem and the other is a printing problem.

The vault refuses the projective descriptionThe same design and the same eye, cast onto a flat floor and onto a barrel vault of radius 5.2 m. The best homography fitted to the floor's marks misses by 1.4e-15 m; fitted to the vault's it misses by 465.4 mm, which is 5.7% of the marks' own extent, and no choice of four marks helps.barrel vaulteye · 1.62 mon the floor1e-12 mmon the vault465.4 mmworst miss of the best homography, log scalevault radius 5.2 m5.7% of the extent, against 4e-14%
Fig. 14 The same surface in the viewing field, where the design is computed to be right from one point. A painted vault solves the dome’s problem by giving up on the second seat entirely, on purpose.

What the curve is actually for

None of this says a curved screen is a bad object, and it is worth saying what the geometry does support.

The curve does one thing exactly: at the centre of curvature every part of the screen is the same distance from the eye. For a display whose focus matters — a monitor at arm’s length, where the eye refocuses between the middle and the edge of a wide panel — that is a real ergonomic property and it has nothing to do with projection at all.

And a curved screen delivers more angular width for the same viewing distance, because the edges wrap toward the viewer. On a cinema screen that is most of the argument — and it is the same argument the cylinder makes against the plane about how much of the world a surface can hold before it runs away.

Where the picture is correct from, and where the reader isA 35 mm lens on full frame is 54.4° across, and the print is correct from focal length × display width ÷ sensor width. On a phone that is 66 mm and readers hold it at 350 — 5.29 times too far. In a cinema it is 11.7 m against a typical 14 m.35 mm on full frame · 54.4° acrossphone5.29×66 mm correctlaptop1.82×301 mm correct27-inch monitor1.12×580 mm correcttelevision2.17×1.2 m correctcinema1.20×11.7 m correct×1 — standing at the station pointhow many times further away the reader is than the picture's own station pointworst is the phone at 5.29×
Fig. 15 The chain the field is built on: a focal length, a sensor, a screen, and the distance the picture is correct from. The curve changes the last term’s meaning and not its arithmetic.

What the geometry does not support is the claim that a curved screen is more correct, or more immersive in any sense that has a number attached. It is correct for a cylindrical picture from the axis, and it is being sent a rectilinear picture and watched from a sofa.

The best seat for an ordinary picture is not in the roomHow far a curved television's picture is from being a projection of anything, against where the reader sits: four picture points fix a homography and the worst of the other forty-five is plotted, in pixels of a 1,920-pixel picture, for a 1,230 mm screen bent to 1500 mm. The minimum is at 6.76 m and leaves 10.2 px; the distance the picture's own field of view asks for is 1.69 m and leaves 40.9 px; the sofa, at 2.6 m, leaves 26.6 px. There is no seat at which the number is zero, because a rectilinear picture on a cylinder is not a picture from anywhere — the best seat is the one that curves least, which is the one furthest away.050100150246how far back the seat is (metres)distance from a homography (px)the sofa: 26.6 pxR = 1500 mmnever zero
Fig. 16 A tighter curve. The residual at every seat rises, so the more pronounced the curve the further the picture is from being a picture — which is the opposite of the direction the feature is sold in.

What this does not say

It does not say the numbers are large. A domestic television’s residual at a normal seat is around ten pixels of a nineteen-hundred-pixel picture, which is under a per cent and is nothing anybody would notice on a moving image.

It does not say what a viewer perceives. This site computes the geometry of pictures and says nothing about seeing, and a reader at the wrong distance in front of any screen usually does not notice and does not care.

It does not treat a screen wrapped far enough round to be a room. The arithmetic here refuses a screen past a quarter turn, because at that point the viewer is inside the surface and the questions change — which is where a set built for one eye takes over.

And it does not say a rectilinear picture is the right thing to send. A picture rendered for the cylinder, watched from the axis, is exact — and the one place that arrangement is routine is a planetarium, which uses a dome and gets the printing problem instead.

A display is a picture surface with a viewer in front of it. Ask which surface, and the question of where to sit stops being a matter of taste and becomes the same question the field has been asking of pictures all along.

A curved screen, from above, with the seat and the axis markedA 1230 mm screen bent to a radius of 4000 mm, seen from above, wrapping through 17.6° of arc. The lower mark is the seat, 2600 mm from the middle of the glass; the upper one is the centre of curvature. From the centre of curvature the screen delivers azimuth proportional to the picture coordinate, to 2.8e-17 radians, so it is exactly a cylindrical picture surface — and from anywhere else it is not, by 0.101 radians at six tenths of the radius. Four picture points fix a homography from the seat and the fifth misses it by 9.8 px of a 1,920-pixel picture, which is what the next figures are about.the seatcentre of curvatureR = 4000 mm · seat at 2600 mm9.8 px off a homography
Fig. 17 The domestic case, at the radius manufacturers actually build: seventeen degrees of arc, a viewer well inside the centre of curvature, and a picture that is not a projection from where they are sitting.
What keystone correction actually costsA projector turned 15° from square throws its rectangular panel as a quadrilateral. Correction cannot add light outside it, so it shrinks the picture until it fits — and 18.7% of the projector's pixels are thrown away. The fraction is measured on the panel rather than on the wall, because turning the projector makes the wall picture larger while making the panel usage smaller.15° of yaw, 6° of pitch, 1.50 throw ratio81.3% of the panel reaches the corrected rectangleouter: the thrown quadrilateral · inner: what correction can keep18.7% of the panel discarded
Fig. 18 The flat case in its own field: a projector off to one side, whose picture is a keystone and is therefore still a homography of the intended one.
One room at 120° across, cast onto six picture surfacesEach panel holds the same angular width of the same scene. The flat plane keeps every straight line straight and pays for it at the edges; the five curved surfaces bend the ground lines by 3.5% to 6.0% of their own length.plane — bend 0.0%cylinder — bend 6.0%stereographic — bend 3.5%equidistant — bend 4.5%equal-area — bend 5.1%equirect. — bend 5.7%120° across in every panelsame scene, same angle, six surfaces
Fig. 19 And the family the screen has joined. A curved display is a member of this set rather than a flat display with a defect, which is what makes every measurement in this essay available.
Where the picture is correct from, and where the reader isA 24 mm lens on full frame is 73.7° across, and the print is correct from focal length × display width ÷ sensor width. On a phone that is 45 mm and readers hold it at 350 — 7.72 times too far. In a cinema it is 8.0 m against a typical 14 m.24 mm on full frame · 73.7° acrossphone7.72×45 mm correctlaptop2.66×207 mm correct27-inch monitor1.63×398 mm correcttelevision3.17×820 mm correctcinema1.75×8.0 m correct×1 — standing at the station pointhow many times further away the reader is than the picture's own station pointworst is the phone at 7.72×
Fig. 20 The chain at a wider focal length, where the picture’s own station point comes in closer and the curve’s residual at that seat is correspondingly worse.

Where this goes next

50 mm on full frame · 39.6° acrossphone3.71×94 mm correctlaptop1.28×431 mm correct27-inch monitor0.78×829 mm correcttelevision1.52×1.7 m correctcinema0.84×16.7 m correct×1 — standing at the station pointhow many times further away the reader is than the picture's own station pointworst is the phone at 3.71× The screen sets the distance Every viewing distance this site has quoted was conditional on an assumed figure width. Replace the assumption with an actual chain — focal length, sensor width, display width — and the same 50 mm frame is correct from 9 cm on a phone, 83 cm on a monitor and 16.7 m in a cinema. Nobody is standing at any of them. fitted conic: 1.467 from being a circlecylinderthe samples, fitted The cylinder, and the price of going all the way round A cylindrical picture can hold three hundred and sixty degrees, keeps every vertical vertical, and bows every horizontal. Its cost is a stretch of sec φ in elevation, which is also the equirectangular surface's cost exactly — two surfaces that are always described as different and are identical in the one respect anybody notices. 15° of yaw, 6° of pitch, 1.50 throw ratio81.3% of the panel reaches the corrected rectangleouter: the thrown quadrilateral · inner: what correction can keep18.7% of the panel discarded A projector is a camera run backwards Turn a projector fifteen degrees from square and it throws a trapezium; keystone correction cannot add light outside it, so it shrinks the picture until it fits and discards a sixth of the panel. And the instrument itself comes back out of the picture it threw — 2880 panel pixels recovered against 2880, by the function the wrong field wrote for hand-drawn cubes.

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Computed from the collection, not written here: the essays that point at this one.

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Area scaleCylindrical projectionDemonstrationDevelopableDisplayHomographyKeystonePicture surfaceViewing distanceViewing position