The hole a rig cannot fill

The hole a scene actually sees settled which point of a lens a picture is actually a projection from: not the physical stop, but the entrance pupil, the point every chief ray is found to cross to a few millionths of a millimetre. A spherical panorama rig carries two such pupils, one per lens, mounted opposite each other so that between them every direction in the sphere falls inside at least one lens’s field.

Every direction, with one exception that has nothing to do with either lens’s optics. The rig is bolted to a tripod, and the tripod’s own column sits directly beneath both pupils, in the one place a camera can never be pointed at itself.

That column blocks a real, computable slice of the sphere from both lenses at once, and no choice of lens, field of view or lens count removes it — because the problem is not that some direction lacks a camera looking at it. It is that the rig’s own body is standing in front of the direction that would show it. An occlusion and a coverage gap look identical in a finished panorama — both are places where the picture has to come from somewhere other than the rig’s own two lenses — and this essay’s job is to show why treating them as the same problem produces an error a coverage gap never would.

The distinction matters commercially as much as geometrically. What a 360 photograph actually is treats the finished equirectangular file as a lookup table of directions rather than a picture in the ordinary sense, and a rig that hands that table a gap has to fill it from somewhere before the file is complete. A manufacturer selling a spherical rig has every incentive to describe the nadir as “requiring a quick patch shot” — a coverage problem with a known fix — rather than as a structural occlusion the rig’s own geometry guarantees, and the difference between those two descriptions is exactly the difference this essay measures.

Coverage is not the problem, and the figure above already proves it #

The hundred-degree fields the hero figure states are not a modest safety margin. Two hemispheres would already meet at the equator with nothing spare; a hundred degrees each gives thirty degrees of overlap on every side, which is the ordinary practice for a two-lens spherical rig and far more than coverage alone requires. Nothing about widening either lens’s field, or adding a third or fourth lens elsewhere on the sphere, touches the hole this essay measures, because the hole is not where a field of view runs out. It is where the rig’s own hardware sits between both pupils and a set of directions that both fields would otherwise happily include.

That distinction is the whole reason this essay exists as a separate measurement from an ordinary field-of-view calculation. A coverage gap is fixed by more glass. An occlusion is fixed by nothing the optics can offer, because the thing in the way is not optical.

A pupil sees around an edge already established the general shape of this fault for a single lens looking past a single obstruction: an entrance pupil of finite size sees slightly further round an edge than a mathematical point would, because different parts of the pupil clear the obstruction at slightly different angles. The rig’s column is that same obstruction problem run twice, once per lens, with the added structure that the two pupils sit on opposite sides of the obstruction rather than both looking past the same edge from the same side — which is precisely what turns two separate blind cones into an intersection smaller than either one.

Spreading the pupils helps, and costs a larger rig to do it #

The hole’s size is not fixed; it depends on how far apart the two pupils sit, and that dependence is itself worth a direct look before asking how the hole is shaped.

Forty-four times less sphere hidden for not quite doubling the pupil separation looks like an unambiguous improvement, and it is one — but it is bought with a rig that is physically larger, which is not free on a device meant to be carried and levelled by hand. It is also bought asymmetrically: the hidden fraction falls very fast at first and much more slowly once the pupils have cleared most of the column’s own width, which the next figure makes exact. Nothing in this trade is a coverage decision; it is a mechanical one, between a compact instrument and a smaller hole beneath it.

There is a second cost to widening the pupils that the hidden-fraction number does not show at all: a wider rig also widens the disagreement between what each lens sees of anything nearby, for the ordinary reason that two cameras further apart triangulate a closer object more differently than two cameras close together do. A rig built only to minimise the nadir hole, with no regard for that second effect, would spread its pupils as far as the housing allows and pay for it in a worse stitch everywhere the scene is not at the rig’s design distance. Every dimension on a rig like this is a compromise stated in the same units as this essay’s own hole, which is why the number is worth having exactly rather than only knowing its direction.

Two blind cones, and the hole is where they overlap #

The single number quoted so far — a percentage of the sphere — comes from a shape, and the shape is worth seeing directly, because the reason two pupils help at all is a geometric fact about where each one’s blind cone sits.

A single, centred pupil’s blind cone is the tripod’s silhouette and nothing more — an exact, unavoidable cap of angular radius that depends only on the column’s width and how far below the pupil it sits. Move the pupil off to one side and that cone does not shrink; it slides, because the column’s silhouette from an off-axis viewpoint is still the whole column, just no longer straight below. What shrinks is the overlap between the two lenses’ slid cones, because each lens now sees past the column on its own side, into territory the other lens’s cone still covers. The hole a finished panorama actually has is that overlap — the directions genuinely hidden from both — which is why it is always smaller than either lens’s own blind cone taken alone, and why the intersection rather than the union is the number that matters.

Getting that arithmetic backwards is an easy mistake to make and worth naming explicitly, because it is the one place in this essay a plausible wrong measurement would land on a specific, wrong number rather than an obviously broken one. Adding the two lenses’ individual blind fractions together, rather than intersecting them, double-counts the region visible to neither by treating it as though it belonged to each lens’s private loss separately — a union dressed up as a sum. The single-pupil closed form the next figure checks against is exactly the tool that catches this: a union-based estimate would not agree with it at zero separation, where the two blind cones coincide entirely and the correct answer is the closed form itself rather than twice it.

Validating the sampler, and where the hole closes to nothing #

The two figures above report a single number at a single pupil separation. What a rig designer actually needs is the whole curve, and a curve computed by sampling directions on a sphere needs its own check before it can be trusted at settings with no simpler answer to compare against.

That agreement at zero separation is the control the whole sweep depends on, in the sense this site always insists on: the one setting where an independent, exact answer exists is checked first, and only once it holds is the sampler trusted at the settings — every offset pupil separation — where no closed form is available to check against. A hole that closes exactly at the column’s own radius is also a clean enough geometric fact to state as a design rule on its own: spreading a rig’s pupils past the tripod’s radius removes the occlusion entirely, and nothing short of that fully does.

The four-figure agreement — 2.5656 sampled against 2.5658 closed-form, over 24,000 directions — is worth reading for what it is rather than assuming it means the sampler is exact. A Monte Carlo estimate over that many samples carries a scatter of roughly the square root of the count, so the two-thousandths-of-a-per-cent gap here is well inside what random sampling alone would produce; a sampler disagreeing by a whole percentage point at this sample count would be the signal of a real bug rather than noise, and this is not that. The check earns its keep by being the kind of comparison that could have failed and did not, which is the only kind worth reporting.

Filling the hole costs a second centre, and that cost is exact #

A rig cannot photograph directly beneath itself from where it stands. The obvious fix — move the tripod, photograph the ground it was just standing on, and paste that patch into the hole — works, in the sense that it produces a picture with no visible gap. It does not work in the stronger sense this site asks of a projection.

Both readings are internally consistent and neither is free of error; they are simply two different choices of which depth in the scene the stitcher declares correct, with everything at another depth paying for that choice. This is the same structure the essay on a stitched seam’s own unavoidable parallax already found on the seam between ordinary rotated frames, arrived at here from a second station rather than a second rotation. A patch is not a worse kind of picture than the rig’s own two lenses produce; it is a different kind, with two stations rather than one, and the difference shows up as a displacement rather than as a blur or a visible seam.

Registering on the floor rather than the far field is not a free correction, either, even though it puts the floor itself exactly right. It trades one wrong answer for another and simply relocates where the error is spent: a picture correct on the ground plane is wrong about anything standing above it, by an amount that grows with that object’s own height, which is exactly what the 20.3-millimetre displacement of a 30-millimetre post reports. A stitcher choosing between the two registrations is choosing which part of the scene under the rig gets to be right, not whether anything does — the same choice a rig is right on one surface makes explicit for the seams between a rig’s own simultaneous lenses, where the optimal depth to be correct at is a closed-form harmonic midpoint rather than a guess.

The zero at no step is worth taking seriously as a control precisely because it is so easy to get for the wrong reason. A method that always reported “no error” would also show zero here, and the only way to tell the two apart is to move the step away from zero and watch the number grow — which it does, smoothly and from an honest arithmetic identity rather than from a numerical fluke. Nothing about a larger step is inherently worse engineering; a bigger step is simply a bigger second station, and the parallax it buys is the exact price of using one.

The near-linearity of the growth is itself informative rather than incidental. An angle that is the arctangent of a ratio is linear only while that ratio stays small, and a photographer stepping a metre or so away from a rig standing over a floor a metre and a half down is squarely in the range where the small-angle approximation and the exact arctangent agree closely — which is why a stitching guide can reasonably say “keep the second station short” as a rule of thumb without needing to quote the arctangent at all. The rule of thumb is not wrong; it is simply the linear part of a curve this essay can also compute exactly at whatever step a real patch shot actually uses.

The same disease in a rig that turns instead of doubling #

A two-lens rig hides its column because two pupils sit apart from each other in space. A single rotating camera stitching a panorama runs into a version of the identical problem for a different reason — its pupil sits apart from the point it is being turned about — and it is worth seeing the two side by side, because the underlying fault is the same fault wearing two different mechanical causes.

Both figures describe a picture assembled from rays that do not all pass through one place. The two-lens rig’s version comes from two simultaneous pupils standing apart in space; the rotating camera’s version comes from one pupil visiting many positions in time, about a pivot that is not quite where the pupil itself sits. Two rays that do not meet is the general statement of what happens next: a bundle of rays that ought to share a single point but do not is resolved by a least-squares fit rather than an exact intersection, and the 9.16-millimetre miss quoted above is that fit’s own residual, reported honestly rather than hidden inside a single “the picture’s centre is here” claim.

Photographers call the point a panorama head should be adjusted to the “nodal point,” and the name is a persistent misnomer worth correcting exactly here: the point that removes parallax from a rotated stitch is the entrance pupil, the same point the hole a scene actually sees identifies as the true centre of projection, and a lens’s nodal points are a different pair of points entirely, generally close to the pupil but not identical to it. A panorama head mounted at the wrong point and a two-lens rig with pupils apart from each other are the same error, stated once for a camera that moves and once for a pair of cameras that do not — and both are errors about where the entrance pupil sits, never about where a nodal point does.

What this does not settle #

The measurement above is specific to a two-lens rig on a tripod, and three things sit outside it deliberately.

It does not show that no rig geometry could ever avoid a hole of this kind. A rig with the tripod’s mounting point routed away from the optical axis entirely — a periscope-style relay, or a mount that attaches from the side rather than from below — would place the occlusion somewhere other than the nadir, and this essay’s arithmetic does not rule that out; it measures the ordinary bolted-underneath arrangement rather than every arrangement a designer might invent. Nor does it rule out a third lens dedicated to the nadir alone, which trades an occlusion for the ordinary registration problem any additional lens brings, rather than eliminating either problem outright.

It does not address whether the hole, once patched, is visible to a viewer in practice. A composite error of a few tens of pixels on an eight-thousand-pixel panorama, at the nadir, where most viewers spend little attention, may be immaterial to how a picture is used even though it is not immaterial to whether it is a single projection. That is a claim about attention and use, not about geometry, and this collection has no machinery for it — the same limit the eye is a picture surface too states for a question about looking rather than about light.

And it does not account for anything moving in the scene between the two photographs a patch requires. A parallax error is a displacement of a static point; a person walking across the ground between the rig’s own shot and the second station produces an inconsistency the arithmetic above has no term for at all, on top of whichever depth was chosen to be correct. That failure mode is common enough in practice — a moving shadow, a pet, a second photographer’s own feet — that it is usually the visible defect in a patched nadir, while the exact geometric parallax this essay measures sits underneath it, present whether or not anything in the scene moved at all.

One centre, then two, and the same arithmetic in both directions #

The essay this collection places beside this one runs the identical geometry the other way round: a projector throwing light onto a surface from a point that is not the surface’s own centre, rather than a rig gathering light from a point that is not where its own body lets it look. A projector that is not at the dome’s centre is what happens when a single centre is required and a device sits away from it on the output side rather than the input side, and the two essays together are the whole of what this collection has to say about a sphere and a point that fails to coincide with it: gathering light from the wrong point loses a fixed cap outright, and throwing light from the wrong point does not lose anything but delivers every ray to the wrong place, by an amount this collection can also compute exactly.