Raytracing Mathematics

Raytracing Mathematics

Core Mechanics

Raytracing simulates the physical behavior of light to generate photorealistic images. It traces paths of light backwards from the camera into the 3D scene.

1. Ray-Sphere Intersection

A Ray is defined mathematically as P(t) = O + tD (Origin + time * Direction). A Sphere is defined as (P - C)·(P - C) = r^2. Substituting the Ray into the Sphere equation yields a quadratic equation (at^2 + bt + c = 0). Solving for t determines if, and exactly where, the ray hits the sphere.

2. Bounding Volume Hierarchies (BVH)

Checking every ray against millions of triangles is computationally impossible. A BVH wraps complex models in simple boxes. If a ray misses the big box, we immediately skip checking all the complex triangles inside it.

Raytracing Flow Map

%%{init: {"theme": "default", "flowchart": {"useMaxWidth": true}}}%%
flowchart TD
    subgraph Camera ["Virtual Camera"]
        A["Shoot Ray through Pixel(x,y)"]
    end
    
    subgraph BVH ["Spatial Acceleration (BVH)"]
        B{"Intersect Root Bounding Box?"}
        C{"Intersect Child Bounding Box?"}
    end
    
    subgraph Math ["Geometry Mathematics"]
        D["Calculate Ray-Triangle Intersection"]
        E["Calculate Surface Normal"]
    end
    
    subgraph Shading ["Light Transport"]
        F["Bounce Ray (Reflection/Refraction)"]
        G["Calculate Color & Shadow"]
    end
    
    A --> B
    B -->|"Yes"| C
    B -->|"No"| H["Render Skybox (Miss)"]
    C -->|"Yes"| D
    D -->|"Hit"| E
    E --> F
    F --> G