An outline reaches a waist and not a dish

Photograph a spool of thread from the side and its waist is the first thing in the picture. The outline curves in from one flange and out to the other, and a reconstruction from a ring of such outlines gives the waist back as a waist. That ought to be a difficulty, because the dish no outline reaches established that a silhouette cannot enter a concavity: a bite in an object’s boundary stays filled at four views and at a hundred and twenty-eight, and the convex hull is the floor under every reconstruction. The spool’s waist is a concavity, plainly, and it lies inside the convex hull, plainly. It comes back all the same.

That essay said in its closing lines that it did not treat three dimensions, and treated the omission as arithmetic. It was more than that. Everything it proved about the convex hull is a fact about lines in a plane, and a solid has lines a plane does not.

A section through the axis hides the lines that do the work #

Cut the spool down its axis and the section has two bites in it, one each side. Inside that section the old result holds exactly: a line lying in the plane of the cut, passing through a point in the left-hand bite, must cross one flange or the other or run into the far side of the body. No view whose rays lie in that plane can enter the bite.

The view that shows the waist is not in that plane. A camera looking at the spool from the side has rays running horizontally, and the ray that grazes the waist on the near side of the object is horizontal and tangent to the circular slice at that height. It passes through the hollow and clears the body. In the section through the axis the ray is perpendicular to the page, and all that remains of it is a dot. The drawing that makes the hollow look like a bite is the one drawing in which the line that reaches the hollow cannot be seen as a line.

So the waist has not escaped the argument about supporting lines. It has satisfied it. Every horizontal slice of a spool is a disc, a disc is convex, and a ring of outlines taken round the axis treats each slice as its own two-dimensional problem. Two outlines are two curves makes the matching point about a ball: the outline of a solid belongs to a curve on its surface, and a different view has a different curve. Slice by slice there is no concavity anywhere for the supporting lines to fail at. The concavity belongs to the profile, and no view in the ring looks along the profile.

The test is whether a line can leave #

What replaces the convex hull is a simple condition, and a surprisingly old one. A point outside a solid is cut away by a silhouette when the view’s ray through that point misses the solid, because then the point projects outside the outline. So a point can be cut away by some view if and only if some straight line through it misses the solid altogether. The set of points with no such line is what Laurentini named the visual hull of an object in 1994: the limit of the reconstruction, taken over every line there is rather than over the rays of any particular rig.

In a plane, this condition and the convex hull are the same thing. A line through a point in a bite that misses the object would have to separate it from the object or thread between two parts of it, and a connected plane figure has no gap to thread. In three dimensions a line can go round. That is the entire difference, and it makes the convex hull too pessimistic: it sets the floor by asking what supporting planes can do, when a silhouette is made of supporting lines, and many more lines touch a solid than planes do.

The measurement is deliberately crude: take a point, walk along a candidate line in both directions, and report whether the walk ever lands inside the solid. Twelve hundred directions over the half-sphere are tried at each point, horizontal ones first. Nothing about the solid enters but a yes-or-no for inside. Keeping the test this plain is the point, since every closed form below is something the plain walk can disagree with.

A dish is different, and the proof is two lines #

Now take a solid drum and press a shallow bowl into its lid. In a section through the axis it looks like the spool’s section turned on its side: one more hollow, inside the same dashed convex hull, no deeper and no stranger.

Take any point in the hollow and any line through it. Follow the line both ways. If it stays level it runs into the raised rim of the dish on either side, because the point is below the rim. If it tilts, then one end rises and the other falls, and the falling end hits the bowl under it. There is no third case. Whatever direction a line takes through that point, one of its two ends is inside the solid, so no line leaves. The walk confirms it at every point sampled: none of the 1,200 lines tried at any of them gets out.

Saying the dish is “inside the convex hull” tells nothing, since the waist is too. The dish is enclosed in a sense the waist is not. It is walled on every side at a height above the point, so that a line can climb out at one end only.

Put both on one object, a spool with a dish in its upper flange, as the figure at the head of this page does, and the result is clean. The convex hull’s excess over the solid is 2.405 units of volume. Of that, 2.136 is the waist and 0.269 is the dish. An unlimited ring of outlines takes the excess down to exactly 0.269: the waist has gone and the dish has not been touched. The two hollows are on the same solid, photographed by the same camera, and their fates have nothing to do with each other.

The waist’s error is the falling term; the dish’s is the flat one #

The old essay’s most useful statement was that a silhouette reconstruction has two errors, one that responds to effort and one that does not, and that they must not be averaged. Both survive, but what sits in each column has changed.

A ring of $n$ views around a spool gives each horizontal slice $n$ pairs of supporting lines, so the slice is cut to a regular polygon with $2n$ sides around its disc. Each slice therefore keeps an excess of $(2n\tan(\pi/2n) - \pi)\rho(z)^2$, which falls as $n^{-2}$, and the integral over the height is the waist’s error. That count of sides is worth stating because it came out wrong the first time here: a view supplies two lines, one on each side, and writing $n$ sides for $n$ views gave 1.133 at four views where the walk, counting points one at a time and sharing no arithmetic, gave 0.223. The closed form with $2n$ gives 0.227. The earlier essay’s own table of views for a stated tolerance carried the same slip, and it now says what it should: nine views for one per cent, not eighteen.

The rate is the one a disc has, because the error is a stack of discs’ errors. So a waist costs no more to recover than a convex body of the same size, and much less than a faceted one, which falls one power more slowly. There is no extra difficulty in the hollow at all, provided the views are in the right place. The next essay takes that proviso seriously.

A lid can be concave one way and still be reached #

The spool and the dish look like two kinds of concavity, and the geometry has a name for the difference. On a surface of revolution with a waist, the middle of the waist curves inward along the profile and outward around the slice, and the two principal curvatures have opposite signs. It is a saddle. The bottom of a dish curves up in every direction, so both curvatures agree. The product of the two is the Gaussian curvature, and at the waist’s narrowest point it comes to $-3.3$ on the spool measured here; a point on the dish has the opposite sign.

The test for that is local and worth stating. At a saddle point of a surface there are always two tangent directions along which the surface does not curve away, and between them directions in which the tangent line stays outside the solid in the neighbourhood of the point. At a point where both curvatures make a bowl, every tangent line immediately enters the solid. Curvature’s sign tells a point whether a line can graze it at all, which is why a curved surface made of straight lines belongs in this discussion, even though it was written for a different reason. A doubly ruled surface is the extreme case: a straight line lying in the surface, through every point.

The cleanest way of seeing it is to change nothing but the sign. Take two square drums. Give one a lid that is a saddle, $z = H + cxy$, and the other a lid that is a bowl, $z = H + c(x^2+y^2)/2 - c$, with the same $c$. Test a point a hundredth of a unit above the lid at each place.

On the saddle every cell has a line that leaves, and the reason takes one sentence. A line at constant $y$ over that lid has a height that changes linearly with $x$, exactly as the lid does along that ruling, so a line parallel to the ruling and one hundredth above it stays one hundredth above it from one edge to the other. Above a saddle, there is always a way across.

On the bowl, the same test finds lines only near the corners.

What the lid would hold as water, no line can reach #

The square bowl also gave the result that surprised this measurement, and it is why the claim here is stronger than a sign. A round dish with a level rim is unreachable in full. But the square bowl is not. Its rim dips: the corners stand at the full height $H$ and the middle of each side is $c/2$ lower, because the bowl’s own curve runs along the edge. And the walk found lines reaching 69 per cent of the hollow between that bowl and the flat ceiling of its convex hull. A concavity with both curvatures the same sign, and most of it reachable.

So what part is not? Pour water into the bowl. It rises until it reaches the lowest place on the rim and then runs out, at the spill level $L$. Now take any point below $L$ and suppose some straight line through it clears the solid. Then the line clears the rim at both ends, so at each end it stands above the rim, and so at or above $L$. But a straight line’s height between two points lies between its heights at those points, so it cannot be below $L$ anywhere between them. That contradicts the point it passes through. Everything a solid’s lid would hold as water, no line can reach. That is the whole proof, and it needs nothing about the shape: not smoothness, not symmetry, not curvature at all.

The bowl with depth 0.35 holds exactly $\pi c/4 = 0.2749$. The walk, knowing nothing of this argument, finds no line from any sampled point below the spill level, on any grid tried. It is not an equality, though, and the measurement says so. Above the spill level there is a further 0.021 of volume, about eight per cent more, that no line leaves either. A line above the spill level must get out through two low places, and on a square rim the low places are in four particular positions, so a point not lined up with any pair has to climb over a higher part of the rim and cannot. That excess held still when the directions tried were made many times finer, so it is a property of the bowl and not of the sampling. What the bowl holds is therefore a floor on the part that cannot be reached, not the exact amount. And that eight per cent is the same on a shallower or a deeper bowl: measured at depths from 0.175 to 1.05, it stays at 7.5 per cent throughout. It has to, since stretching all the heights of a solid by one factor turns straight lines into straight lines, so a deeper bowl is the same problem with another vertical scale.

The round dish in the earlier section is the case where those two agree exactly, because its rim is level: the water rises to the rim, the hull’s ceiling is the rim, and there is no space above the spill level for a remainder.

What this does to the earlier result #

Nothing in the plane changes. The notched disc’s bite is still unreachable by lines in its plane, the convex hull is still its floor, and all of its numbers stand. What changes is the reading of them. The earlier essay spoke of “a concavity” and its object was a cross-section, and a reader was free to think of a notched solid as behaving the same way.

Extrude that notched disc into a long prism and the prism’s notch is a straight channel. A line along the prism’s axis runs down the channel and out of both ends. So a single view along the axis clears the entire notch, in one picture, and every other view of it fails. The two-dimensional result was true because its views were confined to one plane, and that confinement is exactly what a turntable imposes. An experiment with a level ring and an object whose hollows are all channels along the ring’s axis behaves like the section, and looks as though it confirms the convex hull. Close the channel’s ends and it has become a sealed slot with something to hold, and the convex-hull result is finally correct about it, for the different reason given above.

The same reading applies to the objects that essay compared with its notch. Two pictures of a ball and the wall under the paint recover their objects because the pictures carry correspondence, which outlines do not. That contrast stands; only the outline’s side of it has grown. An outline is still a tangent line and nothing more. It is simply that a solid offers it many more places to be tangent. The ball a drawing does not draw round is the limiting case, where every point of the surface is on some outline and the visual hull equals the solid.

What the limit is, stated for a surveyor of objects #

For a practical shape reconstruction, three different statements now need keeping apart.

• What the outlines of a rig reach is set by the rig’s lines: a level ring reaches what a horizontal line can leave, a tilted ring what a tilted line can leave.
• What outlines from anywhere reach is the visual hull in Laurentini’s sense, computed over all lines. It is never smaller than the object and usually larger.
• What outlines can never reach contains everything the object would hold as water, in any orientation, and in general a little more.

The words “in any orientation” are doing work there. The proof above stood a lid the right way up, but it never needed gravity: “height” can be measured along any axis, and the proof goes through for each. A cup holds water only when upright, yet a cup lying on its side is the same solid with the same unreachable interior. The curvature a shadow reports is a different instrument with a different limit, but the same habit helps there: the sharp question is what the instrument physically cannot touch, and a cheap argument often answers it.

Still open: how aimed an outline has to be #

The waist came back in full from a level ring, and that hides an assumption that has not been tested. A level ring contains the lines the waist needs because a spool’s slices are horizontal and the camera has been placed at their height, with no tilt. A spool’s waist is forgiving, because it is wide and open: lines at quite a range of angles get into it. A narrow groove cut round a drum is the same kind of object, a saddle on a solid of revolution, reached in full in principle. But the lines that reach its floor lie very nearly in the groove’s own plane.

The measurement that settles what that costs takes a square-cut groove, computes how deep a line reaches as it tilts out of the groove’s plane, and asks three things: at what tilt the floor stops being reachable; how much of the groove is left to a camera on a turntable raised to the twenty or thirty degrees such cameras are usually raised to; and how many views scattered at random over the sphere would be needed, in place of a well-placed ring, to have a fair chance of a line that gets in at all.

A concavity is not what stops an outline. What stops it is having no straight way out, and the plainest test of that is to ask whether the hollow would hold water.