A projector that is not at the dome's centre

The hole a rig cannot fill measured what happens when a rig gathers light from a point that is not where its own body lets it look. This essay runs the identical geometry in the opposite direction: a projector throwing light onto a dome from a point that is not the dome’s own centre, and a seat receiving that light from a point that may not coincide with either.

From the centre, both steps are trivial. A ray leaving the middle of a sphere meets the sphere along its own direction, and a ray arriving at the middle from any point on the sphere is seen along that same direction — throwing and viewing are the same map read backwards, and composing a map with its own inverse is the identity. Move the projector off centre and neither step is the identity any longer, and they no longer cancel: a beam aimed at one direction is received at another, by an amount this essay computes exactly, and the single-seat correction that fixes it turns out to know nothing whatever about where the projector stands.

A dome projector of this kind is an ordinary fixture of a planetarium or an immersive theatre, and the geometry matters commercially for the plainest reason: the projector itself is a piece of equipment with a housing, a cooling system and a beam path, none of which can simply be teleported to the exact centre of a room built to hold an audience there instead. Every real dome installation puts its projector somewhere the audience is not, and this essay is the arithmetic of what that displacement costs and what a correction can and cannot do about it.

The composition that fails to be the identity #

The mechanism is worth stating in one sentence before any number is attached to it, because the whole essay is one instance of it repeated at different offsets. Throwing from the projector and receiving at the seat are each, individually, an ordinary map from a direction to a point of the sphere and back — the first names a dome point from a direction the projector emits, the second names a direction the seat would need to look in to see that same dome point. From a common centre those two maps are exact inverses of one another, so composing them returns the original direction untouched; from two different points they are each still well defined, but their composition is a new map with no reason to be the identity, and the gap between what goes in and what comes out is exactly the displacement this essay measures.

A projection of a projection already established the general shape of this fact for two flat pictures rather than a sphere: composing two projective maps gives a third projective map, and there is no reason in general for that third map to be the identity even when the two inputs individually look unremarkable. What is specific to a sphere and a centre is only that the identity case is unusually easy to characterise — it happens exactly when both centres coincide — which is what makes “off centre” a single scalar offset rather than a family of unrelated failure modes to enumerate separately.

That gap grows with how far the projector sits from the centre, and it is worth seeing the growth directly before asking what corrects it.

The displacement has a shape, not just a size #

A single worst-case number hides whether the error is scattered randomly across the field or organised in some way a projectionist could anticipate. It is organised, and seeing the shape is the first step toward correcting it.

A fixed point at the projector’s own axis is the detail that makes a correction possible at all. A displacement field with no fixed point anywhere would have to be corrected pointwise with no structure to exploit; one that pivots about a single point, growing smoothly away from it, is a map with an inverse — which is exactly what the next figure computes.

It is worth being precise about which point is fixed and why, because it is not the dome’s own centre and not the seat. It is the single dome point that lies on the line joining the projector to the dome’s centre — the direction the projector would have to aim to send a beam straight along its own offset, which is also the one beam whose throwing and viewing steps stay collinear with the centre even when the projector has moved. Every other direction bends by some amount that grows with how far it lies from that axis, which is exactly the radial-looking pattern the stalks in the figure trace out.

The correction is exact and closed, and it is not free #

If the composition above fails to be the identity by a computable amount at every point, the projector can simply be told to emit the pre-image of what it was supposed to emit — throw at the direction that, once bent through the off-centre geometry, lands where the intended picture actually wanted it.

“Exact and closed” is worth taking literally rather than as a figure of speech. The correction is not an iterative fit, a lookup table built by trial projection, or an approximation good to some stated tolerance; it is the inverse of a well-defined geometric map, computed the same way any other exact recovery on this site is computed, and it is exact at every direction in the field rather than only at a sampled few. What it is not exact about is how much of the projector’s own light lands on a given patch of dome, which the next section returns to.

The correction is also, in this one specific way, an easier problem than the keystone a projector throws against a flat wall it is not square to, where fitting the resulting trapezium back into a rectangle costs part of the panel outright because a projector cannot add light where none was aimed. Nothing here is discarded: a sphere gives every emitted direction somewhere to land, so the pre-warp redistributes the projector’s existing directions rather than cropping any of them away. The cost of an off-centre dome projector is entirely in the second half of this essay, where the light lands unevenly rather than where any of it goes missing.

Exact for one seat, and the correction does not know where the projector is #

The warp above was built for the seat at the dome’s centre. The question every real installation actually needs answered is what a different seat sees through that same correction — and the answer contains the finding this essay exists to report.

That the three curves coincide exactly is the more important fact of the two on this figure, and it is worth being precise about why. The pre-warp is built from one piece of information: the dome point that the design seat’s own line of sight, in each intended direction, actually names. Once that point is fixed, the beam the projector must emit to reach it is determined by the projector’s own position — but nothing about how well the correction works away from the design seat depends on that position at all, because a second seat’s error is a fact about the dome point itself and the direction the second seat sees it from, and the dome point was fixed before the projector’s location ever entered the calculation. Moving the projector changes what it must emit. It changes nothing whatever about what anyone in the room actually receives. That is a stronger and more surprising claim than “the correction degrades gracefully away from the design seat,” and the three-curve agreement is what earns it: a design choice usually thought of as a projector-placement problem is, once the warp is in place, entirely a seat-placement problem instead.

One picture and three people found the flat-screen version of the same fact by a different route: a curved display can be pre-warped for exactly one seat, and there is nothing to be clever about in choosing which one beyond taking the middle of the audience. What this essay adds is that the correction’s independence from the projector’s position was not obvious in advance and needed its own measurement rather than an analogy to carry it.

The seats a screen will accept asks the complementary question for a flat display rather than a dome: given a correction built for one seat, how far can a second seat move before the residual crosses some stated tolerance. The eleven-and-a-half degrees measured here at two metres, in a ten-metre dome, is this essay’s answer to exactly that question for a sphere — and it is a considerably harder number to absorb by sitting slightly more carefully than the pixel-scale residuals a flat screen’s own version of the same question returns, because a dome’s correction is bending an entire hemisphere’s worth of directions rather than a single rectangle.

What the correction cannot buy #

The warp fixes where every ray lands. It says nothing about how many rays land in a given place, and that is a genuinely separate quantity a pre-warp has no lever to move.

Geometry is a matter of aiming, and aiming is exactly what the warp corrects. Resolution is a matter of how much of a finite, fixed supply of light and pixels a given patch of dome receives, and an off-centre projector throws its budget onto a smaller solid angle on its near side and a larger one on its far side regardless of what direction each beam is subsequently retargeted to. No relabelling of directions changes how many photons were in a beam to begin with, so the 3.30-times unevenness measured here survives the warp completely intact — a projector installed off-centre for practical reasons buys a picture that is sharp near itself and soft on the far wall, and no amount of downstream correction in software touches that fact.

Where a surface spends its pixels measured the identical kind of unevenness for a camera looking out through a curved picture surface, and finding the same shape of number here, on the output side of an entirely different piece of equipment, says something about where the fact actually lives. Spending is not a property of lenses, or of projectors, or of any particular instrument; it is a property of the relationship between a point and a curved surface it is imaging or being imaged onto, and it shows up wherever such a pair is not concentric — whichever end of the light path the point happens to sit on.

The same single-point fact, from an unrelated family of screens #

The pre-warp’s whole content — a correction exact for one point and only measurably wrong away from it — is not a fact peculiar to spheres and projectors. It appears with the identical shape on an ordinary curved screen viewed rather than projected onto, computed by a wholly different route.

A curved screen watched from its own centre of curvature and a dome watched from the point a pre-warp was built for are the same statement twice: a curved receiving surface has exactly one point from which it behaves like the simple picture surface everybody assumes it is, and the size of the departure away from that point is a real, computable number rather than a vague sense of things looking slightly off. The screen essay measures it in pixels of homography residual; this essay measures it in degrees of angular displacement. Neither number is an accident of the method used to find it.

The two measurements even agree on which quantity grows the departure. The screen essay’s own 0.229 radians is read at six tenths of the radius away from the seat’s own axis; this essay’s displacement field grows fastest toward the edge of the field, away from the projector’s own axis. Distance from the one privileged direction is what both figures are secretly plotting against, whether the horizontal axis is labelled in metres along a seat’s own sightline or in degrees off a projector’s own aim.

The control this whole essay rests on #

Both major findings above carry their control on the same figure that reports them, rather than needing a separate drawing, and it is worth naming both explicitly. The identity is the on-centre projector: the spend figure’s own flat line reads 1.000000 exactly, which is the arithmetic floor a projector at the centre gives for free, and the seats figure’s own residual at the design seat is that same floor before any distance is added. Both zeros are the case this essay’s machinery would have to get right to be believed at all — a projector and a seat that both sit at a dome’s own centre need no warp, produce no residual, and spend no more of themselves on one direction than another — and the entire remainder of the essay is what changes when either point moves away from it.

What this does not settle #

Three limits are worth being exact about before this essay’s numbers travel anywhere else.

The projector and the dome here are both idealised: a point source throwing rays with no lens distortion of its own, onto a perfect sphere. A real projector has its own barrel and pincushion character on top of everything measured here, of exactly the kind a lens destroys the invariant fits and corrects for an ordinary flat picture, and a real dome is built from panels with seams and small departures from a true sphere rather than the mathematical surface used throughout. Both are second-order corrections to be composed with the exact warp above, not evidence against it; the warp remains the dominant term and the one this essay’s numbers describe.

The warp is built for a single fixed seat and this essay reports what a second, equally fixed seat sees through it — not what a moving viewer experiences turning their head or walking during a show, which changes which direction of the displacement field they are sampling from moment to moment rather than reading one static residual. A seated audience in a planetarium is close to the fixed case this essay assumes; a walkthrough installation where visitors move freely underneath the dome is not, and would need the residual reported here integrated over a path rather than read at a point.

And a real installation frequently uses several projectors covering different parts of one dome rather than one projector covering all of it, which introduces a blending problem at the seams between projectors that this essay’s single-projector arithmetic has nothing to say about — a separate practical difficulty layered on top of the exact geometry measured here rather than replacing it. Each individual projector in such a rig still obeys everything above on its own patch of dome; what a multi-projector system adds is the question of how two patches, each correct for the shared design seat, agree with each other at their common edge, which is a registration problem in the same family as the seam the hole a rig cannot fill measures for a gathering rig rather than a throwing one.

Two essays, one point failing to be where it should #

Between them, this essay and the one it is paired with are the whole of what this collection has to say about a sphere and a point that does not coincide with its centre. Gathering light from an off-centre point loses a fixed cap of the sphere outright, an occlusion no amount of stitching removes. Throwing light from an off-centre point loses nothing — every direction still receives something — but delivers it to the wrong place, by an amount this essay computes exactly and corrects exactly for one seat, at a cost in evenness no correction touches. One point failing to be the centre produces two entirely different kinds of defect, and knowing which kind is in front of a reader is most of the work of deciding what, if anything, can be done about it.

Both essays also share the same instrument for telling a real defect from an artefact of the measurement: a control that must read as an exact identity, checked before any offset case is trusted. There, it was a closed-form cap agreeing with a Monte Carlo sampler to the fourth significant figure. Here, it is a projector at the centre spending exactly as much of itself on its worst direction as its best, and a design seat reading its own residual at the arithmetic floor. Neither zero is a claim about how careful the arithmetic was; each is the one setting at which an independent argument says the answer has to be exactly nothing, which is what licenses reading every other number on either essay’s figures as a measurement rather than as an artefact of whatever produced it.

What the pair does not offer is a general theorem about surfaces that are not spheres. A dome is the one curved receiving surface simple enough that “off centre” has a single scalar meaning — a fraction of one radius, in one direction — and both the rig’s occlusion and the projector’s displacement lean on that simplicity throughout. A surface with no single centre at all, an ellipsoid or an arbitrary curved screen among them, would need the question asked freshly rather than inheriting either essay’s numbers by analogy, because the very idea of “how far off centre” stops being a single quantity the moment the surface stops being a sphere.