# Raytracing Mathematics

> Raytracing Mathematics

- Skill: `j4flmao/raytracing-mathematics` (Agent Skill)
- Install (CLI): `npx skillmds@latest add j4flmao/raytracing-mathematics`
- Raw SKILL.md: https://api.skillmd.com/api/skills/j4flmao/raytracing-mathematics/raw
- Safety review: pending
- Works with: Claude Code, Claude.ai, OpenAI Codex
- Category: Coding & Dev Tools
- Author: j4flmao (https://skillmd.com/u/j4flmao)
- Updated: 2026-09-21
- Page: https://skillmd.com/skills/j4flmao/raytracing-mathematics

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# 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

```mermaid
%%{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
```