How many shadows determine the object

The edge of a shadow is drawn on the object established what one outline is: the set of points where the caster’s own surface turns tangent to a ray from the lamp, drawn onto the floor as a boundary between light and dark. Where a shadow splits in two asked what happens to that boundary’s own topology as the lamp moves. This essay asks the question a single outline raises but cannot answer by itself: how much of the object does an outline actually pin down, and how many of them, taken from around the object, does it take to pin down the rest?

One outline is a weak constraint and it is worth being precise about exactly how weak. A single lamp’s shadow says the object lies somewhere inside the wedge its own two tangent lines bound — an unbounded region, clipped only by however far the floor happens to run — and says nothing whatsoever about what the wedge does not touch. Every additional outline, from a different lamp position, cuts a further wedge out of what is left, and the question this essay measures is what that intersection converges to as the lamp count grows: for a convex object, the object itself, at a rate that can be fitted; for anything with a concavity, something that is not the object, and that stops improving no matter how many further lamps are added.

The question is old — a body reconstructed from its own outlines, taken from around it, is the same idea a sculptor uses turning a block on a lathe against a fixed profile gauge, or a machinist uses checking a part against a set of go/no-go silhouettes from several directions — but stating it as a limit, with a measured rate and a measured floor, turns “outlines are informative” into a claim that can be checked rather than merely believed. What follows treats a shadow’s outline as nothing more or less than a pair of tangent lines and asks exactly what a growing collection of those lines does and does not converge to.

Every outline is one wedge, and nothing else #

A wedge is the plainest object this essay handles, and it is worth stating precisely what it is before anything is done with it, because everything downstream follows from this one fact and no other. From a single lamp, a convex section’s shadow-casting boundary touches exactly two tangent lines running back to the lamp — one on each side — and the object is certain to lie in the infinite wedge those two lines cut out of the plane, because any point outside that wedge would either fail to cast a shadow the lamp sees at all or would cast one on the wrong side of one of the two tangent rays. Nothing about a curved or faceted boundary changes this: whatever the section’s own shape, a lamp sees exactly two extreme rays grazing it, because a line from an external point to a bounded convex region always has exactly two tangents, and the object’s whole shadow is bounded by where those two tangents land.

What one shadow says, and what four say together #

A convex cross-section, seen from directly above with one or a few lamps around it, is the plainest place to see a wedge shrink as outlines are added.

Four lamps turn an unbounded region into a bounded one holding only seventeen per cent more area than the object itself, which is most of the work an outline can ever do, done by a small number of them. The remaining question is what happens to that last seventeen per cent — whether it keeps shrinking toward zero as more lamps are added, or whether some part of it is permanent no matter how many outlines are taken. Two lamps and one map is the place this site first put two point sources to work on one object together, and the finding there was about triangulating a single point from two independent readings rather than about a shrinking region; the wedge-intersection here is the same habit of combining several lamps’ readings applied to a shape rather than a point, and it is worth carrying the earlier essay’s caution forward — combining evidence from several sources narrows an answer, but narrowing is not the same claim as pinning down, and exactly how much narrowing buys is the rest of this essay.

Going from one lamp to four is also the point at which the shape of the remaining uncertainty stops being an unbounded wedge and starts being an honest polygon, which matters for what a reader can do with it. An unbounded region cannot be usefully drawn to scale or compared against a real floor; a bounded one, however loose, can be measured, photographed and checked against other evidence the way any ordinary geometric claim can be. The transition from “somewhere in this cone” to “inside this polygon, 1.17 times the object’s own area” is where a shadow-based reconstruction starts behaving like a measurement rather than like a mere constraint.

The convex case closes #

Pushed to a ring of a hundred and twenty-eight lamps, the intersection of wedges on a convex section stops being merely small and becomes, for any practical purpose, exact.

Nothing about that closing is particular to this section’s own shape. A convex boundary is, at every one of its points, on some supporting line — a line touching the boundary at that point and leaving the whole object on one side of it — and a sufficiently dense ring of lamps supplies a supporting line arbitrarily close to any given direction. The limit of intersecting every supporting line the boundary has, taken over all directions at once, is the boundary itself: this is simply what “convex” means, restated as a claim about outlines rather than about line segments between pairs of points. A hundred and twenty-eight lamps is not a magic number; it is merely dense enough that the gap between the reconstruction and the truth has already fallen well below what a photograph or a machined part would distinguish from zero.

It is worth being clear about what work the lamp count is and is not doing here. Doubling the number of lamps does not double the accuracy of any single tangent line — each one is exact, to the arithmetic floor, the moment it is computed, because a tangent line to a known convex boundary from a known point is an exact construction rather than a fitted or averaged one. What more lamps buy is coverage: each new direction supplies a supporting line the earlier ones did not, and the gap that remains is entirely the gap between adjacent supporting lines, which shrinks as the angular spacing between lamps shrinks. This is a statement about sampling density along a boundary that is already known exactly at every sampled point, and it is why the convergence rate above is set by the boundary’s own curvature and the lamp spacing rather than by anything resembling measurement noise.

The concavity that no shadow reaches #

The identical sweep, run on a section with a bite taken out of it, behaves nothing like the convex case past a certain point.

A wedge is a convex region by construction — it is bounded by two straight tangent lines — and the intersection of any collection of convex regions is itself convex. So the reconstruction this method produces is always convex, whatever the true object’s own shape, which immediately says something sharp about what it can and cannot converge to: it can converge to the object exactly when the object is itself convex, and otherwise it converges to the smallest convex region consistent with every outline supplied, which is a strictly larger set than the object whenever the object has a genuine concavity. The bite in this section is inside that convex hull and outside the object, and every one of the hundred and twenty-eight directions used here has a tangent line running straight past the mouth of the bite without ever dipping into it, because a tangent line by definition touches the boundary and does not cross it.

This is also the point at which counting lamps and counting shadows part company from counting evidence in general. Counting shadows is not counting lamps found that the number of distinct shadow regions on a floor can under- or over-state the number of lamps casting them, because two lamps can produce overlapping shadows that read as one region or one lamp’s shadow can be interrupted into several by an intervening obstacle. The failure measured here is a different one again: every one of the hundred and twenty-eight lamps behaves exactly as expected, contributes exactly the tangent line the geometry says it should, and the count is never in question — the shortfall is not in what is counted but in what a tangent line, however many of them are supplied, is structurally unable to report.

The measured rate: closing, and stalled #

Both behaviours — closing toward the truth, and stalling short of it — have a rate, and fitting it rather than merely watching the two pictures is what turns “gets better” and “does not” into numbers that can be compared.

The fitted exponent of -1.97 is not simply “a number close to two.” It is the rate at which a polygon inscribed by tangent lines converges on a smooth curve as the number of sides grows, which is a fact about how a curved boundary is approximated by straight segments and has nothing specifically to do with shadows — the identical rate governs a regular polygon’s area converging on its circumscribing circle’s as the number of sides doubles. Finding that same exponent here, on an object whose boundary is not obviously the kind of curve that theorem is usually stated for, is confirmation that the wedge-intersection method is behaving exactly as a supporting-line approximation should, rather than as some shadow-specific process with its own unrelated rate.

The dented section’s exponent of -0.36 is doing a different kind of work: it is small enough, and the curve visibly flat enough by a hundred and twenty-eight lamps, that continuing the sweep to a thousand or a million lamps would not meaningfully change the picture. A falling exponent of any size, however small, does technically mean the error keeps shrinking forever in the limit of infinitely many lamps; what makes the dented case a stall rather than merely a slow convergence is that the object being approached is provably not the true object at all — the exponent describes the rate of convergence to the wrong answer, not a slow approach to the right one.

This is exactly the trap a fitted rate can fall into if it is read on its own, without the control beside it, and it is worth stating as a general caution rather than a fact about this one figure. A curve falling at -0.36 and a curve falling at -1.97, shown without labels and without the true object to score against, would both look like a measurement “improving with more data” — the difference between converging correctly and converging to a permanently wrong answer is invisible in the shape of a falling curve and visible only once the curve is scored against ground truth rather than against its own earlier readings. How many lamps a drawing has makes the same point about a different quantity: a criterion that only ever rewards fitting the data slightly better keeps preferring more lamps than the scene actually has, because a better-looking fit and a correct count are not the same measurement without an independent noise level to check the fit against. Both essays are the identical warning — a number that keeps moving in a reassuring direction still needs something outside itself to say when it has arrived.

Different in kind, not degree #

A single pair of numbers, read at the sweep’s own endpoint, makes the distinction sharper than either curve does on its own.

A factor of 675 is not “worse”; it is a different claim entirely, and the earlier convex control exists precisely so that the two can be told apart. On the convex section, 0.10 mm is manufacturing-grade agreement, the kind of number that keeps shrinking as lamps are added and that a reader would be right to treat as “solved, for practical purposes.” On the dented section, 70 mm is not a large residual error awaiting more data; it is the exact, permanent, unrecoverable measure of one particular concavity, and no amount of additional shadow evidence collected from outside the object changes it by a single micron.

The same two terms, from a camera rather than a lamp #

The identical split between a falling term and a permanent one appears when the outlines come from a camera moving around the object rather than from a lamp, which is worth seeing directly because it shows the result belongs to outlines as such rather than to shadows specifically.

A camera’s silhouette and a lamp’s shadow are computed by different machinery — one projects along rays converging on a lens, the other along rays diverging from a point source — but a silhouette and a shadow boundary are the identical kind of object: the set of points where a line of sight, from wherever it originates, is tangent to the caster. The two error terms are the same two terms because the underlying fact is not about lamps or cameras at all; it is about what a tangent line can and cannot certify. An exponent of -1.995 sitting a hair away from this essay’s own -1.97 is not a coincidence needing an excuse — both are measuring the same supporting-line convergence, on two sections chosen independently, and the closeness is the two independent measurements checking each other.

The honest limit #

The exact statement behind all of this is worth writing out in full, because “more shadows help” is true and “more shadows solve it” is false, and the boundary between them is precise rather than a matter of degree. What a collection of outlines determines is the intersection of the wedges each one back-projects, and that intersection is always convex; it equals the object exactly when the object is convex, and otherwise it equals the smallest convex set containing the object — the boundary of which is what every one of these outlines is, individually and collectively, powerless to distinguish from the object’s own boundary wherever the two coincide, and powerless to say anything at all about wherever they do not.

That is a necessary, not sufficient condition, stated plainly: every one of the object’s own outlines is consistent with the convex hull sitting in its place, so agreement with any finite or infinite number of outlines never rules out that possibility. This is not a limitation of the method used here, of the number of lamps affordable, or of the resolution any camera or grid could reach; it is a fact about what a tangent line is. A tangent line touches a boundary and does not cross it, so it can never report that the boundary dips back in on itself between one tangent point and the next, and a concavity is defined by exactly that dip. No refinement of the measurement changes this, because the limitation is in the question being asked of the geometry rather than in how carefully the answer is read off.

It is worth adding what would close the gap, since “outlines alone cannot” invites the question of what else could. A single measurement taken from inside the bite — a probe reaching into the concavity, or a second sensing method that is not built from tangent lines at all — settles the question instantly and completely, because it is not subject to the same structural limit; the limit is specific to information gathered exclusively along lines that graze the boundary from outside. A light far enough away is a different kind of limit on a related family of measurements — there the evidence itself fades smoothly with distance rather than sitting at a hard, permanent floor — and setting the two side by side is a reminder that “the measurement stops improving” and “the measurement was never going to reach this” are not the same diagnosis and call for different remedies.

What this is an instance of #

Where a shadow splits in two found a single tangency, at a single critical lamp height, deciding a discrete fact about one shadow’s own connectedness. This essay’s tangent lines are doing continuous work instead — each one shaves a sliver off an unbounded wedge — but they are built from the identical primitive: a ray or a plane brought into contact with the caster’s boundary rather than crossing it. A dent breaks the terminator puts the same primitive to work on the boundary drawn on the object’s own surface — the attached-shadow curve — rather than on the floor, and finds that a concavity there produces a second curve entirely rather than merely an unreachable region; the two essays are the same warning issued twice, once about what a caster hides from a lamp and once about what it hides from itself.

The wider habit these findings belong to runs through most of what this site calls being determined up to something: a single view fixes a ratio and not a size, two views fix a shape and not a scale, and a shadow’s own outlines fix a convex envelope and not a concavity. In every one of those cases the honest answer is not “not enough data yet” but a qualifier attached permanently to what the data can say — and the lamp and the floor cannot both be recovered is the sibling case where the qualifier is not about a missing concavity at all, but a whole family of scene configurations, none of them privileged by the picture, all producing the identical photograph.