Spherical Harmonics: describe the light instead of following every ray

Light coming from many directions is replaced by a smooth mathematical description.

How the Spherical Harmonics radiation model works Four panels. One: the selected light pattern, how the hot cell emits. Two: in blue, the finest harmonic the model uses at order N, a flower with 2N equal petals that depends only on N. Under the cell, the harmonics the model uses with their solved weights: the model solves for these weights, not for the shapes. In amber, their weighted sum, the model's fit of the selected light, over a dashed outline of the light itself; small dotted lobes mark where the truncated series goes negative. Three: in amber, what the model finds in each cell: the truncated shape of the light arriving from the hot cell, a leaning blob at low order and a narrow lobe at high order. A mesh selector shows that the mesh only changes how blocky the answer is, not how smooth it is in angle. Four: the radiation field the model predicts around a hot spot, drawn as a glow that fades with distance, with the true field shown as a dashed outline for comparison. 1. Selected light pattern how the hot cell emits brightness by direction dashed = reality, as in (4) 2. Model shape (in each cell) set by N, not by the light fit = weighted sum of these 3. Solving in every cell the shape found in each cell lit from the hot cell P₁: leaning blobs everywhere 4. The result the radiation field an even glow dashed = reality · 32×32 cells P₁: smooth and cheap — ideal when the light is already diffuse
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P₁
Fig. 5 — Spherical harmonics in one picture. Instead of tracking individual rays, the method describes how radiation varies with direction using a smooth mathematical shape (2). The blue flower there is the finest harmonic the model uses at that order, the same whatever the light: more petals, finer detail. The model does not solve for that shape; it solves for how much of each harmonic to use — the weights shown under the cell — and the amber fit is their sum. It then solves that description in every cell of the mesh (3), and the result is the radiation field (4): the glow the model predicts around a hot spot, fading with distance, with reality drawn as a dashed outline. A low-order model such as P1 is very smooth and inexpensive — for a diffuse glow it is nearly exact, but it smears a sharp beam into a dim blob. Higher orders bring the beam back, with the small ripples that are the method's signature, at the price of more computation. The mesh only sets how finely that answer is drawn: coarse cells make it blocky, fine cells make it crisp — but no mesh can give P1 a beam back.
Why doesn't a finer mesh fix the blob?

The blurring in PN happens in angle, not in space. The mesh decides how finely the field is drawn across the room; the order N decides how finely the light's direction is described at each point. They are independent knobs. Go from 16 × 16 to 64 × 64 with P1 and the blocky blob becomes a crisp blob — still a blob. Only raising the order brings the beam back, and then a finer mesh is worth having to resolve the sharper field it produces.

Compare with discrete ordinates, where the mesh and the directions fight each other. Here they do not: PN has no ray effect and no streaks to hide, so a finer mesh never makes it look worse — it just costs more cells.