A dome is read as a nearer camera

A head seen in perspective needs a camera, not an ellipse photographed golden seed heads with a pinhole camera close enough to make the near side of a head larger than the far side, and read their divergence from the positions of their organs in two annuli. The correction that had restored a distant photograph — stretching the head back along the short axis of its own second moments — failed once the camera came within a distance set by the tilt. A camera fitted from the organs alone restored it: five numbers, the head’s centre, the tilt, its bearing and the camera’s distance, chosen so that the head they un-project is round, centred and evenly packed. It read seven to ten heads in ten right at every distance down to two head radii.

Every head in that essay was a flat disc, and a capitulum is not. Many are domed, the rim lower than the centre, and some rise to a cone. Seen from the side, a dome does something no flat head does with any camera: it lifts the centre of the head towards the lens, so that in the picture the middle of the head is moved sideways against the rim, however far away the camera is. That essay ended by asking what a dome does to the fitted camera — whether it absorbs a dome as a wrong distance, whether the even-area condition can tell a dome from a camera, and how high a dome must be before the reading mistakes it for a twist.

All three have answers, and the first is the one that matters. The camera absorbs the dome as distance, cleanly enough to be predictable and not cleanly enough to read the head.

A dome with organs packed on its surface #

The heads are the earlier ones: golden Vogel heads of 900 organs, every organ displaced by a tenth of a spacing, ten heads on ten seeds, each photographed across a bearing set by its seed. Each is laid on a paraboloid dome whose height above the rim is a stated fraction of the head’s radius — 0.05, 0.1, 0.15, 0.2, 0.3 and 0.5 — and a flat head is the dome of height nought.

The organs are packed evenly on the dome’s surface, not in plan, because a receptacle packs them on the surface it has. So each organ keeps its angle and moves to the plan radius at which the dome’s area inside it is the share of the whole that the flat head’s area inside its own radius was. Seen from directly above, a domed head is therefore a little crowded towards its rim, where the surface slopes and a patch of it covers less plan; on the highest dome here the organs at the rim stand two fifths more densely in plan than those at the centre.

The camera is the earlier pinhole, now with the dome’s height in it: an organ at height $z$ is nearer the lens by $z\cos\tau$ and lands $z\sin\tau$ further along the bearing, for a camera tilted by $\tau$. Infinitely far away the first term disappears and the second does not. A head counts as read right when its two annuli’s divergences differ by no more than any untwisted flat head’s do square on — the threshold the positions read a twist’s shape by, and every reading since has used. A head misread is therefore a head that would be reported as twisted.

Twenty degrees off, from infinitely far #

On a dome a fifth of the head’s radius high — a gentle dome, drawn to scale beside the photograph — and photographed twenty degrees off the axis from infinitely far, not one of ten heads is read right, either by the moments’ ellipse or through the fitted camera. Lowered by the dial to a tenth of the radius, eight are read right both ways; flat, all ten by the ellipse and nine through the camera. The picture of the head barely changes across the dial. What changes is a sideways shift of the middle of the head against its rim of a fraction of a spacing, which the eye does not see and the annuli do.

The fitted camera, handed the dome a fifth of a radius high, puts itself 7.4 head radii away at a tilt of 20.2 degrees. The true camera is infinitely far and tilted 20 degrees.

Square on, a dome is nearly harmless #

The first thing worth knowing is the safe case. Photographed along its axis from infinitely far, a domed head is read as it comes, about its centroid, with no correction at all, and ten, ten, ten, ten, ten, nine and eight heads in ten are read right from the flat head to the dome half a radius high. The crowding of the outer ring does not move the reading, because the annuli are read by fitting the spacing of the whole head and taking the divergence from the organs’ angles, which a square-on picture of a dome leaves exactly where they were. Recovering the angle from the counts would not notice the crowding either; the counts are read in a band, and a band of a dome is a band of the same spirals.

From closer, square on, the dome’s centre is nearer the lens than its rim and is magnified more, a radial taper that grows with the dome’s height and with the camera’s nearness. At ten radii every dome up to three tenths of the radius high is read right on eight heads of ten or more; at five radii, seven or more. On the dome half a radius high, both near cameras read every head wrong. A photograph taken square on, from a working distance, survives every dome a capitulum is likely to have.

Off the axis, a height times a sine #

Tilt the camera and the safe range collapses. From infinitely far at ten degrees, both corrections read eight heads in ten right or more on every dome up to a fifth of the radius, and two at three tenths. At twenty degrees they hold to a tenth of the radius and read two at fifteen hundredths. At thirty degrees the ellipse reads nine at five hundredths and none at a tenth; the fitted camera, a little more tolerant there, reads six at a tenth and one at fifteen hundredths.

The three tilts fail at three different heights, and the heights fall as the tilt grows, which suggests the obvious single number. The dome lifts its centre by $H R$, where $H$ is its height in head radii, and the picture moves that centre sideways by $H R \sin\tau$ against the rim. So the quantity a photograph cannot hide is the dome’s height times the sine of the tilt.

Drawn against it, the six curves fall together. Every correction at every tilt reads eight heads in ten or more while $H\sin\tau$ is at most 0.035 of the head’s radius, and by 0.05 the ellipse reads two or fewer at every tilt, and so does the fitted camera at ten and twenty degrees. Only the fitted camera at thirty degrees lasts a little longer, failing by 0.075. On a 900-organ head, 0.035 to 0.05 of the radius is about two thirds of a spacing: the reading survives a dome until the picture has moved the head’s middle against its rim by most of the distance between neighbouring organs.

The same number twice #

The perspective essay found a single number deciding when the moments’ ellipse failed on a flat head seen in perspective: the sine of the tilt over the camera’s distance in head radii, the fraction by which the near rim is magnified and the far rim shrunk. The ellipse held where that number was at most 0.035 and failed from 0.058 up, with six heads of ten read right at 0.05 between them.

The dome fails at the same 0.035 to 0.05, measured by a different quantity — a sideways shift of the centre rather than a magnification across the head — and fails both corrections, where the perspective failed only one. A head’s divergence reading tolerates a distortion of its picture of about four hundredths of its radius, whatever the distortion is, and a dome off the axis supplies one that the camera fitted for a flat head cannot take back.

Two thirds of a spacing is not a centre error #

A sideways shift of the head’s middle sounds like the error the centre the spirals give measured, where a 900-organ head read square on tolerated a centre about a sixth of a spacing off before it misread. The dome tolerates four times as much, and the difference is in the shape of the shift. A wrong centre moves every organ’s angle by an amount that falls as one over its radius, so the inner annulus is read through a different error from the outer, which is a twist by construction. The dome moves the organs by an amount that falls smoothly from the middle to nothing at the rim, and the correction — ellipse or camera — takes up most of it, so that what reaches the annuli is the part no correction of its kind can model.

That part grows with the dome’s height and the tilt’s sine together, and nothing else: at a tenth of the radius and twenty degrees, and at a fifth and ten degrees, $H\sin\tau$ is 0.034 and 0.035, and the two cells read eight heads in ten right by both corrections. A dome twice as high seen at half the tilt is the same photograph, as far as the reading can tell.

The camera puts itself nearer #

What the fitted camera does with a dome is not random. It finds the tilt: the median over ten heads is within a degree of the truth on every dome, at every tilt and distance read, and within 1.1 degrees on the highest dome from five radii, though single heads stray by up to four. And it puts itself too close. A flat head photographed from infinitely far is read as a camera sixty to a hundred and fifty radii off, which is as near to infinity as the organs can say. A dome five hundredths of a radius high is read from 31 radii at twenty degrees, a tenth from 13.6, fifteen hundredths from 9.0, a fifth from 7.4, three tenths from 4.7 and half a radius from 3.0.

The products of height and distance sit between 1.3 and 1.7 at every tilt, near 1.4 at twenty degrees, and the rule that fits them is a reciprocal one: the camera reads itself nearer than the truth by about 0.7 times the dome’s height, counted in reciprocal head radii. It holds at true distances too. A dome a tenth of a radius high, photographed from ten radii at twenty degrees, is read from 5.8 radii — the reciprocal of a tenth plus seven hundredths — and a dome a fifth of a radius high from ten radii, from 4.2. Across every dome and every camera the fitted camera was handed, the median excess of its reciprocal distance over the truth is between 0.58 and 0.86 times the dome’s height.

That is the camera absorbing the dome as a wrong distance, which the question asked about, and the reason it can is visible in the picture. A near camera and a dome both make the middle of a tilted head land further along the bearing than an affine squash would put it, by an amount that grows with the square of the distance from the middle. To that order a dome is a camera nearer than it is, and the five numbers have no sixth with which to say otherwise. It is the same kind of confusion as a shell’s cut photographed with a squash along its axis, which is exactly the picture of another shell — except that there the impostor is exact and here it is only close, which is what the next section turns on.

Absorbed is not read #

Absorbing the dome as distance would be harmless if the nearer camera un-projected the head round. It does not. The dome and the camera agree in the term that dominates, and disagree in the rest: a near camera shrinks distances across the bearing as well as along it, a dome does not, and a dome’s organs were packed on a surface the flat-head camera cannot know about. The head the camera un-projects is close enough to a Vogel head to satisfy its fit and far enough from one to read as twisted.

At twenty degrees the fitted camera reads nine heads of ten right on a flat head from every distance; on a dome a tenth of the radius high, eight from infinitely far, nine from ten radii and five from five; on a dome fifteen hundredths high, two, two and one. The ellipse, which already failed on a flat head at five radii, fails on every dome there and on every dome from a tenth up at ten radii. A near camera and a dome add, in what the fitted camera reads and in what it gets wrong.

The even-area condition seldom tells #

The fitted camera does not only return five numbers; it returns how far the head it un-projects still is from round, centred and evenly packed. If a dome left a misfit no flat head leaves, a worker could at least be told the camera had been fooled. The test is strict and fair: a domed head is flagged when its misfit exceeds the largest misfit any flat head leaves at the same tilt, at any of the distances read.

At ten degrees the flag finds two heads in ten on the highest dome and at most one on any other. At twenty it finds one head on the domes a tenth and fifteen hundredths high — where eight heads in ten are already misread — two on a fifth, and five of ten on the domes three tenths and half a radius high. Only at thirty degrees does it catch them: four or five heads at the two lowest domes, seven at fifteen hundredths, and every head from a dome a fifth of the radius high. So the condition that made the camera work cannot, at the tilts a careful photographer would use, tell a dome from a camera; at the tilts it can, the reading had failed long before.

What a person with a photograph should do #

The practical advice is narrower than it was for a flat head. A camera fitted from the organs is still the right correction for a flat head seen in perspective, and it is still right for a head that is domed and photographed square on, where nothing needs correcting. For a domed head photographed off the axis there is no correction here that works: past $H\sin\tau$ of about four hundredths of the radius, every head in ten is at risk.

Two things follow. Photograph a domed head as nearly along its axis as possible, which a head photographed from the side already recommended for a different reason: at five degrees off, a dome would have to be more than four tenths of the radius high before the band is reached. And treat a fitted camera that reads itself much nearer than the lens was as a reading of the dome rather than of the camera, since a camera distance is something a photographer knows and the organs are reporting a different one. Neither is a correction; both are ways of not needing one.

Paraboloids and pinholes #

A capitulum is not a paraboloid, and its organs need not be packed evenly on its surface: a receptacle that grows faster at its rim, or a cone with a flat top, would distort the picture by a different taper, and nothing here measures one. Every head is a 900-organ golden head with a tenth of a spacing of displacement; a larger head has more spacings in its radius, and the band of four hundredths of the radius is then more spacings wide, which may make a larger head more tolerant of the same dome. The camera is a pinhole, and a real lens adds its own radial distortion, which a dome would add to.

Readings that would undo it #

A square-on photograph of a dome up to three tenths of the radius high misread on more than three heads in ten. A domed head photographed off the axis read right on eight of ten past $H\sin\tau$ of 0.05 by the ellipse, or by the fitted camera at ten or twenty degrees. A fitted camera whose median tilt is off by more than about a degree on a dome, or one whose reciprocal distance does not exceed the truth by between 0.55 and 0.9 times the dome’s height. A misfit that flags most domed heads at twenty degrees. Each is checked against the measured heads whenever they are read.

The head’s shape, read as the camera’s #

A dome photographed off the axis moves the middle of a seed head sideways against its rim by its height times the sine of the tilt, and once that is about four hundredths of the head’s radius — two thirds of a spacing on a 900-organ head — the divergence reading calls the head twisted. The camera fitted for a flat head finds the tilt and explains the dome as nearness, reading itself closer by about 0.7 times the dome’s height in reciprocal radii, and the head it un-projects is still misread. The even-area condition that made that camera work cannot tell the difference at the tilts that matter. Square on, a dome is harmless.

Still open: a sixth number for the dome #

The flat-head camera has five numbers and no way to say that a head is curved. A sixth — the dome’s height, entering the un-projection as a lift of each organ by its plan radius — would let the fit separate a dome from a near camera if anything in the picture separates them, and the misfit’s failure at small tilts says that little does. The next measurement fits the six-number camera to the domed heads here and asks whether it recovers the dome’s height and the true distance together, or only their sum; and, if it recovers both, whether the reading it hands on is right again past the band where every five-number reading failed.