Picture surface — where it appears
Named by 39 essays across 7 fields — each of them below, with the objects they name alongside it.
No picture surface keeps everything
A picture has to be cast onto something, and every candidate surface destroys something different. Six of them are measured here on the same three properties, and the corner of the plot where a surface pays nothing is empty — not because nobody has thought of one, but because a theorem says there is none.
A straight line in a scroll is a hyperbola
Under a pushbroom the image of a straight world line is a Möbius function of the paper coordinate, which is a rectangular hyperbola. It is straight exactly when the line holds its depth — so a curve in a handscroll is a depth signal rather than a stylistic one, and the sag is computable in pixels.
When the picture surface is not flat
A flat picture plane keeps straight lines straight and stretches the edges without bound. A cylindrical one spreads the stretch evenly and bends every straight line that is not through the axis. Neither is the distorted one — they are answers to different questions, and the choice decides what a wide view can be.
The sky inside a cone
From under water the whole sky — every direction out to the horizon — arrives inside a cone of 48.61°. Outside it the surface is a mirror. That cone is a picture surface, and it has a distortion no surface in the curved field has — an area scale that runs to zero.
A scroll is not a panorama
Both draw straight world lines as curves, and one of them is a projection. A rotating eye keeps its centre exactly however far it turns; a translating eye has none at all. Curvature and centrelessness are independent properties, and conflating them is the standard mistake about both objects.
The arcs a curvilinear drawing uses
The taught way to draw a very wide view by hand is to run every straight edge of the world as a circular arc. That recipe has been repeated for sixty years without a surface attached to it, and it turns out to name one exactly — fitting a general conic to the image of a straight line returns a circle to nine decimal places under stereographic projection and returns nothing like a circle under any of the other standard picture surfaces.
One parameter between two surfaces
Wide architectural views are usually made on a projection with a number attached to it — a family running from the flat plane at one end toward the cylinder at the other, with everybody using the value one. That value has never been given a geometric defence. Measured across the family with the same battery of tests as every other surface, the worst angular error over the field has a minimum, and the minimum is at 1.04.
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.
Conformal is not undistorted
The most distorted-looking picture in ordinary circulation is the little planet — a 360 photograph re-projected from below, with the ground curled into a ball. Its worst angular error over 160 degrees of the sphere is 4.4e-8 degrees, which is arithmetic noise. Every crossing in the original crosses at exactly the same angle in the result, and what has gone is area, over a factor of 255.
The screen that names the seat
A flat screen shows a homography of the intended picture from every seat in the room, and an observer's own framing is free to be a homography too — so a flat screen's picture is consistent with every seat there is. A curved one is not, and the seat comes back out of the picture in all three directions, in units of the screen's own radius.
A pond in a scroll is not an ellipse
A pinhole draws a round pond as an exact ellipse whose widest row is 3.58 px off the row of the pond's centre — the drawn-circle error every perspective textbook warns about. A handscroll draws the same pond widest exactly on its centre's row, and draws it as a quartic that no conic fits: the best ellipse misses it by 3.08 px. Each keeps what the other loses, and off to one side the pinhole's pond leans 9.67 px while the scroll's does not lean at all.
Six flat pictures of everything
There is one way to photograph the whole sphere and keep every straight line straight, and it is to stop using one surface. Six flat pictures at ninety degrees cover everything, each of them a perfect pinhole, and the price is paid entirely at the seams — where a straight line does not bend but kinks, by an angle that reaches 45 degrees and is exactly zero for the lines lying in the seam's own plane.
The seats a screen will accept
Collect the seats whose picture is within a pixel of the one intended and the result is a solid — half a cubic centimetre in front of a curved desk monitor, a litre in front of a curved television. 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 lines a surface leaves alone
Only the plane draws every straight line straight, which is easy to measure and which gets read as though a curved surface bent everything. It does not. Every curved picture surface here keeps a one-parameter family of world lines exactly straight, and the family is a curve on the sphere of lines rather than a region of it — so a grid of samples finds none of it, which is what the first version of this measurement reported.
One picture and three people
A curved screen can be pre-warped for one seat, and the search over which seat to choose returns the middle one to a quarter of a per cent — there is nothing to be clever about. What the correction buys the sofa as a whole is six per cent, and the worst seat grows at fourteen pixels for every metre of audience, with no width at which it is zero except one person.
A curved screen is eight flat ones
A projection matrix is a plane and nothing else, so a curved display cannot be rendered — it has to be driven as several planes and assembled. The gap between chord and arc is the whole error, it goes as the square of the angle each piece spans, and the piece count therefore goes as the inverse root of the tolerance — three for eight pixels, eight for one, fifteen for a quarter.
The ceiling that is not a plane
Paint the same design for the same eye onto a floor and onto a barrel vault, then fit the best possible homography to each set of marks. On the floor it misses by femtometres, because the map is a collineation and four marks determine every other. On the vault it misses by half a metre, and no choice of four marks helps — which is where every projective construction made for a floor stops applying.
What the two eyes are sent
A reader's eyes are two seats sixty-three millimetres apart, so a curved screen delivers each of them a different map — and the part of the difference no homography absorbs is binocular evidence of the glass. Turned into a depth it comes back as the screen's own sag, 49 millimetres against 47 on a television, by a route that never saw the radius.
The evenness a curve buys
A curved screen is sold on evenness, and evenness turns out to be three quantities that disagree. On pixel pitch the curve wins from every seat; on the angle the glass is turned through it wins until three and a half metres along the sofa; on the plain distance from eye to glass — the reading the argument is usually made in — it gives up before half a metre.
A projector in the viewer's eye
A projector paints a wall along its own rays, so from the projector's own position the wall's shape is invisible — exactly, on a cylinder, on a dome and on a plane alike. Move a hand's breadth away and the residual is pixels. The one place a projector can stand and ignore the shape of what it is throwing onto is the eye of the person watching.
Matching buys one seat
Feeding a curved screen its own picture surface makes the picture exact at one point and worse everywhere else than the flat picture it replaced. Forty centimetres along the sofa the two cross, and past that the matched picture is the worse of the pair — because a flat picture is mediocre everywhere and a matched one is perfect at a point and falls away from it faster.
The pivot that is not the eye
A camera bolted to a tripod turns about the screw under its baseplate, and the light crosses somewhere else. Every ray of the panorama that results passes the pivot by e sin γ — the offset times the sine of how far off its own frame's axis the ray points — so the picture has a radius where a projection would have a point.
The parallax you cannot shoot away
A stitched panorama's parallax has two halves and they do not behave alike. The one across the seam falls as the sine of half a frame spacing, so more frames buy it off; the one up the frame is the sine of half the frame's own height, and no quantity of shooting touches it. They cross at π over β, which has no pivot error in it at all.
The camera that is a cylinder
A swing-lens camera turns its lens about its own entrance pupil and sweeps a slit across film bent into a circle concentric with it. Compute where the light lands, unroll the film, undo the pinhole's inversion, and the result is not similar to the cylindrical picture surface — it is the same map, to the arithmetic floor. What it pays instead is detail, and a shear on anything that moves.
A rig is right on one surface
Several cameras looking outward have several centres, and no warp registers all of a scene. The residual is a disparity, so the depth the stitch is computed for has an exact optimum — the harmonic midpoint of the depth range, not the middle of it — and the arithmetic middle costs a factor of 2 z_far over the sum, which tends to two.
What an eye can paint
A flight of steps has eighteen faces and no eye reaches more than fifteen. Pointed at a cluster of blocks, a seating rake and a corridor with a doorway in it, the same measurement finds 8 of 21, 7 of 13 and 6 of 7 — and the plane, which offers its whole self to every eye, is the control that makes the law a law rather than a fact about stairs.
Facing the reader is not being reachable
A face turns toward the eye or it does not, and that is a dot product any reader can compute. Whether the eye’s rays actually land on it is a different question with a different answer — on a seating rake, three faces of ten that face the reader receive nothing, and they are 27 per cent of the facing area. On a corner the same test loses nothing at all, which is what makes the gap occlusion rather than arithmetic.
Drawn for the cylinder, shown on the cylinder
Four essays in this collection have named the arrangement where a picture's surface and a screen's surface are the same surface, and none has run it. It is exact — a tenth of a millionth of an arcminute — and the exactness is not the finding. Three separate things have to be true at once, and a curved television's own maker prints a sitting distance where the first of them is out by five degrees.
The ball at the edge of the frame
A ball photographed near the edge of a wide picture is not drawn as a circle. It is an ellipse, longer along the radius from the centre of the picture than across it, and the centre of that ellipse is not the image of the centre of the ball. Both are properties of the flat sheet the picture is on, and an exact pinhole produces both.
Three conditions, and three prices
A matched picture needs the seat, the horizontal scale and the vertical law all at once. Each is broken alone here with the other two held, and all three turn out to be linear in the mismatch — the unforgiving case, with no margin at all. The vertical law costs a third of an arcminute on a television and eighty-four on a dome, because it is the difference between an angle and its tangent and that difference is cubic in the picture's vertical field.
A design that lands in two rooms
Cast a design down a corridor with a doorway in it and four per cent of the picture goes through the opening and lands on a wall three metres beyond, at 9.4 metres from the eye against the end wall's 6.4. The picture is continuous across the edge of the doorway and its scale jumps by half again, which is a corner anamorph's discontinuity with a gap in the middle of it instead of a fold.
The surface a screen wants
Six picture surfaces laid on one screen and viewed from the seat its curvature names, each with its own extents fitted so the ranking is about shape rather than scale. Each screen's own surface is exactly right on it and nothing else is — and on a curved television the runner-up is a fifth of an arcminute behind, which nobody can see.
A picture that can be printed
Two screens are fed the surface that is exactly right for each, so nothing about the viewer is left in the answer. What remains is whether the picture can be made flat before it goes up — and a cylinder unrolls while a sphere does not, so the dome's picture is stretched by eighteen per cent between its middle and its rim before anybody sits down.
One flight, two pictures
From the top of a descending flight every riser faces away, so the design lands entirely on treads. From the foot of the same flight the risers take 68 per cent of the design at a median stretch of 1.4, and the treads take the rest at a median of 2.6. Give each reader the faces the other cannot use and one staircase carries two pictures, with no face asked to hold both.
Where a surface spends its pixels
A picture surface is a budget before it is anything else, and the six named ones distribute the same marks over the same directions quite differently. The flat plane lays 25.0 times as many on a square degree at the edge of a 70° field as on one at the centre; the equal-area fisheye is flat to 8.3e-6 per cent.
The horizon's shape belongs to the surface
The horizon is one great circle of directions whatever draws it, and at zero tilt all six named surfaces draw it straight. Tilt the camera and they separate — and the cylinder, not the equirectangular surface, is the one whose horizon is exactly a cosine, to 9e-16 against 8.3e-3.
A pole is a line
An equirectangular surface sends one direction to a whole edge, so an 8° cap of sky at the pole takes 4.50 per cent of the marks against a 0.49 per cent share of the world. The worst singularity is not the pole at all — the equidistant fisheye's antipode costs ×17.9 — and only the cube map, which never holds a sphere in one chart, is bounded.
The kink at a seam
A cube map is six flat pictures, so every great circle is drawn exactly straight inside a face — to 3e-15 of its chord — and breaks at the join. The break is a kink and not a bend, it is exactly zero at a seam's midpoint whatever the slant, and it is bounded by 2·atan(½) = 53.130° at the corner.
Shot on one surface, shown on another
Two picture surfaces are two charts of the same pencil of rays, so a reprojection between them is a change of coordinates and loses no geometry at all — bit-exact at all 408 sampled directions. What it costs lies elsewhere — 70 per cent of the source has nowhere to go, and the target wants ×5.49 the marks at its edge.
Named alongside it
The objects these essays reach for when they reach for this one.
Area scaleDemonstrationEquirectangularAnisotropyCylindrical projectionfield of viewViewing positionHomographyAnamorphosiscentre of projectionConformalFisheye