A soft shadow on a curved floor is not the lamp's image

The penumbra is the lamp’s image settles the flat case: a lamp with a size, seen past a straight edge, does not cast a hard-edged shadow but a soft one, and the soft part is nothing more exotic than the lamp’s own image, cast through the edge as though the edge were a pinhole’s aperture. Its two boundaries come from two rays — one from each rim of the lamp, both grazing the edge — and where those rays land is where full darkness gives way to full light. On a flat receiving plane that is checked against a closed form to the limits of double-precision arithmetic, and nothing about it is in doubt.

Nothing in that construction mentions the floor’s shape, which is exactly the gap this essay closes. The two rays are cast identically whatever lies beneath them — a rim, an edge, a landing point, the same for a plane, a dish, a ridge or a step — so the projection itself does not know the floor is not flat. What changes is only where the rays land, and that is the whole of what a curved or broken receiver does to a penumbra: the band is still the lamp’s image cast through a pinhole, and the image has been dropped onto a surface that is not the pinhole’s own image plane.

Four receivers carry the same lamp and the same card through the rest of this essay — the flat plane already settled, a dish curved in both horizontal directions, a ridge curved in one, and a floor with a step in it — and each departure from the plane’s result is a number a real projection prints rather than an assertion about curvature in general.

The mechanism moves the band, and moves it unevenly #

The scene above is worth looking at as a section before it is trusted as a number, because a stepped floor is the case where “the width of the band” stops being one quantity.

The mechanism is not subtle once it is drawn: a step has a vertical face, and a ray that lands on that face has travelled further, along the surface, than its horizontal displacement would suggest. Nothing about the light bends; the floor bends, in the one place where the receiving surface is not a graph of height over ground position at all but very nearly a second, vertical floor glued to the first. The riser does not distort the rays — they are still two straight lines from the lamp’s rims through the edge — it distorts the ruler a reader would use to describe where they landed, and a ruler laid along the surface reads a longer distance than a ruler laid across the ground beneath it.

That distinction between “across the ground” and “along the floor” is not a nicety. A builder checking a stepped landing with a tape measure gets 2.488 m; a drawing of the same scene from directly above, or a plan with the step’s outline flattened onto it, gets 0.990 m. Both numbers are the same band, correctly measured two different ways, and reporting either one without saying which is a category error waiting to be made on an actual staircase. The gap between them, 1.498 m, is itself a number worth sitting with: it is larger than the whole band on a flat floor, so a reader told only “the band is however wide” with no word about which ruler produced that figure has been told something that could be either an accurate report of a modest band or a wildly wrong one, with no way from the number alone to tell which.

Why a curved receiver needs a search rather than a formula #

A flat floor’s closed form exists because a ray meeting a plane is one linear equation, solved by dividing one number by another. None of the other three receivers admit that shortcut, and the step is the case that shows why a search — rather than a smarter formula — is the right tool for all of them.

A dish and a ridge are each a graph of a quadratic, so a ray meeting one of them is a quadratic equation in the ray’s own parameter, still solvable in closed form if a little more work is spent finding which of two roots is the physical one. A step is not a quadratic at all: its height is one constant on one side of a line and a different constant on the other, with no derivative at the seam. A solver written to expect a smooth curve — Newton’s method chasing a slope, say — has nothing to chase at the one place a step’s own argument turns on, which is exactly where a card’s shadow is most likely to fall given how this essay’s geometry is arranged. The bisection actually used here needs no derivative and no case-split for the corner: it merely asks, at a candidate point along the ray, whether the ray is above or below whatever the receiver’s height function returns there, and narrows the bracket until the two agree to machine precision. The same fifteen lines of logic that solved the dish and the ridge solve the step without alteration, which is the reason four visibly different floors can share one routine and one number of significant digits.

The shape departs before the width does #

A dished floor is a gentler case, and it is gentler in a way that is itself informative: the band’s overall span barely moves, and the shape of what happens inside it moves regardless.

The reason a dish bends the profile without moving its ends far is that a dish’s height runs as the square of distance from its own centre, so equal steps across the lamp’s rim do not project to equal steps on the receiver — the map from “which point of the lamp” to “where its ray lands” is curved rather than affine, and a curved map applied to something that was evenly spaced comes back unevenly spaced. On a flat floor that map is exactly affine — a plain ratio of two heights — which is the only reason “the penumbra is the lamp’s image, spread evenly” was ever a clean sentence to write.

This is a genuinely separate failure from the step’s. The step moves where the band’s ends land and leaves what happens between them alone wherever the floor beneath it stays flat; the dish leaves the ends close to where a flat floor would put them and rewrites everything between. A single number called “the width of the band” cannot report both kinds of departure, which is the argument for measuring width and shape as two different questions rather than one.

The 0.0453 is a fraction of the lit fraction itself — how much of the lamp a given point can see — rather than a length, and turning it back into a length is instructive. Read off the four-floor comparison below, the dish’s own band along the floor comes out at 1.396 m, barely different from the flat floor’s 1.440 m. A shape error of 0.045 inside a band that width means a point roughly an eighth of the way across it reports a brightness that belongs, on the flat floor’s own ramp, to a point several centimetres further along — a small absolute distance, and not a small fraction of what the reading was supposed to mean, since the ramp is the only thing standing between “how bright is this point” and “how far along the band is it,” and the dish has just shown that relationship is not the straight line it looks like on a flat floor.

Four floors, and the one that changes nothing #

Measuring one curved floor against the plane leaves open whether curvature in general is the cause, or whether the dish and the step each have some narrower defect of their own. Four floors settle it, because one of the four is curved and changes nothing at all.

The ridge is the control this essay needs, and it earns the word rather than merely supplying it. A ridge is curved — its height runs as the square of one horizontal coordinate exactly as the dish’s does — and it takes the identical code path as the dish and the step: the same two rays, the same bisection against the same kind of surface, differing only in which function computes the height at a point. It returns the flat answer to machine precision because the curvature it has runs across the band rather than along it, and the two limiting rays that define the band both run along the source’s own line, where a ridge’s height never changes. So the ridge is not a case where curvature happens to do nothing; it is the case that proves the earlier two numbers are about the geometry and not an artefact of asking a curved-floor solver to answer a question about a flat one, because the identical solver, asked the identical question one axis over, refuses to invent a departure that is not there.

That the ridge and the dish share a height law and disagree so completely on the outcome is the cleanest statement available of what this essay is actually about. It is not “curved floors distort penumbras” as a blanket rule — a curved floor distorts a penumbra exactly where the curvature has a component along the direction the band’s own rays run in, and not otherwise. A designer laying a ridged floor beneath a line of straight-edged shelving, with the ridge’s own axis running parallel to the shelves, would see none of this; the same ridge with its axis turned across the shelves instead would put its curvature back along the rays and would show a departure of its own by the identical mechanism, though that fifth arrangement is not one this measurement casts and no number is claimed for it here. The control is not a footnote to the finding, in other words; once it is stated this way it is the finding, and the step and the dish are two ways of making the component non-zero rather than two unrelated defects.

What the closed form loses when the floor bends #

The plane’s own answer for the band’s width is not just a number, it is a straight line: hold the lamp and the card fixed, raise or lower the card, and the width comes out exactly proportional to the card’s height over the gap between lamp and card. A curved floor is what breaks that proportionality, and the break is a length rather than a percentage.

“The width scales with the distances” is, on a flat floor, a single ratio doing all the work: raise the card and every point of the band moves in lockstep with it. On a step, the ray landing on the low floor obeys that ratio and the ray landing on the riser does not, because the riser sits at a fixed height regardless of where the card is — so the two ends of the band are governed by two different relationships between the card’s height and where a ray lands, and no single straight line fits both. A miss of 1.047 m against a band that is itself only 1.44 m wide on a flat floor is not a rounding matter; it is the statement that the linear law has stopped applying to at least one of the two rays entirely.

Two things are worth separating in that number. The flat floor’s own worst departure, 8.9e-16 m, is not zero for the uninteresting reason that no floating-point arithmetic on a real machine returns exactly zero after several divisions and a square root; it is the reading a straight line gets when the model behind it really is straight, and every essay on this site that fits a line to a genuine proportionality reports a residual of roughly that size. The step’s 1.0468 m is fourteen orders of magnitude larger, and the ratio between the two numbers is doing more work than either number alone: it says that whatever the plane’s own fit was measuring — noise in the arithmetic, nothing else — the step’s departure is not more of the same thing at a larger scale, but a different phenomenon that happens to be reported by the same line of code.

A different measurement pays the same cost #

The construction so far is one particular route — two projected rays and the width or shape between their landings — and a reader is entitled to ask whether the whole finding is an accident of that route. A completely different measurement, run on the same four floors, is not.

That is a homography fitted from four correspondences, which shares no arithmetic at all with a penumbra’s two limiting rays — and the floor that is not a plane finds the identical shape of failure: exact on the surface that is genuinely flat, exact at whatever four points the fit was told about on every surface, and wrong everywhere else on a curved one. The residual has a shape is the essay that goes on to characterise what that “everywhere else” error looks like as a function of position, and it is worth reading as the general statement of what this essay has just found for one particular construction: a curved receiver does not fail a fit or a projection randomly, it fails it in a pattern set by the surface’s own curvature, which is why a four-point fit and a two-ray penumbra land on the same four floors in the same order.

The parallel runs further than shape-of-failure. A floor cannot fake a second lamp shows that a curved receiver’s departure from a plane cannot be mistaken for a second light source no matter how it is dressed up, which matters here precisely because a reader seeing a penumbra’s shape distort on a dish might otherwise reach for “as if there were two lamps” as an explanation. There is one lamp, one card and one curved surface in every figure in this essay, and the distortion is entirely the receiver’s. A homography and a two-ray penumbra are different enough constructions that the same distortion showing up in both is not a coincidence of one particular fit; it is a statement about the four floors themselves, and any measurement built on straight lines projected onto them should expect the identical shape of departure.

The honest limit #

Four floors were chosen because each has a closed form for its own height, which is what let the ridge serve as a control and the step’s riser be stated in exact centimetres rather than estimated. A real floor — uneven flagstones, a worn stone step, a ramp with a lip — has no such closed form, and this measurement would have to be redone against whatever surface actually receives the shadow rather than interpolated from these four. Nothing here claims that a dish and a real floor with a comparable dip behave identically; only that the kind of departure — a shift in where a ray lands, translated into a shift or a bend in what a straight edge’s shadow reports — is general, because it follows from where two straight lines happen to intersect a stated surface and not from any property peculiar to a paraboloid or a step.

Nor does anything here say how visible 0.045 of the lamp’s own width actually is to an eye. That fraction is a fact about the geometry of two rays and a surface; whether a person notices it depends on the lamp’s absolute size, the viewing distance and the eye’s own acuity, none of which this measurement touches. What a point in shadow can see of the sky uses the identical machinery — the same ray-casting count that produced the band above — turned toward a different question, how much of an extended source a place can see rather than how a card’s edge images one, and it says the same thing explicitly: only the geometry is computed, and how bright any of this looks is somebody else’s subject.

What is settled is narrower and cleaner than either of those cautions weakens: a penumbra’s width and shape are a genuine record of the lamp only when the receiver is flat, or flat along the one line the band actually runs on. The lamp’s size over its distance, and nothing else takes that flat-floor record and asks exactly what it can and cannot recover about the lamp itself, which is the natural next question once this one is settled. How many lamps make one lamp and the lamp a low shadow cannot locate both build on the same flat-floor construction from two different directions — one asking what stands in for the lamp, the other what stands in for the floor’s own straight edges — and neither would have a clean starting point without this essay first showing which floors are safe to build it on. Counting shadows is not counting lamps is the same caution again in a different register: a floor’s own shape can make one lamp’s shadow look like it needs two lamps to explain, exactly as a dish’s shape can make one lamp’s penumbra look like it needs a different lamp to explain, and in both cases the extra complexity belongs to the receiver rather than to the source.