A corner is not a wall

Everything else in this field is spectral. The reflectances are functions of wavelength, the illuminants are functions of wavelength, and the arguments are about what happens when functions of wavelength are multiplied together.

The geometry enters through exactly one quantity, and it has no wavelength in it at all.

The claim #

A form factor $F_{ij}$ is the fraction of everything leaving surface $i$ that arrives at surface $j$, it depends on the geometry and on nothing else, and two identities it must satisfy are worth more than the numbers themselves.

The identities are reciprocity, $A_i F_{ij} = A_j F_{ji}$, and closure, $\sum_j F_{ij} = 1$ for a surface in a closed cavity. Both are guaranteed by the physics and neither is guaranteed by the arithmetic, which is what makes them useful — the same reason every claim on this site is given a test it could fail.

What the number is #

For two Lambertian surfaces, the fraction of one’s output reaching the other is

$$F_{ij} = \frac{1}{A_i}\iint \frac{\cos\theta_i \cos\theta_j}{\pi r^2}\,\mathrm{d}A_j\,\mathrm{d}A_i$$

with $\theta_i$ and $\theta_j$ the angles between the surface normals and the line joining the two points, and $r$ the length of that line. The $\cos\theta_i$ is Lambert’s cosine law at the sending end; the $\cos\theta_j$ is the receiving surface’s foreshortening; the $1/r^2$ is the inverse square law; and the $\pi$ normalises a Lambertian emitter’s hemisphere.

Nothing in it is wavelength dependent, which is the reason a room’s equilibrium colour is a property of its paint rather than its shape — as the previous rung derives — while everything about how fast that equilibrium is reached is geometric.

For a cube the two distinct values are 0.199825 for the opposite face and 0.200044 for each of the four adjacent ones. Those are close enough that they are often quoted as “a fifth each”, and the near-equality is a coincidence of the cube. Stretch the box to $2\times1\times1$ and the floor sees its ceiling with 0.2859 and each long wall with only 0.1164. Make it tall — $1\times3\times1$ — and the floor sees the distant ceiling with 0.0330 and each wall with 0.2418.

That last case is the interesting one for colour. In a tall room the floor barely sees the ceiling at all, so almost everything reaching the floor has come via a wall, and the walls’ paint dominates the floor’s illuminant almost completely. A tall narrow room is spectrally a much stronger version of the effect than a cube.

The two identities #

Reciprocity says $A_i F_{ij} = A_j F_{ji}$. It follows immediately from the integral being symmetric in the two surfaces apart from the $1/A_i$ out front, and it catches transposition errors — an index swapped, a dimension passed in the wrong order — which are otherwise silent because the resulting matrix looks entirely reasonable. It is checked here to $10^{-9}$ on three differently-shaped boxes.

Closure is the more valuable of the two, and the reason is worth stating carefully.

A radiosity solve with a face missing does not throw, does not produce a negative number, and does not look wrong. It looks like a slightly darker room.

That is the failure mode this site keeps finding, in different clothes each time. Ticks that were absent rather than misplaced. A purity calculation that reported every sample as fully saturated under nine captions each quoting 1.00. A caption strip that said the same thing on 204 placements. In every case the output was plausible, no gate objected, and the symptom was absence.

Closure catches it because a face with a missing neighbour has a row summing to less than one, and there is no way to arrange five faces of a box so that the sums come out right. It is asserted on three box shapes, and the gate feeds it a matrix with the front wall deleted and requires the assertion to refuse.

Two routes, and they converge differently #

The closed forms for two aligned parallel rectangles and for two perpendicular rectangles sharing an edge are standard results. Evaluating them is not the point; the point is having a second, independent route to the same number.

So the double integral is also done numerically, on a grid, by the midpoint rule — slow and obvious, which is what a check against a closed form should be. Two independent derivations disagreeing beats one number against a table, which is the habit that caught a 1% chroma error in this site’s appearance model when every structural check had passed.

The check found something about itself.

The left column is second order and the right column is first. Doubling the sample count divides the parallel error by four and the perpendicular error by two.

The cause is the shared edge. The integrand carries $1/r^2$, and on two faces that touch, $r$ goes to zero along an entire line of the four-dimensional domain. A midpoint rule never samples that line and never resolves it either — the singularity is integrable, so the answer converges, but it converges at the rate a non-smooth integrand allows rather than at the rate the rule would otherwise give.

Why the tolerance was not simply loosened #

This is the part worth carrying to another problem.

The obvious response to a check that misses by 5.6% is to set the tolerance at 6% and move on. That would have been wrong, and not for aesthetic reasons.

The quadrature costs $n^4$ — samples on both surfaces, two axes each. Reaching one part in a thousand on the perpendicular pair by brute force needs about 700 samples per axis, which is $2.4\times10^{11}$ point pairs. Not a computation anybody runs. So the choice was between a check accurate to 5% and no check at all.

A check accurate to 5% cannot distinguish a correct closed form from one that is 5% wrong, and a 5%-wrong form factor is precisely the error the second route exists to catch. Loosening the tolerance would have left a verification that passes on the failure it was written for — the worst possible outcome, because it also reports success.

The repair is Richardson extrapolation, which costs one additional grid: evaluate at $n$ and $2n$, and combine according to the known order. That lands within $5\times10^{-7}$ of the closed form on the parallel pair and $1.4\times10^{-4}$ on the perpendicular one, both far inside a useful tolerance.

One detail in that repair is load-bearing. The order used is chosen from the geometry — whether the two rectangles touch — rather than fitted from the data being extrapolated. Fitting an order from the same three numbers being extrapolated produces a wrong answer with a tight-looking error bar, which is a worse failure than the one being fixed. Whether two faces share an edge is a question about the box and is answered by sampling the distance between them.

The assertion also checks that the un-extrapolated quadrature is already within 5% — so the agreement cannot be an artefact of an extrapolation tuned to the answer it was checking.

Which wall is which in the two-by-one box #

The stretched box is quoted as the floor sees its ceiling with 0.2859 and each long wall with only 0.1164, and both numbers are right about the box while the label on the second is on the wrong wall.

A 2 × 1 floor under a ceiling at height 1 has four walls of two kinds: two long ones, 2 wide, and two end ones, 1 wide. Evaluating the perpendicular closed form for each:

The row sums to exactly 1, which is closure doing its job on the corrected assignment and would not have done on the other one — 0.2859 + 2(0.1164) leaves 0.4813 for two surfaces, or 0.2406 each, which is the long walls’ own value arrived at backwards.

The correction points the same way the essay does and further. The two long walls together take 48 per cent of the floor’s view against the ceiling’s 29, so even a box only twice as long as it is wide has already handed the floor’s illuminant to its walls — while a cube splits it evenly and the tall room gives the walls 97 per cent. The sequence 29, 80, 97 across three shapes is a sharper version of a tall narrow room is spectrally a much stronger version of the effect than a cube than the cube and the tall room alone provide.

The convergence table is exactly first and second order #

The claim that one column is second order and the other first is the essay’s finding about its own check, and the four rows support it to two digits rather than approximately.

Taking the ratio of successive relative errors against the ratio of successive sample counts: the parallel column gives 4.04, 4.01, 2.78 where a second-order rule predicts 4.00, 4.00 and 2.78; the perpendicular gives 1.98, 1.99, 1.66 where first order predicts 2.00, 2.00 and 1.67. No exponent was fitted and none of the eight predictions is out by more than one per cent.

That precision matters for the repair rather than for the diagnosis. Richardson extrapolation needs the order to be right, not approximately right, and the essay is explicit that the order is taken from the geometry rather than fitted. The table is the evidence that the geometry gives the correct order: an integrand with an integrable singularity along a line of its domain converges at exactly first order here, and one without converges at exactly second.

Extrapolating the last two rows on those orders lands at 0.19982487 against a true 0.199825 — an error of 6 × 10⁻⁷ — and at 0.20005866 against 0.200044, an error of 7 × 10⁻⁵. Both are inside the essay’s quoted figures, and both are three to four orders better than the raw quadrature they came from.

The brute-force alternative checks too. First order means the error falls as one over the sample count, so reaching a part in a thousand from 1.7 × 10⁻² at forty samples needs 680 per axis and 2.1 × 10¹¹ point pairs. One extra grid buys what a four-hundred-fold increase in sampling would have cost, which is the whole argument for extrapolating rather than loosening.

What was computed, and how #

The closed forms are evaluated for arbitrary box dimensions, not just for a cube, and every figure names the box it drew. A check that only ever ran on a cube would miss any error that happens to cancel when all three dimensions are equal, and the cube is exactly the case where the two distinct values nearly coincide.

The perpendicular factors are computed individually rather than by dividing the remainder. Closure would allow the four adjacent faces to be filled in as $(1 - F_\text{opposite})/4$, which is correct for a cube and wrong for every other box — and would make closure pass by construction, destroying it as a check.

The figures assert closure themselves rather than relying on the library having done so. A figure is seen alone, and the fleet rule is that a generator asserts what is true of its own arguments.

A deep box is the shape a corridor or an alcove actually is, and the form factors there are nothing like a cube’s.

What the geometry does to the colour #

The form factors are colourless and they decide how coloured a room is, which sounds contradictory and is not.

The equilibrium cast of a uniform cavity does not depend on shape, as the previous rung shows. What the shape decides is the mixture at any particular surface — how much of the light there came direct and how much came via a wall. That mixture is what a camera sees at a point, and it is entirely geometric.

The extreme cases make the dependence concrete. In a cube, a fifth of the ceiling’s output reaches the floor directly, so the floor’s light is a substantial fraction first-hand. In a room three times as tall, that drops to 0.0330 — the floor’s light is almost entirely second-hand, and the walls’ colour dominates it. Same paint, same lamp, same gain, and a visibly different result at the floor.

Two more shapes say that the effect is a property of the enclosure rather than of any particular room, and one of them shows the geometry the numbers come from.

This is why colour bleeding in a rendered image is a property of the modelling as much as of the materials, and why an architect’s visualisation of a room can be spectrally wrong in a way that has nothing to do with the paint specification: get the ceiling height wrong and the mixture at the floor changes.

What the pictures cannot show #

A uniform radiosity per face is an averaging assumption and the figures inherit it. The real gradient — bright at the crease, falling away from it — is the visible signature of interreflection in a photograph, and a six-patch solve has none of it. Every swatch here is a face’s average, and a reader looking at a photograph of a real corner sees something the model cannot draw.

The matrix figures show numbers, not light. They are the one family in this field with no swatch in them, because a form factor has no colour. That is unusual enough on this site to be worth flagging: the caption strip names the box rather than an observer, since changing the colour-matching functions would move nothing in the picture — and that required a new band in the observer index to classify honestly rather than being filed under a stated condition.

Nothing here is measured. Both routes to a form factor are computations. There is no experimental check on this page, and the confidence comes entirely from two derivations agreeing plus two identities holding — which is a weaker kind of evidence than a measurement and is stated as such.

Turning the same box on its side changes which faces are near each other without changing its volume, which is the cleanest way to see that the matrix is about shape.

Where the model stops #

Lambertian surfaces only. A form factor is defined for ideal diffusers. Real gloss sends light in preferred directions, and there is no fraction-arriving-at that is independent of where the light came from. Everything in this field would have to be re-posed as a full four-dimensional transport problem, and the spectral conclusions would survive unchanged because they are about how many times $\rho$ appears.

Flat faces, so no self-view. The diagonal of the matrix is zero because a plane sees none of itself. A curved concave surface sees itself, and its diagonal is not zero — which is why the enclosure parameter used in the metamer essay can be treated as a single surface’s self-view without contradicting anything here.

No occlusion. In a convex box every face sees every other face entirely. Put furniture in the room and the integral needs a visibility term, which is where the real cost of radiosity lives and which no closed form covers. It is also what makes a real shadow a place with a smaller form factor to the source rather than none at all, which is why a shadow still has an illuminant.

Six faces is a coarse discretisation. A real solve subdivides surfaces into thousands of patches, because the assumption that radiosity is uniform across a face is badly wrong near a corner — the crease is much brighter than the middle of a wall. Six faces gets the average right and the gradient not at all.

The generalisation #

Two habits transfer, and the second is the more useful.

Prefer a check that a plausible bug would fail. Closure is worth more than reciprocity here not because it is deeper but because the bug it catches — a missing surface — produces output that no other check would question. Choosing between candidate assertions by asking which plausible failure each one catches is more productive than choosing by which is easiest to state.

A verification’s precision has to be better than the error it is looking for, or it is theatre. This is obvious written down and easy to violate in practice, because the natural response to a failing check is to widen it until it passes. The correct question is whether the widened check can still fail for the reason it was written, and if it cannot, the check should be made more accurate or abandoned honestly — not kept as decoration.

Who found it, and when #

View factors are older than radiosity by sixty years. They come from radiative heat transfer, where they are called configuration factors or angle factors, and the closed forms for rectangles were being tabulated in the 1920s and 1930s — Hottel’s work is the standard reference, and the tables in his and Sarofim’s book are still the ones cited.

The reciprocity and summation relations were understood from the start there, because thermal engineers were solving networks of surfaces and needed the arithmetic to close. Radiosity in graphics, from Goral and colleagues at Cornell in 1984, took the whole apparatus across essentially unchanged and then spent a decade on the problem the thermal literature had not had to solve at scale: computing the factors for complicated geometry with occlusion, which is what the hemicube and later methods are for.

The convergence-order observation on this page is not a discovery — the behaviour of quadrature near an integrable singularity is textbook numerical analysis. What is worth recording is that it was found by checking, and that the naive repair would have silently disarmed the check.

Where the ladder goes next #

The neighbour on this rung is the room the form factors are for — what colour bleeding actually amounts to once the solve is run, and why the bleeding saturates rather than running away.

The rung above leaves transport behind. Paint is not a filter asks what happens when two pigments are mixed rather than two surfaces bounced between, and the answer is that the multiplication model is the wrong one entirely.