How a surface redistributes light

BRDF lobe explorer

The BRDF answers one question: light arrives from ωi — what fraction leaves towards ωo? It is bidirectional, so the answer changes when you move either direction, which is exactly what a static picture of a lobe cannot show. Drag the incoming ray and watch all three panels respond at once.

The lobe, in the plane of incidence —
Drag the incoming ray
incoming ωi surface normal mirror direction the lobe: how much leaves each way
The same lobe in 3D —
Drag to rotate

The distance from the origin in any direction is the value of the BRDF for that outgoing direction. A perfect mirror is a spike; a Lambertian surface is a hemisphere of constant radius.

What it looks like —

The same BRDF evaluated per pixel over a sphere, lit from the direction you are dragging. This is the appearance the lobe on the left predicts.

Incident angle θi
—
Peak outgoing direction
—
Peak BRDF value
—
Energy reflected ∫f·cosθ dω
—
Controls the surface
-48°
0.28
0.85
0.25

Roughness controls the Phong exponent. At 0 the glossy lobe collapses towards the mirror spike; at 1 it spreads out until it is almost indistinguishable from diffuse — which is the notes' point that glossy is a combination of the two, sitting on a dial rather than being a third thing.

— what this model says

Why the energy readout matters. The notes call the BRDF a probability distribution, and the constraint that follows is that a surface cannot reflect more light than it receives: ∫ f·cosθo dωo ≤ 1 for every incoming direction. Push reflectivity to 1 and the readout sits just under 1 — a perfect, lossless reflector. Anything above 1 would be a surface that manufactures light, and in a path tracer that shows up as an image which gets brighter the longer you let it run instead of converging. Real BRDFs also obey reciprocity: swap ωi and ωo and the value is unchanged, which is what makes it legal to trace paths backwards from the camera at all.