A projector in the viewer's eye

A projector is a camera run backwards treats a projector as what it is — a centre of projection with a picture inside it — and prices keystone correction on a flat wall: turn the machine fifteen degrees from square and correcting the trapezium it throws costs a sixth of the panel.

Everything there assumes the wall is flat. This is what happens when it is not, and the answer is not a degradation of the first result. It is a different result with a different shape.

The exact point #

Start with the sentence that makes the rest of the essay follow.

A projector paints the wall along its own rays: each panel pixel travels in a straight line until it lands, and where it lands is where the wall happens to be. So a viewer whose eye is at the projector’s own centre looks back along those same rays, and receives each mark in exactly the direction the panel emitted it. The shape of the wall decided how far along each ray the mark sits, and how far along a ray a point is is precisely what a projection cannot see.

Measured on three surfaces — a curved television’s cylinder, a dome, and a flat panel — the residual from that position is at the arithmetic floor on all three, and the three are the point of the check. The exactness is not a property of gentle curvature; it is a property of the coincidence of two centres, and a dome is as exact as a plane.

Move the viewer 150 millimetres to one side and the residual is 3.7 pixels. So the exactness belongs to the point rather than to its neighbourhood, and a projector nearly at the viewer’s eye is nearly right rather than right.

Which is the definition of an anamorph #

Stated in the viewing field’s vocabulary, the paragraph above is not a new fact. It is the definition of an anamorph with the words in a different order.

Anamorphosis is only a viewpoint describes the construction: take an intended picture, run rays from a chosen eye through it, and mark where they land on whatever surface is there. That is a projector, with the panel as the intended picture and the projector’s centre as the chosen eye.

So every projected picture on a surface that is not flat is an anamorph, and its design eye is the projector’s lens. Pavement paintings, cathedral vaults and the picture a projector throws onto a rippled curtain are one construction, and this collection has now measured it from both ends: the pavement, the corner, the vault, and the projector.

The difference is only that a painter chooses where the reader will stand and a projector installer usually does not think of themselves as choosing anything.

Why keystone correction works on a plane and cannot work otherwise #

Keystone correction is a homography applied to the panel before the light leaves. On a flat wall that is exactly the right tool, and the reason is a composition that closes.

The map from the panel to a flat wall is a homography, because both are planes and the projector is a centre. The map from that wall to a viewer’s picture is a homography, for the same reason. So the whole chain — panel to wall to eye — is a homography, and a homography applied at the panel can cancel it. Whatever the viewer’s seat is, some correction makes the picture rectangular for them; the only cost is the panel it throws away, which is the earlier rung’s sixth.

Curve the wall and the first link stops being a homography. The chain is then not a homography, no map applied at the panel is its inverse, and the correction cannot exist. Not “is harder to compute” — does not exist, in the same sense that no polynomial straightens a curve that is not a polynomial.

What a correction can still do is make the picture right for one seat, by warping the panel so that the marks land where that seat needs them. That is the general anamorph again, and it is the same one-seat result the audience essay reaches for a curved television from the other direction.

How far a projector may stand from the eye #

The residual grows linearly with the offset between the projector and the viewer, which makes the practical question a division.

On a wall of a curved television’s radius, at a three-metre seat, the picture departs by one pixel at fifty millimetres of offset. On a dome it is twenty-five.

Fifty millimetres is the distance between a projector’s lens and the bridge of somebody’s nose if they are holding it against their face. So the practical reading of the exact result is that it is unattainable: the one place a projector could stand is occupied by the person watching.

That is not a defeat, and the reason is worth spelling out. The number is a tolerance on being a projection, which is the same quantity every other rung in this row reports, and a projected picture that departs by a few pixels is in exactly the same position as a curved television watched from the end of a sofa — geometrically not a picture of anything, and perfectly acceptable to look at. What the number does is settle who the design eye is. A projector installed at the ceiling has painted an anamorph for a viewer at the ceiling, and everybody in the room is at a wrong eye by two metres.

The wall that is nearly flat #

Between a plane and a dome there is the case every installation actually has, which is a wall that is flat to within a few millimetres, and the measurement above prices it without any extra work.

The residual is linear in the offset between projector and eye and, for a gently curved wall, linear in the curvature as well — so a wall bowed by a centimetre over three metres, with the projector two metres from the viewer, contributes a departure a reader can compute by scaling. What that arithmetic says is that ordinary building tolerances are invisible and deliberate curves are not, which matches what installers report and is worth having as a number rather than as experience.

It also says which walls are dangerous, and the answer is the ones that are locally flat and globally not: a wall with a shallow pilaster, a curtain with a fold in it, a room whose two halves meet at a slight angle. Those are piecewise planar, so each piece is separately correctable by a homography and no single correction handles two of them — which is the corner anamorph’s structure and produces the same visible signature, a picture that is right on one side of a join and sheared on the other.

The two centres are interchangeable #

The tolerances in this row and the tolerances in the curved-screen row are the same number measured twice, and noticing that saves deriving anything here.

A surface carries marks put there by one centre and read from another. The intended picture and the received one differ because the marks sit at different depths, and a depth difference seen from two places separated by a baseline $b$ at range $d$ shows up as a parallax proportional to $b \times \text{relief}/d^{2}$. That expression is symmetric in the two centres. It does not know which of them held the lamp.

So the closed form the curved-screen row derives — departure $\approx Nbw/8Rd$ for a surface of width $w$ curved at $R$, on a picture $N$ pixels across — applies here with the baseline reinterpreted. There it is how far the reader has moved from the design seat. Here it is how far the projector stands from the reader’s eye, and the fifty millimetres this essay measures on a wall of a curved television’s radius is the thirty-three millimetres that essay measures for a seat on a sofa, scaled by the difference in how wide the picture is and how far away.

Three things follow without further measurement.

The anisotropy transfers. A projector offset toward the wall costs about $0.3w/d$ of what the same offset across it costs, so hanging a projector directly behind a viewer’s head is far cheaper than hanging it beside them — a ceiling mount two metres above the audience is at full price, and a mount two metres behind them is at roughly a quarter of it. That is the one degree of freedom an installer usually has, and it is the good one.

The overlap of two projectors has a number. Two machines a metre apart, throwing a metre-wide picture at three metres onto a wall curved at four, disagree by about twenty pixels where their throws cross — the same expression with the baseline between the two lamps. That is why the second projector is coverage and not a second chance: the two pictures are each exact for their own centre and differ from each other by a quantity nothing at either panel can remove.

And the ceiling mount can be priced. Two metres between the lamp and the audience on that same wall is forty pixels, which is not a subtle effect and is invisible for the reason every other number in this row is invisible: nobody in the room has seen the picture the projector meant to throw.

Projection mapping, named #

There is an industry that does this deliberately, and it is worth naming because it is the one case where the geometry is understood by the practitioners and not by the audience.

A picture thrown onto a building, a car or a set of white boxes is designed so that the marks land where the designers want them from a chosen viewpoint — usually a camera position for the recording, and sometimes a seat for a live audience. The technique is called projection mapping and its geometry is the anamorph’s exactly: the projector is one centre, the design eye is another, and everything the audience sees is right in proportion to how close the two are.

Two consequences follow immediately from the measurement above and neither is folklore.

The recording looks better than the room. A camera placed at the design eye receives the exact picture and everybody else receives an anamorph read from the wrong place, which is why the same show is spectacular on video and merely interesting in person — the same asymmetry pavement paintings have.

Two projectors are not two chances. Adding a second projector at a different place adds coverage and brightness and does not add a second design eye: the two throws are both anamorphs for their own centres, and where they overlap the surface carries two pictures that agree only where the surfaces are flat.

The dome, where the audience is the argument #

A planetarium is the one place where this geometry is confronted head-on rather than lived with, and its answer is instructive: it moves the audience rather than the projector.

A dome’s projector sits near the centre of the dome, and the seats are arranged around that centre — so every viewer is close to the design eye in the only direction that is cheap to be close in, and the residual for a seat two metres from the projector on an eight-metre dome is far smaller than it would be for the same offset on a flat wall at the same distance, because a sphere seen from near its own centre is nearly the same from any of those places.

The tolerance measured above is twenty-five millimetres for a dome viewer at four metres, which sounds fatal and is a worst case over the whole surface. What it says exactly is that the picture on the dome is a projection of a scene for one point and a smooth distortion of one for the rest of the room, and that domes work for the same reason curved televisions work: nobody in the audience has ever seen the undistorted version.

What the wall gives back #

The measurement so far reads as a list of things a projected picture cannot be. It also has an inverse, and the inverse is the reason this rung is in the collection rather than in a manual.

Because the departure is a function of the offset between projector and eye, a photograph of a projected picture names that offset — the same recovery the curved screen’s seat makes, with the projector’s position taking the place of the intended content. And because the residual is zero at the design eye and grows linearly away from it, the measurement is well conditioned in exactly the region where somebody would want it.

So a room with a projector in it is an instrument: it measures where the camera is, in units of the wall’s radius, provided the wall’s shape is known. That is not a useful instrument — a tape measure is easier and always available — but it is the same instrument as the mirror ball’s and the dome port’s, and it returns the same kind of answer for the same reason: a ratio, with the size supplied by the room.

Why the projector and the camera are the same measurement #

One more symmetry is worth naming, because it is what lets this rung reuse the wrong field’s machinery unchanged.

A projector and a camera are the same object with the light going the other way, and every construction this collection has for one works for the other. The projector’s focal length comes out of the quadrilateral it throws by the function written for hand-drawn cubes; the wall’s shape enters exactly as a receiving surface enters a shadow problem; and the design eye of the resulting anamorph is the centre of projection of the machine.

That last identity is the one to carry. This collection has three constructions that place a picture on a surface from a point — a camera projecting a scene, a lamp casting a shadow, and an eye designing an anamorph — and they have been treated as three subjects with three fields. A projector is the fourth, and it makes the family’s shape hard to miss: the differences between them are about what is known and what is being solved for, and never about the geometry.

The lamp is the second eye makes the same argument from the shadow end, using a lamp as a second centre of projection to recover depth. A projector is a lamp whose shadow is a picture.

A projector is a lamp with a picture in it #

The last thing worth saying is a piece of vocabulary, because it explains why this rung was cheap to write.

Everything above uses machinery built for shadows. A projector is a point source; the wall is a receiving surface; the picture on the wall is what the source puts there along its rays; and where along each ray the mark lands is decided by the surface and by nothing else.

That is the shadow field’s own sentence, and it means the two fields’ results transfer without translation. A shadow’s mark is exact in rays and its plan reading is not — so a projected picture’s direction from the projector is exact whatever the wall, and any reading that involves where the mark is on the wall depends on the wall’s shape. The same division, in a room with a projector instead of a lamp.

It also means the projector inherits the shadow field’s refusals. A wall that is not flat breaks any reading of the picture that assumes a plane; a second projector is a second centre and its picture is a second, independent anamorph; and the surface cannot be recovered from the picture alone without a second view, because one centre and one surface leave a family of surfaces that would have produced the same marks.

That last point is the practical one for anybody trying to calibrate a projection onto an unknown surface: it takes a camera somewhere other than the projector, and the geometry of that pair is the two-view field’s rather than this one’s.

The short version #

A projector’s own position is the one place from which the shape of what it is throwing onto cannot be seen. That is exact, on a plane, a cylinder and a dome alike, and it fails at a hand’s breadth.

On a flat wall it does not matter, because a flat wall is a homography from every position and keystone correction can cancel the whole chain. On any other wall it decides everything: the picture is an anamorph, its design eye is the lens, and every viewer is at a wrong eye by however far they are sitting from the machine.