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Phase 1 — Math for Graphics
- Everything a path tracer needs from mathematics. This is not a survey — go deep on each concept until you can derive it from scratch.
- Parent: PathTracer Learning
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1.1 Vector Algebra
- PathTracer Learning - Concept - Dot Product
- Measures alignment between two vectors
- Used everywhere: lighting, reflection, cosine-weighted sampling
dot(a, b) = |a||b|cos(θ) — when both are unit vectors, this is just cos(θ)- Key insight:
dot(N, L) > 0 means light is on the same side as the normal
- PathTracer Learning - Concept - Cross Product
- Produces a vector perpendicular to both inputs
- Used to build orthonormal bases (tangent, bitangent, normal = TBN matrix)
cross(a, b) = |a||b|sin(θ) * n̂- Order matters:
cross(a, b) = -cross(b, a)
- Normalization
normalize(v) = v / length(v)- Always normalize direction vectors before using in dot/cross products
- Unnormalized normals are a common source of subtle shading bugs
- Vector reflection formula
reflect(v, n) = v - 2 * dot(v, n) * n- Assumes
n is unit length
- Used in specular BRDF evaluation
- Vector refraction (Snell’s law)
n1 * sin(θ1) = n2 * sin(θ2)- GLSL:
refract(v, n, eta) where eta = n1/n2
- Total internal reflection when
1 - eta² * (1 - dot(n,v)²) < 0
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- World space vs local space vs tangent space
- World space: global scene coordinates
- Local space: relative to an object’s origin
- Tangent space: aligned to a surface normal (used for normal maps, BRDF sampling)
- Building an orthonormal basis from a normal
- Given
N, construct T (tangent) and B (bitangent) such that {T, B, N} are mutually perpendicular unit vectors
- Frisvad / Duff et al. method (numerically stable):
- Used to transform sampled directions from tangent space to world space
- Homogeneous coordinates
- 4D representation:
(x, y, z, w) where w=1 for points, w=0 for directions
- Allows translation to be expressed as matrix multiplication
w=0 vectors are unaffected by translation — correct for normals and ray directions
- Normal transform rule
- Normals do NOT transform with the model matrix
M
- They transform with the inverse-transpose:
N_world = transpose(inverse(M)) * N_local
- If
M is orthogonal (rotation only), then inverse(M) = transpose(M) so it simplifies to just M
- See PathTracer Learning - Concept - Normal Mapping for tangent-space details
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1.3 Radiometry — The Language of Light
- PathTracer Learning - Concept - Radiometry
- Radiant flux, irradiance, radiance — the quantities path tracing computes
- Understanding units prevents factor-of-π errors
- PathTracer Learning - Concept - Solid Angle
- The 2D angle measure on a sphere —
dω = sin(θ) dθ dφ
- Full sphere =
4π sr, hemisphere = 2π sr
- Essential for understanding PDFs and the rendering equation
- Key radiometric quantities
- Radiant flux
Φ — total power (watts)
- Irradiance
E — power per unit area arriving at a surface (W/m²)
- Radiance
L — power per unit area per unit solid angle (W/m²/sr)
- Radiance is what cameras measure and what path tracing computes
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1.4 Probability and Statistics for Rendering
- PathTracer Learning - Concept - Monte Carlo Integration
- Estimating integrals by random sampling
- The rendering equation is an integral — Monte Carlo is how we solve it
- PathTracer Learning - Concept - Importance Sampling
- Sample more where the integrand is large
- Reduces variance dramatically for glossy BRDFs
- PathTracer Learning - Concept - MIS
- Multiple Importance Sampling — combine BRDF and light sampling optimally
- Power heuristic, balance heuristic
- Probability Density Function (PDF)
p(x) — probability of sampling value x- Must integrate to 1 over the domain:
∫ p(x) dx = 1
- For uniform hemisphere sampling:
p(ω) = 1 / (2π)
- For cosine-weighted hemisphere:
p(ω) = cos(θ) / π
- Expected value and variance
E[f(x)] = ∫ f(x) p(x) dx- Monte Carlo estimator:
(1/N) Σ f(xᵢ) / p(xᵢ) converges to E[f]
- Variance decreases as
O(1/N) — doubling samples halves variance, not error
- Quasi-Monte Carlo
- Low-discrepancy sequences (Halton, Sobol) instead of pseudo-random
- Converges at
O(1/N) for smooth integrands vs O(1/√N) for random
- Blue noise sampling — perceptually better distribution of samples
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1.5 The Rendering Equation
- Kajiya 1986 — the foundation of physically-based rendering
L_o(x, ω_o) = L_e(x, ω_o) + ∫_Ω f_r(x, ω_i, ω_o) L_i(x, ω_i) (N · ω_i) dω_i
L_o — outgoing radiance (what we want to compute)L_e — emitted radiance (light sources)f_r — BRDF (how the surface scatters light)L_i — incoming radiance (recursive — this is why it’s hard)(N · ω_i) — Lambert’s cosine law∫_Ω — integral over the hemisphere of incoming directions
- Why it’s recursive
L_i(x, ω_i) is the outgoing radiance from whatever surface the ray hits- That surface also has its own rendering equation
- Path tracing solves this by tracing paths of finite length and using Russian roulette to terminate
- Light transport equation (LTE)
- More general form: accounts for participating media (fog, smoke)
L(x→y) = L_e(x→y) + ∫ f_s(x, ω_i, ω_o) L(x'→x) G(x, x') V(x, x') dAG(x, x') — geometry term (cosines and distance)V(x, x') — visibility (0 or 1)
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Phase 1 Checklist
- TODO Can derive dot product formula from first principles
- TODO Can build an orthonormal basis from a normal vector
- TODO Understand why normals use inverse-transpose transform
- TODO Can explain Monte Carlo integration and why it works
- TODO Can write the rendering equation from memory and explain each term
- TODO Understand the difference between uniform and cosine-weighted hemisphere sampling
- TODO Can explain radiance vs irradiance and why the distinction matters
- TODO Understand solid angle and how
dω = sin(θ) dθ dφ is derived
- TODO Can explain MIS power heuristic and when to use it