Ray Intersection

Ray–Surface Intersection

An interactive geometry primer

A ray tracer renders images by casting rays from a camera through<br>every pixel and finding where each ray hits a surface. This tool explores the<br>core mathematical question: how do you compute that intersection?

Choose a geometry — sphere , cylinder , or<br>torus — and explore it across three lenses:

📐

Theory

Derive the intersection polynomial step by step

🎮

Intuition

Aim a ray interactively and watch the roots move

💻

Code

Edit live GLSL — see the shader render in real time

Start exploring →

🔵 Ray–Sphere Intersection

🔵 Sphere<br>🔷 Cylinder<br>🍩 Torus

drag near-plane · orbit 3D · click shape to transform

Theory<br>Intuition<br>Code

§ 1 — Ray Casting

drag to orbit

drag to orbit

drag to orbit

§ 2 — 🔵 Sphere — Degree-2 Polynomial

drag to orbit

§ 3 — 🔷 Cylinder — Degree-2 Barrel + Caps

drag to orbit

§ 4 — 🍩 Torus — Degree-4 Polynomial

drag to orbit

Parameters

Radius R

2.00

Minor radius r

0.50

Ray X (near plane)

0.00

Ray Y (near plane)

0.00

Near Plane (click/drag to aim ray)

Quadratic f(t)

f(t) = 0 ⟹ ray hits sphere at parameter t