A head seen in perspective needs a camera, not an ellipse

A head photographed from the side photographed golden seed heads a few degrees off their axis and read their divergence from the positions of their organs in two annuli, the way the positions read a twist’s shape. Read as if seen square on, a 900-organ head was misread as twisted past twelve to fourteen degrees. Stretched back first along the short axis its own second moments find, every head was read right to thirty degrees.

Every photograph in that essay was affine: the camera far enough away that every distance along one bearing is shortened by the cosine of the tilt and every distance across it is left alone. A camera close to a head does not take that picture. The near side of a tilted head is nearer the lens and larger in the picture than the far side, so the squash is not one factor but a factor that changes across the head, and the organs’ centroid, which found an affine head’s centre to a hundredth of a spacing, is pulled towards the near side. That essay ended by asking how close the camera can come before the corrected reading fails, whether the centroid is the reason, and whether a correction for perspective can be fitted from the organs the way the second moments fitted the squash.

A pinhole camera #

The heads are the ones the earlier readings used: golden Vogel heads of 900 organs, every organ displaced by a tenth of a spacing, ten heads on ten seeds, each tilted along a bearing set by its seed so that the ten see ten directions round a half turn. The camera is a pinhole at a stated distance from the head’s centre, in head radii, looking along the line to the centre, with the head’s plane tilted against the picture by 10, 20 or 30 degrees. As the distance grows the picture becomes the affine squash of the earlier essay, and the camera infinitely far is read beside the others for that reason.

A head two centimetres in radius photographed from twenty centimetres is ten radii away; from ten centimetres, five. Those are ordinary close-up distances, not extreme ones.

Each picture is read four ways. Square on, the organs about their centroid. Ellipse, stretched back along the short axis of the second moments about the centroid — the correction that restored an affine photograph. True centre, the same stretch about the head’s real centre, which no photograph gives, to separate what the centroid costs from what the taper does. And fitted, un-projected through a camera fitted from the organs, as below. A head counts as read right when its two annuli’s divergences differ by no more than any untwisted head’s do square on.

A head from five radii #

Photographed from five radii at twenty degrees, the first head reads 137.5064° in its inner annulus and 137.5090° in its outer when read square on, and 137.5069° and 137.5013° when stretched back by its second moments — the stretch has made the two annuli disagree more, not less. Through the fitted camera it reads 137.5083° and 137.5078°, against the golden angle’s 137.5078°, and the fit puts the camera at 5.15 radii and 19.9 degrees. Of the ten heads photographed this way, one is read right square on, none by the ellipse and nine through the fitted camera. Moved by the dial to ten radii, the ellipse reads every one of them right again.

How close is too close #

The ellipse holds to a distance that falls as the tilt grows. At ten degrees it reads ten, nine, nine and ten heads right from the affine far camera in to five radii, and none at three. At twenty degrees it reads ten at ten radii and none at five. At thirty it reads nine at twenty radii, six at ten and one at five. Inside those distances it reads at most three heads in ten right, and at every tilt by two radii it has failed.

Read square on, without the stretch, a head at ten degrees holds as long as the stretched one, since ten degrees is inside what an affine squash costs nothing; at twenty and thirty degrees the square-on reading fails even with the camera infinitely far, as the earlier essay found, reading four heads in ten right.

The difference it makes to the annuli #

The failure is not marginal. The twist threshold, the largest difference between the annuli any untwisted head shows square on, is about one thousandth of a degree. Corrected by the ellipse, the heads photographed inside the failing distances differ by a median of two and a half to twenty-six thousandths — two to twenty-six times the threshold — while where the stretch holds they sit at four to six tenths of it. A perspective photograph read with the affine correction does not make a head look slightly twisted; it makes it look as twisted as a head turned at its rim by a large fraction of a degree.

Is it the centre? #

The centroid does move. Infinitely far away it sits on the head’s centre to a hundredth of a spacing at every tilt, as an affine picture must leave it. As the camera comes in it drifts towards the near side, roughly as the sine of the tilt over the distance: at twenty degrees by 0.06 spacings from twenty radii, 0.13 from ten, 0.26 from five, 0.44 from three and 0.68 from two. The centre the spirals give found that a 900-organ head tolerates a centre about a sixth of a spacing off before it misreads, and the drift crosses that at about the distance the ellipse fails, which makes the centroid the obvious suspect.

It is not the reason. Taken about the head’s true centre, which no photograph supplies, the same stretch fails at the same distances: none of ten read right at three radii and ten degrees, none at five radii and twenty, one at five and four at ten radii at thirty degrees. Wherever the stretch about the centroid reads two heads or fewer right, the stretch about the true centre reads at most four. Giving the reading the right centre buys almost nothing; the stretch itself is the wrong correction.

One number decides it #

The three tilts fail at three different distances, but at one value of a single number. A pinhole at distance L from a head of radius R tilted by τ magnifies the near rim and shrinks the far rim by about a fraction R·sin τ/L either way, so the taper’s strength in head radii is sin τ divided by the distance. The stretch holds at ten degrees from five radii, where that number is 0.035; at twenty degrees from ten radii, 0.034; at thirty degrees from twenty radii, 0.025. It fails at ten degrees from three radii, 0.058; at twenty from five, 0.068; at thirty from five, 0.100. The one cell in between, thirty degrees from ten radii at 0.050, reads six heads in ten right.

So the rule a person needs is one inequality: the second moments are enough while the sine of the tilt over the distance in head radii stays under about a twenty-fifth, and are the wrong correction above about a twentieth. A head photographed straight down its axis has no taper at any distance, and a tilt of a few degrees can be taken from very close.

The failure is a cliff rather than a slope: between the distance where every head is read right and the one where none is, the grid holds no cell that reads half, except at thirty degrees. That is what a threshold reading does. The positions read a twist’s shape set the reading to compare two annuli and declare a twist when they differ by more than any untwisted head’s do, and a taper that turns one side of every ring against the other crosses that line all at once, at the distance where the difference it makes between the two annuli first exceeds what a displacement of a tenth of a spacing makes on its own.

The counts never notice #

None of this shows in the spiral counts. In every one of the 540 readings made here — three tilts, six distances, ten heads, read square on, stretched and through the fitted camera — both annuli count two consecutive Fibonacci numbers, so a person counting spirals on a photograph taken from two radii would see nothing wrong at all. Recovering the angle from the counts alone gives an interval of divergences far wider than the thousandths of a degree at issue here, which is why the counts are untroubled and why they cannot settle the question. The positions can, and the positions are what the camera moves.

So the failure here is the quiet kind. A reading that refused, or a count that changed, would warn a person off; a twist declared on a head that has none, with every count looking right, does not. A twist is a divergence found the counts’ round trip refusing rather than misleading under displacement; the positions’ reading under perspective misleads rather than refuses, and only the fitted camera’s own consistency — a head it un-projects that is round, centred and evenly packed — says anything has been corrected.

The taper is the problem #

What the stretch cannot represent is that a perspective picture shrinks the far side of the head more than the near side — the step past a section seen from the wrong angle, which read a shell’s squash as the same factor everywhere. An ellipse stretches every part of the picture by the same factor along one direction, so it can undo a squash that is the same everywhere and cannot undo one that changes across the head. Read in annuli, a taper makes one side of each ring wider than the other, which moves every organ’s angle by an amount that goes once round the head — a pattern the twist reading, comparing two annuli, takes for a change of angle with radius.

The second moments do not see the taper either. They describe the head’s spread by three numbers, and a taper adds a lopsidedness that three numbers about the spread have no room for. Correcting perspective needs something the second moments discard.

A camera fitted from the organs #

A Vogel head offers three things a correct un-projection must give back: its centroid at its centre, equal second moments in every direction, and an even area per organ across the head, since each organ adds the same area. A picture taken in perspective breaks all three — the near side’s organs are spread out, so their area per organ is larger, and the spread falls off across the head as the cube of the magnification.

So the camera is fitted as five numbers — the picture of the head’s centre, the tilt, its bearing, and the distance — chosen so that the head they un-project has its centroid at the chosen centre, equal second moments, and no slope in the logarithm of its local area per organ, read from each organ’s sixth-nearest neighbour away from the centre and the rim. The fit is told nothing about the true camera.

It finds the camera well. Its tilt is within 1.6 degrees of the truth on every head at twenty and thirty degrees and within 1.4 at ten except from two radii, where one head’s is 3.3 degrees off. Its bearing is within a degree and a third at the median at twenty and thirty degrees and within six at ten, where a shallow tilt gives the bearing little to fix it by. Its distance is within six per cent at the median from ten radii in; from twenty radii at ten degrees it is eighteen per cent short, because there the taper is too slight to measure and a far camera and a farther one take nearly the same picture. And it places the head’s centre within about a fifth of a spacing everywhere, inside the sixth the reading tolerates at every distance but the nearest.

What the fitted camera reads #

Un-projected through the fitted camera, seven to ten heads in ten are read right at every tilt and every distance down to two radii: at ten degrees nine, ten, nine, nine, eight and ten from the far camera in; at twenty, nine at every distance to five radii, eight at three and seven at two; at thirty, ten, nine, nine, eight, nine and eight. In every cell where the stretch reads two heads or fewer right, the fitted camera reads seven or more. The median difference between its annuli sits at a quarter to four fifths of the threshold everywhere.

It is not free. With the camera infinitely far, where the stretch is exactly right, the fitted camera reads one head in ten fewer at ten and twenty degrees: three of its five numbers describe a taper the picture does not have, and fitting them adds noise the stretch never pays. On a photograph known to be taken from far away, the stretch is the better reading; on one taken from closer than about ten head radii, it is the wrong one.

What a person with a photograph should do #

The working rule is simple. A seed head photographed from more than about twenty radii can be corrected by its own second moments, at any tilt up to thirty degrees. From closer than ten radii at twenty degrees, or five at ten, it cannot, and the error it leaves is several times the size of the twist a reading is looking for. There, fit the camera from the head itself, or photograph from further away with a longer lens, which is cheaper.

The fitted camera asks the head to be a Vogel head — round, centred, evenly packed — and it is only as good as that assumption. A head displaced before it is counted found real displacements of the order used here harmless to the counts; a head whose organs genuinely thin out towards one side would read to this fit as a camera tilted towards it.

A flat disc and a pinhole #

That a real head is flat. A capitulum is domed, and seen obliquely its centre and its rim sit at different depths, a relief this pinhole picture of a flat disc does not have and which would add a taper of its own. That a lens is a pinhole: barrel and pincushion distortion are a further radial distortion. And that ten heads a cell is enough to put the failing distances finer than a factor of two; the grid of distances is 20, 10, 5, 3 and 2 radii.

Readings that would put the failure back on the centre #

A distance inside which the stretch fails and the stretch about the true centre reads more than four heads in ten right. A fitted camera that reads fewer than seven heads in ten right at any tilt and distance read, or is off in tilt by more than two degrees at twenty or thirty. An affine photograph that the stretch reads worse than the fitted camera.

The stretch, the taper and the fitted camera #

The second-moment stretch that restores an affine photograph of a seed head fails once the camera is close enough for perspective — inside about ten head radii at twenty degrees, five at ten — and the error it leaves is two to twenty-six times the twist threshold. The centroid’s drift towards the near side is not the cause: the same stretch about the true centre fails at the same distances. The taper is, and a five-number camera fitted from the organs’ own centring, roundness and even packing recovers it and reads seven to ten heads in ten right down to two radii, at the cost of one head in ten when the camera was in fact far away.

Still open: a head that is not flat #

Every head here is a flat disc. A real capitulum is domed, its rim lower than its centre, and seen from the side the dome foreshortens its outer rings more than its inner ones even with the camera infinitely far — a radial taper that no tilt or distance of a pinhole produces on a flat head. The next measurement lays the same heads on domes of stated height, photographs them at the same tilts and distances, and asks whether the five-number camera 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 twist reading mistakes it for a twist.