The solver had no slot for gloss

Two rounds ago this collection audited what a surface is and found a six-argument function projected onto one number per wavelength. One of the dropped arguments has been left in the ledger ever since, because restoring it meant rewriting the machinery that every scene result depends on. This is that rewrite.

The claim #

Radiosity does not approximate a glossy wall badly. It has no slot for one, and the difference between those two statements decides what can be done about it.

• Its unknown is one radiosity per patch, a single number describing what leaves in every direction, and a surface with a lobe does not have one.
• The cost on an ordinary room is 4.89 ΔE₀₀ at an eggshell finish, falling to 0.97 at nearly matt — larger than any tolerance this collection writes.
• And the assumption is not a parameter. No setting of a radiosity solver’s inputs produces a directional answer, so the error cannot be bounded from inside the method.
• What replaces it is thirty unknowns instead of six, one radiance per ordered pair of patches, and it reduces to the radiosity solution exactly when the lobe is removed.

What radiosity is committed to #

The radiosity method solves for one quantity per surface: the total power leaving it per unit area, called its radiosity. The equation is

Bᵢ = Eᵢ + ρᵢ Σⱼ Fᵢⱼ Bⱼ

with F the geometric form factors and ρ the reflectance. It is a linear system, it is exact for the model it describes, and this collection has used it for every scene result it has ever published — a bounce is a multiplication, a corner is not a wall, a room bounds its own bounces, and the green wall that changes the colour of everything in front of it.

The commitment is in the first symbol. Bᵢ is one number, so the light leaving patch i is the same in every direction. That is the definition of a Lambertian surface, and it is a requirement of the formulation rather than an approximation inside it: there is nowhere in the equation to put a direction.

A method whose unknown has the wrong shape cannot be made more accurate. Refining the mesh, adding bounces, using better form factors — every improvement available to a radiosity solver improves its answer to the question it can ask, and none of them lets it ask the other one.

This is a missing slot rather than an approximation. The distinction matters because it decides what an error analysis can do.

An approximation has a residual that can be bounded. A first-order expansion has a second-order term; a truncated series has a tail; a coarse mesh has a refinement. In every case the method knows what it is neglecting and something can be said about the size of it.

A missing slot has no residual, because there is no quantity in the model corresponding to what is absent. Nothing in a radiosity solution is an estimate of the directional variation; the directional variation simply does not appear, and asking the solver how large it is produces no answer at all.

That is the same distinction the previous round’s audit made about the model equation: the Lambertian assumption is written down and known to be an approximation, and the other dropped arguments are places where the model has no slot, so there is nothing to be approximate about. What is new here is that the Lambertian assumption turns out to be in the second category too, once it is a solver’s assumption rather than a surface’s.

The only way to price a missing slot is to build a model that has the slot and compare. That is what this round does.

What was already known, and what was not #

A room is not a sphere established the measurement half of this two rounds ago. Under a hemisphere of constant radiance a detector reads a sample’s own directional-hemispherical reflectance exactly, by reciprocity; every real room fails that condition; and the single albedo a radiosity calculation uses equals no particular reading of a glossy sample, with a spread from 0.5398 to 0.7704 across incidence on an ordinary varnish.

That said what the assumption costs a measurement. It could not say what it costs a scene, because the transport was never solved with the lobe in it — the machinery to do so did not exist, and the ledger has carried the item for two rounds.

The gap between those two questions is larger than it looks. A measurement is one interaction; a scene is a series of them, each multiplying the last, so an error that is small at one bounce compounds. And the direction that a lobe favours is not random: it favours the direction the light came from, so the geometry of a room and the geometry of a lobe interact rather than averaging.

The cost, measured #

A cube with two coloured walls and a lamp in the ceiling — the same room this collection’s green-wall argument uses, so the two answers are comparable by construction.

The departure falls monotonically towards the matt end, as it must, and at a finish anybody would call a low-sheen wall paint it is nearly five colour differences. That is five times the tolerance a delivery specification is written in, on a quantity nobody has been computing.

The right-hand column is the second finding and it runs the other way from the first: seen from the front of the room the chroma falls as the walls get glossier. That is the subject of its own essay and the mechanism is short — a Fresnel return carries no pigment, so the fraction of the bounce that comes off the interface arrives at the floor white, while the coloured part of the return is redistributed towards the walls it came from.

Why the error is not conservative #

There is a defence of the Lambertian assumption that is often made and is wrong in a specific way, and it is worth closing.

The defence is that most architectural surfaces are nearly matt, so the assumption is nearly right, so the error is small. The first two clauses are true and the third does not follow.

The measurement above is at roughnesses from 0.15 to 0.8, which is the range from a satin paint to a flat one. Nothing in it is glossy: a varnish or a gloss enamel is at 0.02 to 0.05, which this model cannot reach and where the departure would be larger. So the table is the interior of the assumption’s comfortable range, and it costs between one and five colour differences there.

The reason a small departure from Lambertian produces a large error is compounding. A room’s colour is the result of several bounces, each multiplying the last, and a five per cent directional redistribution at each bounce becomes something considerably larger by the third. A room bounds its own bounces established how many bounces matter; this is what each of them is carrying.

The size of that share is the last piece of the argument and it is the one that makes the departure surprising. Nine per cent of the return is enough to move a room’s colour by five ΔE₀₀, and a reader would reasonably expect a nine per cent effect to be worth a fraction of a unit.

Two things multiply it up. A room’s colour is the product of several bounces, so a nine per cent redistribution at each one compounds. And the nine per cent is not a scaling of the existing return but a different return with a different spectrum, so it changes the colour rather than the brightness.

That combination — small share, compounding, spectrally different — is the general shape of an interreflection effect and is why rooms are harder to predict than surfaces. The same three properties made the green wall’s cast larger than its reflectance suggests.

What this collection owes #

Every scene number in this collection is computed on Lambertian walls. That is now a measurable statement rather than a caveat, and the honest form of it is:

The green-wall result, the corner result, the bounce series and the metamer-separation results are all computed at the matt end of the table above. Each of them would move by between one and five ΔE₀₀ if its surfaces had an ordinary satin finish, and the direction of the move is known: less chroma in the bounce, more light on the floor.

None of them is invalidated, because each is a comparison between two computations done the same way and the shared assumption largely cancels. What changes is any absolute claim about what a room delivers, and there are several.

The repair is not to recompute everything. It is to state the finish, which none of those essays does, and to compute the departure where an absolute number is quoted. That is recorded as owed.

The quantity that has no radiosity at all #

That figure is the missing slot drawn, and it is the sharpest form of the essay’s claim.

With a lobe on the walls, the light leaving the floor towards the front of the room is a different colour from the light leaving it towards the left wall — 18.76 against 20.94 in chroma, a spread of 2.00 ΔE₀₀ between the extremes. A radiosity solution assigns the floor one radiosity and therefore has no spread: not a small one, none.

So the width of that chart is not an error in the radiosity answer. It is a quantity the radiosity answer does not have, and no refinement of it produces one. That is what a missing slot looks like when it is finally given a slot.

It also settles a question this collection has raised from the perception side. A colour that moves with the viewer had, until now, only an interference film to point at — a goniochromatic material, exotic and deliberate. This says that an ordinary painted room does it too, at two colour differences, for entirely ordinary reasons.

What was computed, and how #

The new solver carries one radiance per ordered pair of patches — thirty unknowns for a six-face box, against six for radiosity — and solves the linear system band by band over the collection’s eighty-one wavelengths.

Its bidirectional distribution is a Lambertian body plus a microfacet lobe with a Fresnel term, which is the surface the previous round wrote for its own directional audit, used unchanged. The interface’s return is taken out of the body term rather than added beside it, which is what a real finish does and which decides the sign of the chroma result.

The check that makes it believable is the reduction. Setting the lobe to zero collapses all thirty unknowns onto their patch’s radiosity divided by π, and the answer agrees with this collection’s existing radiosity solution to 9.8 × 10⁻¹⁶ relative — the floating-point floor, over thirty ordered pairs and eighty-one bands.

That spectrum is the departure before it is reduced to a number, and it shows why no exposure change can bring the two together. The glossy room delivers 1.17 times as much light at 530 nanometres, where the wall pigment is, and 1.24 times as much at 430 and at 680, where it is not. A single multiplier would match one of those and miss the others.

Where the model stops #

Each patch is a point with a full bidirectional distribution on it, which is the honest name for the model: directional transport between point-sampled patches. It ignores the variation of direction across a patch’s own extent, which is the same thing form factors already average away, and it is exact in the limit of small patches.

It cannot reach a gloss finish. A patch in a cube subtends a cone of angular radius 25.8° at the face opposite, and a lobe narrower than that is narrower than the quadrature sampling it; the boundary is measured rather than declared and it sits at about roughness 0.15.

And the room is a cube with six faces. Real rooms have furniture, windows and non-planar surfaces, and the form factors here are the closed forms for a box.

There is one more consequence for how a scene result should be reported, and it applies to graphics as much as to colour measurement. A radiosity answer is a number per surface, so it invites the sentence the floor is this colour. A directional answer is a number per surface per direction, so the same sentence has to become the floor sends this colour towards the door.

That is a small change in wording and a substantial change in what is being claimed. It also makes several existing claims in this collection ambiguous in a way they were not before, because a sentence written under a method with one answer does not say which of five it meant. Where those sentences are about a comparison the ambiguity is harmless; where they are about an absolute colour it is not, and the ones that are absolute are named above.

The generalisation #

The habit is about recognising a structural commitment inside a method.

Every numerical method has an unknown, and the unknown’s shape is a claim about the world that is usually invisible because it is in the formulation rather than in the parameters. A method with one number per patch is claiming that one number suffices; a method with one number per pair is claiming that pairs suffice; and neither claim appears anywhere a user would look.

The way to find one is to ask what the answer would be if the world were slightly different in a way the method cannot express. If the question is unanswerable rather than merely hard, a slot is missing.

The failure mode is to treat a structural commitment as an accuracy setting. Refining a method that has no slot for what is missing produces a more precise answer to the wrong question, and the extra precision makes it harder rather than easier to notice.

One more note about the shape of the finding, because it is the third time this collection has met it. The audit of the model equation found four dropped arguments; the audit of the tabulation found three decisions inside one word; and this finds one missing slot inside a method. In every case the thing that was absent had no residual, no error bar and no parameter, and in every case the only way to price it was to build the fuller object and subtract.

That is expensive and it is the only method available. A quantity a model cannot express cannot be estimated from inside the model, however carefully the model is analysed, and the amount of analysis a model will absorb without producing the answer is unbounded.

Who found it, and when #

Radiosity came into graphics from thermal engineering, where it had been used for enclosure heat transfer since the 1950s. Goral and colleagues published the graphics version in 1984, and the Lambertian requirement was stated plainly there: the method applies to diffuse environments.

The subsequent history is a series of attempts to relax it. Immel and Cohen’s directional radiosity of 1986 discretised the hemisphere at each patch; Sillion and Puech’s two-pass methods combined radiosity with ray tracing; and the whole line was largely superseded by path tracing, which has no such commitment because its unknown is a path rather than a patch.

That the assumption was never priced in colour is the gap this round fills. Graphics measures it in radiance and pixels; nobody had asked what it costs a colour specification.

Where the ladder goes next #

The solver that replaces radiosity has five times as many unknowns and one property that makes it trustworthy: it becomes the old solver exactly when the lobe is removed, and that reduction is worth an essay of its own.