The lamp's size over its distance, and nothing else

A soft shadow on a curved floor is not the lamp’s image settled which floors let a penumbra be trusted as a record of the lamp at all. On the flat floors that survive that filter, the next question is sharper: trusted to say what, exactly, about the lamp?

A single straight-edge shadow answers one question and one only — how wide the lamp looks from where the edge is. It does not answer how wide the lamp actually is, and it does not answer how far away it is, because those two quantities enter the construction only as a ratio. Scale a lamp and its distance together by the same factor and the band it casts does not change at all: the same picture on the floor is compatible with an 18 cm lamp at 1.5 m, a 36 cm lamp at 3.0 m, or a 90 cm lamp at 7.5 m, and nothing in one shadow can tell those apart.

The reason is the same reason a nearer coin and a farther dinner plate can subtend the same angle at an eye. Every ray this construction casts leaves a point on the lamp and grazes the same edge, so the triangle formed by a rim, the edge and the floor has one angle fixed by the geometry and the lamp’s half-width and height above the edge entering only through their ratio — double the half-width and double the height and every one of those triangles is congruent to the one before, just carried out along rays that are twice as long. A shadow is silent about anything that a similar triangle is silent about, and a similar triangle is silent about absolute scale by definition.

That this holds for a straight edge and stops holding as soon as a card of finite width is substituted is not obvious in advance, which is why the family above is drawn rather than merely argued for: an edge has no length of its own to compare a lamp’s size against, and a card does, so the same rescaling that leaves a straight-edge shadow unchanged moves a card’s shadow by an amount fixed by the ratio of the lamp’s size to the card’s own — a ratio that does not rescale even when the lamp and its distance both do.

The degeneracy is exact, and it is a fact about the edge #

A difference of 0.0e+0 sounds like the kind of claim that ought to be taken on faith or hedged as “negligible.” It is neither, and the reason is worth making precise before anything is built on top of it.

The three rows are not three degrees of the same effect; they are a control and two ways of breaking it. The straight edge has no width of its own, so nothing in the geometry distinguishes “far and large” from “near and small” — every ray that would tell them apart passes through the same one-dimensional edge regardless. A card of finite width does have a length scale that does not participate in the lamp-and-distance rescaling, and the moment that second length exists the five members stop agreeing, all the way up to 1.000. And scaling only the lamp, leaving its distance fixed, obviously changes the picture on the floor — the exact reading a person expects — which is there to confirm that the family really is being redrawn each time and the zero above is not an artefact of the code returning a cached answer. A method that reported exactly nothing on all three rows would be broken; one that reported exactly nothing only on the row with no second length is behaving.

The 1.000 is worth reading as a fact about the card rather than about the lamp. It is the largest value the underlying quantity can take at all — a place that sees the whole lamp behind one arrangement and none of it behind another — so the card family is not merely “different,” it is as different as two profiles compared this way can be, at some point on the floor. The 0.296 sits between the two extremes for a reason that is itself worth noting: holding the distance fixed and scaling only the lamp changes both the lamp’s absolute size and, through it, the umbra’s own extent, but it does not introduce a second length scale the way a card does — it is simply the ordinary, expected sensitivity of a shadow to the size of what is casting it, present at every rung of this site and not a new phenomenon in its own right. What makes the top row the interesting one is not that it differs from the bottom two, but that a straight edge is the one arrangement, among the three, for which the natural expectation — “of course a bigger, farther lamp looks different from a smaller, nearer one” — turns out to be false.

What one flat measurement can actually say #

The angular size a straight-edge shadow reports is a real, well-defined quantity, and it is worth putting a number on before asking what more is needed. A source of known width, at a known distance from an edge, images through that edge exactly as the lamp above does.

The penumbra is the lamp’s image is where that closed form is first stated, and this is the same formula read for what it withholds rather than what it supplies: a width divided by a distance is an angle, and an angle is exactly the information a single flat measurement contains. Nothing about adding more precision to one photograph changes that. A perfectly sharp edge, measured to an arbitrary number of decimal places, still yields only the ratio, because the ratio is the only combination of the lamp’s size and its distance that appears anywhere in the equation relating them to the band’s own width.

The one number that depended on which card came first #

That angular size is not merely a ratio in the abstract; it is a ratio read from somewhere, and where it is read from turns out to matter in a way this essay’s own machinery got wrong before it got right.

An angular diameter is not a property of a lamp on its own — it is a property of a lamp and a place to stand, exactly as apparent size always is — so reporting “the” angular diameter of a lamp recovered from two cards requires saying which of the two heights it is reported from. The first version of this measurement did not say so: it reported the angle from whichever card happened to be listed first in an array of measurements, which meant the identical pair of cards, handed over in the other order, described a different lamp — 10.29 degrees against 33.40, both arithmetically correct and neither the same question, because “how wide does the lamp look from 1.00 m” and “how wide does it look from 2.40 m” are two different quantities that happen to share a source. That is now reported from the lower of the two cards by convention, with the general formula available for any other height a reader wants it evaluated at. What was never in doubt through any of this is the recovery of the lamp’s actual size and height, which is the pair of numbers the rest of this essay is built on and which do not depend on an ordering at all.

The distinction matters because it is easy to conflate two claims that sound alike and are not. “The lamp’s angular size cannot be recovered without ambiguity” is false — an angular size is perfectly well defined the moment a place to measure it from is named, and this essay’s own opening figure computed one, 33.40 degrees, without any trouble at all. “The lamp’s actual size and distance cannot be recovered from one angular measurement” is true, and is the degeneracy this essay is about. The ordering bug conflated the two by making the first claim look shaky — an angle that changed when nothing about the lamp did — when the actual fault was silence about which of many equally legitimate angles was being reported. A recovered lamp has one height and one half-width; it does not have one angular diameter, because it can be looked at from more than one place, and the fix is to say from where rather than to treat the angle as if it were the lamp’s own property.

Breaking the degeneracy: a second occluder at a different height #

The family figure showed what one straight edge cannot do. A second edge, at a different height from the first, is what supplies the length the single measurement was missing.

The reason two heights suffice where one cannot is worth stating exactly rather than left as “more data helps.” A card at height y leaves a band whose width w satisfies w = 2R·y/(h − y), where h is the lamp’s height and R its half-width — one equation in two unknowns from one card, and one equation in two unknowns has a whole line of solutions rather than a point. A second card at a different height is a second equation, and two equations in two unknowns generically have one solution. The height’s presence in the denominator is what makes a single card insufficient in a way that is not obvious from the picture alone: it is not merely that one number under-determines two, it is that the one number available is a nonlinear mixture of both, and no amount of precision on that single mixture separates its ingredients.

Why a second card rather than, say, a second reading of the same card taken more carefully is worth being explicit about too, since the two are easy to run together. Reading the same band twice, however precisely, supplies the same equation twice — it can shrink the noise on w, and the conditioning reading below is exactly about how much that noise matters, but it cannot add a second independent equation, because y has not changed and neither has the relationship between y, h, w and R. What breaks the degeneracy is not more measurement of the same thing; it is measurement of the thing at a second, distinct value of the one quantity — the occluder’s own height — that the equation treats as known rather than as one of the two unknowns being solved for.

Why two points determine a line, and one point determines nothing #

Rewriting that same relationship linearises it, and the linear form is what makes the “one card is not enough” claim visible as a picture rather than an argument about algebra.

A single card is a single point in that picture, and a single point has every line in the plane running through it — which is the geometric statement of what “one measurement, two unknowns” means, made visible rather than merely asserted. Offered one card’s band alone, the least-squares fit this essay’s recovery uses does not guess at a slope; it refuses outright, on the stated ground that a lamp’s size and height are two unknowns needing two measurements and it was given one. That refusal is not a failure of the method — it is the method correctly reporting that nothing in a single band determines a line through it, which is the same fact the family figure showed by exhibiting five different lamps a straight edge cannot distinguish. A solver that quietly returned some height and width from one card, by defaulting to an arbitrary slope or picking the nearest previously-seen answer, would be worse than one that refuses, because it would look exactly like a measurement.

What five millimetres of misreading costs #

A refusal at one card and a clean answer at two is not the end of the story, because the two cards are not equally good at supplying that second point depending on how far apart they are.

The line picture from the previous reading explains this without anything further needing to be measured. Two marks close together fix a line’s intercept (the lamp’s height) very badly, because a small error in either mark’s position swings the whole line through a large arc at the axis; two marks far apart fix it firmly, because the same small error in position is now a small fraction of the line’s own run. The condition number — the ratio the least-squares fit reports from the two heights alone, with no noise involved — states the same fact in one figure available before a single trial is drawn: 491 for cards a hand’s width apart, 37 for cards a metre and a half apart, a factor of thirteen. The scatter itself moves further still over the same range, 4.11 m down to 0.026 m, a factor of 158 — the condition number is the cheap warning a reader can read off the arrangement before touching a shadow, and the trials are what confirm that the warning understates rather than overstates how much the separation matters. A reader with two cards and a tape measure has the warning available immediately, simply by asking how far apart the two heights are.

The honest limit #

Everything this essay recovers depends on having two occluders at two different heights in the same photograph, and a photograph that shows only one straight edge cannot be made to supply the second no matter how carefully its single band is measured — sharpening the edge, correcting the lens, averaging many frames, none of it manufactures the missing length, because the missing length was never encoded in that single band to begin with. This is not a statement about measurement noise; it is the same statement the family figure already made, that the shadow itself carries no such information for a solver to extract however good the solver is.

That limit has a practical shape worth stating plainly: two cards at two heights are not always available. A single doorway, a single window frame, a single railing — one straight edge in the whole scene — leaves this recovery with nothing to solve, and the honest response is the same refusal the solver gives when handed one measurement rather than a wrong number produced by assuming a plausible distance. A reader with only one edge is not stuck with a worse version of this measurement; they are stuck with a different, strictly smaller one — the angular size and no more — and the two should not be presented as though the second were merely a noisier version of the first.

Nor does this essay’s construction generalise past the strip source it uses without further work. A lamp with a size that is not effectively linear — a disc, a window, an irregular fixture — leaves a penumbra whose closed form is not the one line this essay linearises, and the two-measurement argument would need to be rebuilt from that shape’s own equation rather than borrowed unchanged. What would carry over unchanged is the shape of the argument rather than its algebra: any source with an extent has some closed-form relationship between an occluder’s height and the width of the band it leaves, that relationship has the source’s size and the source’s distance entangled in it exactly as the strip’s does, and breaking that entanglement will need a second occluder at a second height for the same reason it does here — one measurement of one ratio-like quantity, however that quantity is shaped, is one equation in two unknowns.

And nothing here says anything about the brightness of either band, only its width. A photograph’s own exposure, the lamp’s actual wattage, how the eye perceives a soft edge against a hard one — all of that is closer to the honest limit a soft shadow on a curved floor is not the lamp’s image already stated for a different measurement on the same construction: the geometry of where a ray lands is computed here, and whether either band is bright enough to find on a real photograph is a separate question this site does not take up. The lamp, out of the picture, a light far enough away, the lamp comes out in rays and not in plan, the drawing does not run out of lines and a lamp behind the camera recover a lamp’s position from an entirely different construction — the lines drawn through objects and their shadows rather than the width of a soft edge — and the lamp a low shadow cannot locate measures exactly how that other construction degrades as the lamp gets low, in the same spirit this essay measures how the penumbra construction degrades as the two cards get close. The two families answer different questions about the same lamp — where it is, against how large it is — and neither substitutes for the other.

How many lamps make one lamp and what a point in shadow can see of the sky both start from the same flat-floor band this essay has just finished characterising, one asking what stands in for a source with a size and the other asking what an extended source leaves visible from underneath it. Both inherit the finding here without restating it: a single flat measurement of a soft edge is an angle, is only ever an angle, and needs a second measurement at a second height before it can become a size.