What a removed wall costs that a removed roof does not

What the removed roof buys and measuring a room off the page both treat fukinuki yatai’s two editing operations — lift the roof, delete the near wall — as one convention with one set of costs, applied to two different occluders. That is right for what those essays measure, coverage and area. It stops being right the moment the question is not how much floor a reader can see but what kind of picture is left once the surface is gone.

A building with its near wall deleted is still, exactly, a projection of the modified building from wherever the eye stood — fitting a single centre to its drawn marks returns that eye to the arithmetic floor. A building with its roof lifted off is not a projection of anything from anywhere, and the identical fit does not return a large number in that case: it refuses outright, because the marks were drawn along a fixed direction rather than from a point at all. Two edits, one convention, and one of them survives being asked where it was drawn from while the other cannot be asked the question.

What the fit is actually doing #

The routine behind every number in this essay is not new to it. pictureFrame, drawFrom and fittedCentre are this site’s standing answer to “is this a central projection, and from where” — counting the eyes needs the room uses the identical machinery to detect a second eye, and the underlying least-squares solve is the same one two rays that do not meet runs on a stereo pair.

Handed a set of drawn marks and an assumed picture frame, it draws one ray from a trial centre through each mark and asks where those rays come closest to meeting. If the marks really were drawn from one point, every ray passes through it exactly and the closest-meeting point is that point, to the limits of floating-point arithmetic. If they were not, the rays still have a closest-meeting point — skew lines almost always do — and the routine returns it regardless, along with how far the rays actually miss it. The residual is the whole instrument: a number a genuine photograph makes vanishingly small and a fabricated scene need not.

The picture frame itself is a plane stood some stated depth in front of the trial centre, and drawFrom is what turns a world point into a mark on it — the same construction a real picture plane performs, run here as a computation rather than photographed. Nothing about that construction cares whether the “eye” it is run from is the real one; it is perfectly happy to draw a consistent-looking set of marks from a trial centre that has nothing to do with how the scene was actually built, which is exactly why the fit afterwards is doing real work rather than confirming something already given. A routine that could only succeed when handed its own assumptions back would prove nothing about the marks; this one is handed marks from fukinukiScene and framingEye — an ordinary three-room building and the finite eye a painter would frame it from — with no privileged knowledge of either passed to the fit.

The wall removed: still exact #

The near wall of the centre room is the occluder measuring a room off the page and what the removed roof buys both charge 44 percentage points of floor coverage. Deleting it and re-running the fit answers a different question: does the picture that remains still belong to one eye.

2.96e-15 m is not evidence of anything subtle. It is what the fit returns when nothing about the underlying geometry has changed at all — every mark still lies on a ray from the one eye that framed the building, because deleting a face of a solid does not move any of the vertices that remain. The wall’s absence changes which surfaces are available to draw; it does not touch the map from world point to picture point for the surfaces that are left, so a routine built to test that map reports, correctly, that nothing about it has been disturbed.

The same edit, a different elevation #

A single reading proves less than two that agree for a reason the essay can state, so the wall-removed edit is worth repeating at a second elevation before leaning on it.

3.19e-15 m at 66° against 2.96e-15 m at 52° is not “close” in the sense two measurements of a real quantity are close; both are floating-point noise, and the fact that the noise itself moves slightly with elevation is exactly what a residual with no real geometric content should do. What the second reading actually confirms is the scope of the claim: the wall-removed picture is an exact projection of the modified building from whichever eye drew it, not merely from the one elevation the lead figure happens to use.

Why the roof removed is not merely large #

The temptation with a fit that reports 2.96e-15 m in one case is to expect the other case to report some larger but still finite number — a picture drawn along a fixed direction, the reasoning might go, is simply a very bad approximation to a picture drawn from very far away, and ought to fit an eye a long way off with a correspondingly large residual.

That is not what happens, and the reason is worth being precise about. roofRemovedResidual draws every ray not from a trial point but along one fixed direction — the elevation the roofs were removed at — so every ray in the bundle is, by construction, exactly parallel to every other. Two parallel lines in three dimensions do not converge at a large distance; they do not converge at all, which is a qualitatively different fact from converging far away. closestPointToRays is built to notice the difference rather than push through it: handed a bundle with no finite common point, it reports exactly that — “every ray is parallel — the closest point is at infinity” — instead of returning some enormous but finite coordinate that would read, to an incurious glance, like an ordinary large residual.

That refusal is the measurement. A number, however large, would still say “there is a point out there, just a bad one.” The refusal says there is no point to be a bad estimate of, which is the correct description of a bundle of parallel rays and a much stronger claim than any residual could carry.

It is worth naming the plausible wrong implementation this refusal is guarding against, because the site’s own libraries have made the opposite mistake before on a differently-shaped input: a solver asked to intersect a bundle of rays that all lie in one plane can find its three-dimensional least-squares system singular and divide by a pivot of zero rather than noticing that no unique answer exists. A bundle of exactly parallel rays is a stricter case of the same trap — every ray shares one direction, so the system a naive solver would set up is not merely singular in the sense of admitting many equally good answers, it has no finite answer to admit at all. A routine that silently returned whatever a zero pivot produces would report some number, and a caption quoting it would look exactly like every other residual on this page while meaning nothing. The explicit refusal is what stands between “no answer” and “a wrong answer offered as though it were a small one.”

The control: seeing through is not removing #

The contrast between a wall deleted and a roof removed could, so far, be read as being about which surface is edited rather than about what removing a surface does. The control separates those two readings by editing nothing at all.

A roof made transparent rather than deleted is still, geometrically, the identical finite-eye projection as the wall-removed reading — every mark on every surface, including the ones an opaque roof would have hidden, still lies on a ray from one real point in space. The fit returns 1.52e-15 m, another arithmetic floor, because there is nothing here for it to disagree about. What isolates the roof-removed reading’s refusal, then, is not “seeing into a building that is normally closed” — the glass roof does exactly that and fits perfectly well — it is the specific choice, in the roof-removed drawing, to project along a fixed direction instead of from a point. Seeing in is free. Discarding the centre is what costs the picture its status as a projection of anything.

All three, side by side #

The bar drawn at height 1 for “roof removed” is worth flagging explicitly, because a chart is exactly the format that tempts a reader into comparing three heights as though they were three readings of one quantity. They are not: two of the three bars are genuine residuals, both at the arithmetic floor, and the third is a placeholder standing in for the absence of a residual at all. Reading it as “the roof-removed picture is a thousand trillion times worse” would be taking the wrong lesson from the right figure — the correct reading is that two of the three edits left behind a number and one of them did not, which is a difference in kind rather than in scale.

That is also why the figure draws all three edits against the same building rather than against three separately staged ones. Holding the rooms, the elevation and the framing eye fixed across all three means the only thing that varies between the bars is which surface was touched — a wall deleted, a roof made transparent, a roof deleted — so whatever separates the results is attributable to the edit and to nothing else on the page. A comparison that changed the building along with the edit would leave open the possibility that some incidental difference in geometry, rather than the edit itself, was doing the work; holding everything else constant is what lets the single differing bar be read as a fact about roofs and walls rather than about the particular building chosen to demonstrate them.

A different kind of refusal #

It would be easy to conclude from the figure above that a fitting routine either reports the arithmetic floor or refuses cleanly, with nothing in between. A different construction on this site, borrowed here for the contrast, shows a third behaviour that is worth setting beside the first two.

That is the third possibility this essay’s own family never produces. A two-view reconstruction handed a vanishingly small baseline and noisy readings does not know to refuse the way roofRemovedResidual does — it has no bundle of exactly parallel rays to notice, only a bundle of rays that are merely close to parallel, and “close to” is not a condition any solver can test for cleanly. So it returns an answer, a confident and badly wrong one, with nothing in its output to distinguish it from a good reconstruction of a real scene.

Set against that, the roof-removed refusal is the better-behaved failure of the two, and it is worth being clear about why. The rays in this essay’s roof-removed reading are not merely close to parallel; the construction makes them exactly parallel, by drawing every one along one stated direction rather than toward any point, however distant. Exact parallelism is a condition a solver can test for outright, and this one does. A photograph never quite achieves it — real light rays from a real distant eye are parallel only in the limit — so the clean refusal measured here is a property of a drawn convention rather than something a camera pushed far enough away would ever trigger for itself. The building blocks of both failures are the same rays-and-residual machinery; what differs is whether the degeneracy is exact or merely approached, and only the exact case can be caught before an answer is returned.

The honest limit #

Nothing here measures whether a fukinuki yatai painting was actually drawn from a point, a direction, or by no consistent rule at all — real paintings are hand-drawn, and the fit this essay runs assumes marks precise enough that a genuine central projection would land at the arithmetic floor and a genuine parallel one would trigger the exact refusal. A hand’s own inconsistency would smear both readings toward some finite, unrefused, moderately sized residual that this essay’s clean pair of cases does not examine, and telling that residual apart from measurement noise in a real drawing is a different, harder problem than anything solved above.

It also says nothing about which of the two conventions is the better one to draw with — that comparison belongs to what the removed roof buys, which measures coverage and uniformity rather than projective status, and a convention that fails to be a projection from anywhere is not thereby a worse picture, only a different kind of object. A parallel drawing was never claiming to be a photograph; this essay’s contribution is to say precisely what it is instead, and to show that the identical fitting routine can tell the two apart without being told which is which in advance.

Nor does the clean dichotomy — refuse or land at the arithmetic floor — generalise to every editing operation a painter might make. Removing a wall left every remaining vertex on its original ray because deletion does not move anything that survives it; an edit that moved a surface, rather than simply removing one, would generally break that guarantee and land somewhere in the wide middle ground this essay’s two cases skip past entirely. The building used throughout is also a simple one — flat floors, vertical walls, right-angled rooms — and nothing here says whether the same clean separation would survive a building whose rooms were not boxes to begin with.

And the fit throughout assumes the picture frame’s own depth and orientation are known, which in a real drawing they are not; recovering them alongside the centre is a harder, jointly-solved problem this essay sidesteps by fixing the frame and searching only for where the eye sits within it. What is measured here is the best case for detection — marks precise enough and a frame given enough that the only question left is whether a finite centre exists at all — and a real drawing offers a fitting routine none of those conveniences.

What this joins #

Counting the eyes needs the room runs the same fitted-centre machinery to ask whether a picture holds one eye or two; this essay runs it to ask whether a picture holds an eye at all, which is the same instrument pointed at a more basic question. The room a divergent picture is a photograph of asks a structurally similar question of two constructions sharing an edge rather than of one building under two edits, and reaches the same kind of answer by a different route: consistency with a single projection is a real, checkable property, and it can fail cleanly or hold exactly, with very little occupying the middle when the geometry is stated precisely enough. Two stations in one picture is the family member where a routine that ought to refuse does not, because its two halves are both individually well-posed rather than one of them lying at genuine infinity — the three essays together cover a refusal, a disagreement between two answers, and an answer that should not exist but is returned as though it does, which is a fuller map of what a fitting routine can do than any one of them alone.

The transferable form is short enough to state once. A single fitted number never says, on its own, whether it measures a real quantity or the shape of an unrefused failure; the fitting routine must be tested at the exact configuration it is meant to refuse, and its behaviour there — refuse cleanly, return the arithmetic floor, or return a confident wrong answer — is what tells the rest of its output apart from noise.

It is also the same lesson the two edits themselves teach about fukinuki yatai, read back from the geometry to the convention. A wall is a surface standing in front of the subject; deleting it removes an obstruction and changes nothing about the eye behind it. A roof is not merely a surface in this convention’s own logic — the whole device is drawn along a direction precisely because no single eye, however placed, would show three rooms on equal terms, which what the removed roof buys measures as a coverage problem and this essay now confirms is, underneath the coverage, also a projective one. The convention was never trying to be a photograph with the roof missing; it was always a different kind of drawing, and the refusal measured here is that difference stated in the fitting routine’s own terms rather than in a painter’s.