A groove is reached only along its own plane

An outline reaches a waist and not a dish turned a question about concavity into a question about lines: a point can be cut away by some silhouette exactly when some straight line through it misses the solid. That condition is a yes or a no. It classifies hollows as reachable or not, and it says nothing about how hard a reachable hollow is to reach. A spool’s waist happens to be an easy one. A level ring of cameras contains lines that enter it from every side, and the error falls with the view count at the same rate a plain disc’s does.

Cut a narrow groove round a drum and the classification does not change. The groove’s floor is curved outward around the drum and is walled above and below; a horizontal tangent line leaves from any point in it. By the test of whether a line can leave, it is the same class of hollow as the waist. It behaves nothing like it, and the way it differs is the subject here. The lines exist, and hardly any camera holds one.

The object throughout is a drum of radius 1 with a square-cut groove 0.25 deep and 0.06 wide. That is the proportion of a retaining-ring groove, made deeper so the numbers are easy to read.

A line has to stay between the walls for its whole length #

Take a point on the groove’s middle plane, at a depth $x$ below the drum’s surface, so at radius $r = 1 - x$. Any line through it that leaves the solid has to travel from there to outside the drum’s radius, in both directions, without touching the side walls. The shortest such journey is along the tangent to the circle of radius $r$, where the half-chord inside the drum is $\sqrt{1 - r^2}$. If the line tilts out of the groove’s plane by an angle $\tau$, then along that half-chord it rises by $\sqrt{1 - r^2}\tan\tau$. That must not exceed half the groove’s width.

$$\sqrt{1 - r^2}\,\tan\tau \;\le\; \frac{w}{2}$$

That is the whole of the geometry, and it has two consequences that the picture does not make obvious. One is that the depth reached falls quickly with tilt, since the half-chord grows as the square root of the depth, and so a small tilt with a short allowed chord gets only a little way in. The other is that the view count does not appear in the condition. A line at the tilt $\tau$ either fits or does not. Taking more views at that tilt adds more azimuths of the same kind of line.

Setting the left side to the depth of the floor gives the tilt at which the floor stops being reachable: $\tan\tau = w / (2\sqrt{1 - (1-d)^2})$, or 2.60° for this groove. Below that tilt a line reaches the floor. Above it the reach collapses, to 72 per cent at 3°, 24 per cent at 5°, 5.8 per cent at 10° and half of one per cent at 30°. The search is checked against this at every tilt listed and they differ by no more than one step of the search grid. The check matters because the tangent’s optimality is an assumption until something else has found the same limit: a non-tangent line through a mid-plane point has a longer chord and a longer rise, but it was still possible that some asymmetric route did better, and over 720 azimuths none does. Off the middle plane an asymmetric route does help, because the point is nearer one wall than the other; the section figure below counts those, and they do not rescue the floor.

A turntable has a height, and the height is a decision #

Hardly anyone shoots an object on a turntable with a level camera. A level camera looks straight across and does not see the top of the object, so it is usual to lift the camera by twenty or thirty degrees and shoot down, and a ring of views at that height is what a turntable capture usually means. Two outlines are two curves already notes that a different view is a different curve on the surface; what the raised ring adds is a constraint on which curves are available. Every ray in the set has the same tilt.

Three degrees is barely above the floor tilt and already 56 per cent of the groove’s section stays filled. Move the slider and it goes fast. This figure and the first are not two measures of one thing: the first gives the deepest reach anywhere on the middle plane, and this one shows the whole section, which is less. A line near a wall has less rise to spend, so the section cleared always falls before the middle reach does.

The figures at 20° and 30°, 0.7 and 0.2 per cent, are small enough to require a second reading. On a turntable at a normal height, this groove is not reconstructed as a shallow or blurred groove. It is reconstructed as a line drawn on a smooth drum, or as nothing, and reconstruction software has no way of reporting the difference, because a smooth drum satisfies every outline perfectly. The dish no outline reaches said the same of its notch, and it was a fair warning there. The difference here is that the groove really is reachable, and a capture from slightly different heights would have found it. The limit was chosen by whoever set up the tripod.

Counting views does not help, and it can be shown by counting them #

The usual reflex in the face of a poor reconstruction is to shoot more pictures. The argument above says this cannot help beyond a point, and it can be measured, because a finite ring has its own shortfall over the limit: it may have no view in the right direction at the worst place round the drum. So take $n$ views round the turntable at a stated height, find the worst azimuth, and ask how deep a line from that ring gets at that azimuth.

The level ring goes from 75 per cent at four views to all of it, as the polygon around the floor’s circle closes on the circle: with $2n$ lines around a disc of radius 0.75, its corners stand $0.75(1/\cos(\pi/2n) - 1)$ beyond the floor, which is 0.062 at four views and under 0.001 at thirty-two. Nothing about the groove makes that hard. At 5° the ring has reached 21 per cent by sixty-four views and 23 per cent by two hundred and fifty-six, against a limit of 24.3. At 20° it reaches a third of one per cent by sixty-four and a whole one per cent by two hundred and fifty-six, against a limit of 1.4. Those last gains are real, and they are gains in a small number. At a shallow reach the directions that work are a narrow lens, so a raised ring has to be dense before it has one in the lens at every azimuth, and all that density buys is a little closer to a limit that was fixed before the first picture.

That is a different kind of statement from the falling and flat terms of the earlier essays. The falling term there responds to the count and the flat one to nothing. A groove’s error has a limit that responds to something, namely the height, which is a cheap variable to change, and an approach to that limit which responds to the count. Reporting a capture’s quality by its number of frames would call a sixty-four-view ring at 5° much better than a twelve-view ring at the level. For this object it reaches nearly five times less far.

The same ordering holds for a spool’s waist, if not nearly as sharply. A spool’s waist is wide, so the tilt that loses a large part of it is large. What varies from one object to another is how wide the set of useful lines is, and the shape of a hollow fixes that: a wide open waist has a wide set and a narrow groove a narrow one. The classification gave a yes for both.

The lines that work, drawn on a map of all the lines #

The tilt limit concerns a single number. The full answer for a given point is a set of directions, and it is short work to draw it. Take the point at depth 0.1 on the middle plane, and for every direction test whether a line through the point leaves the solid.

The set is two small lenses, one on each side of the tangent direction, and they are the same lines read in the two senses. At zero tilt the lens’s width in azimuth comes from the floor: an azimuth too far off the tangent makes the line dip below the floor’s circle and hit the bottom. Its height comes from the chord, by the formula above, and reaches 3.9° at its tallest. Each lens is about 66° wide by under 4° high on this map, so most of what a lens loses comes from the tilt side. A camera has plenty of azimuth to choose from, and very little height.

That shape is why a ring works and a scatter does not. A ring puts many directions along one row of this map, and if that row passes through the lens at zero tilt, then a view every sixty degrees or so is enough at this depth. A set of views distributed evenly over the sphere spreads the same number of pictures over the whole map, and most land in the great majority of the directions where nothing leaves. The difference in cost has a precise size.

What views at random would have to number #

Suppose the camera positions are simply scattered, as in a walk around an object with no plan, and suppose the object’s groove is not known in advance. A direction is then a uniform point on the sphere, and the chance that a single view can cut away a given point in the groove is the area of the two lenses against the sphere’s. For a point near the floor it is a little under one per cent. The number of views needed for a stated chance of at least one working view is then fixed by arithmetic: $\ln(1 - P)/\ln(1 - p)$.

108 views for a point just inside the rim, 192 at 0.1, 330 at 0.2, 685 at 0.24. These figures are for one point. To carve out the groove all the way round, at every depth, the same random directions must serve every azimuth at once, and those at different azimuths need different directions, so the true number is larger. A level ring of eight reaches 94 per cent of the depth at its worst place, and a ring of thirty-two reaches 99. That is a very large ratio, and not an argument against scattering views in general: on a ball or a convex lump, where the useful directions are all directions, scattering is at no disadvantage whatever.

The sharp cost arises only if the object has hollows whose useful lines lie in a narrow set. A photographer who knows about the groove aims at it. A capture that does not know has to pay for all the directions a groove might want, or lose it.

What separates this from the falling term, and why it is the same as the flat one #

It may be tempting to write this up as a third term alongside the two the earlier essays separated. It is better understood as the same flat term, with a condition on it. The flat term in the dish no outline reaches is the set of points with no line that leaves, and the wall under the paint shows what a different measurement does about it. Take a rig as given, restrict the lines to those it contains, and the same definition applies: a rig has its own visual hull, computed over its own lines. For a level ring that hull is almost the true one for this object. For a ring raised 20° it has almost the whole groove in it, flat in the view count, exactly as a dish is for every rig.

So there is one idea in play, seen at three levels. Over all lines, the flat term is what the object holds as water, or a little more. Over the lines a rig contains, the flat term is larger, sometimes much larger, and it depends on the rig. And the rig’s own falling term goes down toward its own flat term at a rate set by the number of views. One picture of a ball gets to skip almost all of this because a ball has no hollow and needs only a single outline; most other things do have to answer the question.

A practical test follows, and it uses no knowledge of the object at all. Compute the hull twice, from two rigs: the capture as shot, and the same capture with half its frames dropped. If the shape does not change, the capture has converged, which is the standard reading. That reading is compatible with being converged on the wrong thing. To tell, compute the hull from a capture at a different height. A change there says the rig’s limit was the deciding term, and it is the one difference that more frames at the original height could never reveal. This is the same move as a survey tested against one it did not use: the instrument cannot report on its own limit, so the check has to come from somewhere else.

What a wider groove and a square cut were hiding #

Two choices were made about this groove and both should be declared. The groove is 0.06 wide, and the formula is linear in the width: a groove twice as wide has a floor tilt about twice as large, until the small-angle behaviour stops being a good approximation. Moving the slider on the tilt figure shows this directly. At 0.14 wide the floor tilt is 6°. A groove has to be very wide before an ordinary turntable height reaches its floor, because 30° needs a width of about 0.76 on this drum, and at that size it is no longer a groove but the kind of waist a spool has.

The groove is also square-cut. A V-shaped or rounded groove has walls that lean back from its floor, so a tilted line has a little more room near the surface, and the section cleared has a different shape. It would change the near-rim numbers somewhat, and nothing else, because the floor’s own condition is still set by the narrowest part. The argument here about a set of lines is unaffected and so is the disagreement between the ring and the scatter; the square cut is used because it gives a closed form to test the search against. A drum with a helical thread instead of a groove is a different and harder case, taken up below.

Two more things this does not settle. The rays were parallel, as they were in the earlier essays on purpose. A camera at a finite distance sees the groove at slightly different tilts from one end of the frame to the other, so a raised camera at close range has a small band of tilts, and that could include a useful one for a groove near the frame’s edge. And no real silhouette is sharp: a pixel’s width of outline noise blurs the reach near the rim, where the limits above are smallest, and would probably eat most of the tiny percentages at 20° and 30°. So the figures there are best read as upper limits.

Still open: what a thread does #

A groove is closed: its plane is a plane, and a line can be laid in it, much as a line can be laid along a surface made of straight lines. Cut the same hollow as a helix, as a screw thread, and there is no plane. A line through the channel can follow the helix for a short distance only. The classification still says that some line leaves from every point, because near any point the thread’s two walls are almost parallel and locally it is a groove with a small slope; so a level ring ought to fail, and a ring raised to the helix angle might succeed on one side of the drum and fail on the other.

The measurement that settles it cuts a thread of stated pitch and depth in the same drum, finds at each point the set of lines that leave, and asks three things: whether a single ring at the helix angle clears the channel on the near side and leaves it on the far; whether two rings, raised above and below the level by that angle, clear all of it; and, since a thread of many turns has many channels side by side that interfere only through the helix’s own rise, at what pitch the screw stops being a set of grooves and starts being a rough cylinder with a slightly smaller radius.

A rig has a visual hull of its own. Once the count of views has done its work, the reconstruction has converged on that hull and not on the object, and only a view from a different set of lines can tell the two apart.