Concept: Solid Angle
Definition
- A solid angle measures “how much of the sphere” a region subtends
- Analogous to 2D angle (radians) but for 3D directions
- Unit: steradians (sr)
- Full sphere:
4π sr
- Hemisphere:
2π sr
- A small patch on the unit sphere:
dω = sin(θ) dθ dφ
Derivation of dω
- Parameterize the unit sphere with spherical coordinates
(θ, φ)
θ — polar angle from +Z axis, range [0, π]φ — azimuthal angle around Z axis, range [0, 2π]
- A small patch at
(θ, φ) with size (dθ, dφ):
- Width along φ:
sin(θ) dφ (circle at latitude θ has radius sin(θ))
- Height along θ:
dθ
- Area:
dω = sin(θ) dθ dφ
- Verify:
∫₀^π ∫₀^{2π} sin(θ) dθ dφ = 2π * [-cos(θ)]₀^π = 2π * 2 = 4π ✓
Projected Solid Angle
dω⊥ = cos(θ) dω = cos(θ) sin(θ) dθ dφ- This is the solid angle projected onto the surface plane
- Appears in the rendering equation as
(N · ω_i) dω_i
- Hemisphere integral of projected solid angle:
∫_Ω cos(θ) dω = π
- This is why Lambertian BRDF has
1/π — to normalize over the hemisphere
Solid Angle of a Sphere
- A sphere of radius
r at distance d subtends solid angle:
Ω = 2π * (1 - cos(θ_max)) where sin(θ_max) = r/d- For small angles:
Ω ≈ π * r² / d² (area / distance²)
- Used for: sampling area lights, computing light PDFs
Converting Between Area and Solid Angle
- A surface patch of area
dA at distance r, angle θ to the ray:
- This conversion appears in NEE when computing the PDF of sampling a light
p(ω) = p(A) * r² / cos(θ_light)p(A) = 1 / area_of_light (uniform sampling)
Hemisphere Sampling PDFs
- Uniform hemisphere:
p(ω) = 1 / (2π)
- Verify:
∫_Ω (1/2π) dω = (1/2π) * 2π = 1 ✓
- Cosine-weighted hemisphere:
p(ω) = cos(θ) / π
- Verify:
∫_Ω (cos(θ)/π) dω = (1/π) * π = 1 ✓
- GGX NDF sampling:
p(h) = D(h) * cos(θ_h)
- Verify:
∫_Ω D(h) cos(θ_h) dω = 1 (by definition of NDF)