Phase 2 — CPU Ray Tracing

• Build a software path tracer before touching Vulkan. Understanding the algorithm on CPU makes the GPU version much clearer.
• Reference: “Ray Tracing in One Weekend” series by Peter Shirley
• Parent: PathTracer Learning

2.1 Camera and Ray Generation

• PathTracer Learning - Concept - Camera Model
  • Pinhole camera: rays through pixel centers
  • Thin lens: depth of field via aperture sampling
  • Anti-aliasing: jitter ray within pixel footprint
• Generating a camera ray

  Ray generateRay(Camera cam, float u, float v) {
      // u, v in [0,1] — pixel UV with jitter for AA
      vec3 horizontal = cam.right * cam.viewport_width;
      vec3 vertical   = cam.up    * cam.viewport_height;
      vec3 lower_left = cam.origin - horizontal/2 - vertical/2 - cam.forward * cam.focal_length;
      vec3 target = lower_left + u * horizontal + v * vertical;
      return Ray(cam.origin, normalize(target - cam.origin));
  }

2.2 Ray Definition and Scene Intersection

• PathTracer Learning - Concept - Ray Definition
  • A ray is a parametric line: P(t) = origin + t * direction
  • t > 0 means in front of the origin
  • t_min and t_max define the valid interval (avoids self-intersection)
• PathTracer Learning - Concept - Ray-Triangle Intersection
  • Möller–Trumbore algorithm — the standard for real-time and offline rendering
  • Returns t, u, v barycentric coordinates
  • Used to interpolate normals, UVs, and other vertex attributes
• PathTracer Learning - Concept - AABB
  • Axis-Aligned Bounding Box — the building block of BVH
  • Slab method intersection: test 3 pairs of parallel planes
  • t_enter = max(t_x_min, t_y_min, t_z_min), t_exit = min(t_x_max, t_y_max, t_z_max)
  • Hit if t_enter <= t_exit && t_exit > 0

2.3 Acceleration Structures

• PathTracer Learning - Concept - BVH Construction
  • Bounding Volume Hierarchy — tree of AABBs
  • Naive O(N) per ray → BVH O(log N) per ray
  • SAH (Surface Area Heuristic) for optimal splits
  • Binned SAH: O(N) per level, K=32 bins
• PathTracer Learning - Concept - BVH Traversal
  • Recursive descent: test AABB, if hit recurse into children
  • Leaf nodes contain actual triangles
  • Ordered traversal: visit closer child first for early exit
  • Iterative traversal with explicit stack (GPU-friendly)

2.4 Radiometry and Shading

• PathTracer Learning - Concept - Radiometry
  • Radiance L is what path tracing computes per ray
  • Irradiance E = ∫ L cos(θ) dω — what arrives at a surface
• PathTracer Learning - Concept - Normal Mapping
  • Perturb shading normal using a texture
  • TBN matrix transforms tangent-space normal to world space

2.5 Materials and BRDF

• PathTracer Learning - Concept - BRDF
  • Bidirectional Reflectance Distribution Function
  • f_r(ω_i, ω_o) — ratio of reflected radiance to incident irradiance
  • Must be energy-conserving: ∫ f_r cos(θ) dω ≤ 1
  • Must be reciprocal: f_r(ω_i, ω_o) = f_r(ω_o, ω_i)
• PathTracer Learning - Concept - Microfacet Theory
  • GGX NDF, Smith G term — the theory behind rough specular surfaces
  • Why α = roughness² (perceptual remapping)
• PathTracer Learning - Concept - Fresnel Effect
  • Schlick approximation: F(θ) = F0 + (1-F0)(1-cosθ)^5
  • Metals vs dielectrics: F0 interpretation
  • Why everything is more reflective at grazing angles
• Lambertian (diffuse) BRDF
  • Scatters light equally in all directions
  • Combined with cosine-weighted sampling: f_r * cos(θ) / p(ω) = albedo
  • The π and cos(θ) cancel perfectly — very clean implementation
• GGX Microfacet BRDF
  • D — Normal Distribution Function (GGX/Trowbridge-Reitz)
  • G — Geometric shadowing/masking (Smith)
  • F — Fresnel term (Schlick approximation)
  • GGX NDF: D(h) = α² / (π * (dot(N,h)² * (α²-1) + 1)²)
• Disney Principled BRDF
  • Artist-friendly: baseColor, metallic, roughness, specular, anisotropic, sheen, clearcoat, transmission
  • Combines diffuse + specular + clearcoat + sheen layers
  • Used in Godot’s PBR material system

2.6 Path Tracing Core Loop

• PathTracer Learning - Path Tracing Algorithm
  • The full algorithm with all components
• PathTracer Learning - Concept - Monte Carlo Integration
• PathTracer Learning - Concept - Importance Sampling
• PathTracer Learning - Concept - MIS
  • Combine BRDF sampling and NEE without double-counting
• PathTracer Learning - Concept - Next Event Estimation
  • Direct light sampling — connect each path vertex to a light source
  • Dramatically reduces variance for scenes with small light sources
• PathTracer Learning - Concept - Russian Roulette
  • Probabilistic path termination
  • Unbiased way to handle infinite recursion
• The path tracing loop (pseudocode)

  color trace(ray r, scene s):
      hit = intersect(r, s)
      if no hit: return sky_color(r.direction)
      if hit.material.is_emissive: return hit.material.emission
      
      // Russian roulette
      survival_prob = max(hit.albedo)
      if random() > survival_prob: return vec3(0)
      
      // Sample new direction (BRDF importance sampling)
      new_dir = sample_brdf(hit.normal, -r.direction, hit.material)
      pdf = brdf_pdf(hit.normal, -r.direction, new_dir, hit.material)
      
      // Evaluate BRDF
      brdf = hit.material.eval(-r.direction, new_dir, hit.normal)
      
      // NEE: direct lighting
      L_direct = sample_lights(hit, s)
      
      // Recursive trace
      incoming = trace(ray(hit.point + hit.normal * 0.001, new_dir), s)
      L_indirect = (brdf * incoming * dot(hit.normal, new_dir)) / (pdf * survival_prob)
      
      return L_direct + L_indirect
  

2.7 Image Output and Tone Mapping

• PathTracer Learning - Concept - Tone Mapping
  • HDR radiance values → LDR display values
  • Reinhard, ACES filmic, exposure control
• PathTracer Learning - Concept - Anti-Aliasing
  • Jittered sampling within pixel footprint
  • Temporal anti-aliasing (TAA) for real-time
• PathTracer Learning - Concept - Environment Map
  • HDR environment maps for image-based lighting (IBL)
  • Importance sampling the environment map

2.8 Accumulation and Convergence

• PathTracer Learning - Concept - Temporal Accumulation
  • Average N samples over time: color = (prev * (N-1) + new) / N
  • Converges to ground truth as N → ∞
• Fireflies
  • Extremely bright pixels from rare high-energy paths
  • Caused by low-probability paths with high contribution
  • Mitigation: clamp radiance, use MIS, better importance sampling
• Convergence rate
  • Monte Carlo converges at O(1/√N) — need 4x samples to halve noise
  • Importance sampling reduces variance without changing convergence rate
  • Denoising (DLSS, OIDN) compensates for slow convergence

Project

• PathTracer Learning - Project - CPU Path Tracer

Phase 2 Checklist

• Implement ray-AABB intersection
• Implement Möller–Trumbore ray-triangle intersection
• Build a BVH with SAH splitting
• Implement Lambertian BRDF with cosine-weighted sampling
• Implement GGX microfacet BRDF with importance sampling
• Implement Fresnel (Schlick) and metallic workflow
• Implement Russian roulette path termination
• Implement next event estimation (direct light sampling)
• Implement MIS for NEE + BRDF
• Implement HDR environment map with importance sampling
• Implement tone mapping (ACES or Reinhard)
• Render a Cornell box scene

Navigation

• Previous: PathTracer Learning - Phase 1 - Math for Graphics
• Next: PathTracer Learning - Phase 3 - GPU and Vulkan