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

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

1.2 Coordinate Systems and Transforms

• 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):

    void buildONB(vec3 N, out vec3 T, out vec3 B) {
        float s = sign(N.z);
        float a = -1.0 / (s + N.z);
        float b = N.x * N.y * a;
        T = vec3(1.0 + s * N.x * N.x * a, s * b, -s * N.x);
        B = vec3(b, s + N.y * N.y * a, -N.y);
    }

  • 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

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

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

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') dA
  • G(x, x') — geometry term (cosines and distance)
  • V(x, x') — visibility (0 or 1)

1.6 Camera Model

• PathTracer Learning - Concept - Camera Model
  • Pinhole camera, thin lens model, depth of field
  • How to generate primary rays from pixel coordinates

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

Next

• PathTracer Learning - Phase 2 - CPU Ray Tracing