Vectors, Matrices, Quaternions, and Transforms with Three.js

Read before you buy: the PDF here is a free sample — the first 20 pages of the 472-page book (payments, sales and reviews on this demo site are sample data). The full e-book is coming soon to the thenexbooks store.

Stop guessing at rotations and cameras. Learn the math behind 3D — with code.

Sooner or later every 3D developer hits a wall made of math. An object rotates the wrong way, a camera points somewhere unexpected, a model appears in the wrong place, or motion jitters as the frame rate changes. Libraries such as Three.js hide much of the math, but they cannot hide it completely — and when something goes wrong, understanding is the only reliable fix.

This book teaches the math you actually need for 3D on the web, with code instead of proofs. Every idea is introduced with a picture, written as a small JavaScript function so you can see exactly what it does, checked with real numbers, and then connected to the Three.js classes you will use every day: Vector3, Matrix4, Quaternion, Euler, Plane, Ray, Box3, and the curve classes. High-school algebra is enough to start.

• 472 pages · 28 chapters + 4 appendices
• Coordinate systems, vectors, and dot and cross products
• Matrices, transforms, the scene graph, and local versus world space
• Euler angles, gimbal lock, quaternions, lerp, slerp, and easing
• Cameras, projection, rays, and intersections with planes and volumes
• Normals, curves, procedural geometry, integration, collisions, springs, and noise
• Floating-point pitfalls, plus a camera rig, a solar system, and a drone flight simulator

Table Of Contents

Part I — Foundations

• Ch 1 · Why 3D Math Matters for Web Developers
• Ch 2 · Coordinate Systems, Axes, and Units
• Ch 3 · Vectors I: Points, Directions, Addition, and Length
• Ch 4 · Vectors II: Dot and Cross Products
• Ch 5 · Angles, Trigonometry, and Radians

Part II — Transforms

• Ch 6 · Matrices and Linear Transformations
• Ch 7 · Translation, Rotation, and Scale with 4×4 Matrices
• Ch 8 · Composing Transforms and the Scene Graph
• Ch 9 · Local Space, World Space, and Converting Between Them
• Ch 10 · Euler Angles, Rotation Order, and Gimbal Lock

Part III — Rotations and Cameras

• Ch 11 · Quaternions Made Practical
• Ch 12 · Interpolation: Lerp, Slerp, and Easing
• Ch 13 · Cameras: View Matrices and lookAt
• Ch 14 · Projection: Perspective and Orthographic
• Ch 15 · From 3D to Screen and Back: Projecting, Unprojecting, and Rays

Part IV — Geometry

• Ch 16 · Lines, Planes, and Intersections
• Ch 17 · Bounding Volumes: Boxes and Spheres
• Ch 18 · Normals, Tangents, and the Math of Lighting
• Ch 19 · Curves: Bézier, Catmull-Rom, and Splines
• Ch 20 · Procedural Geometry with Math

Part V — Motion and Physics

• Ch 21 · Velocity, Acceleration, and Numerical Integration
• Ch 22 · Collision Detection Basics
• Ch 23 · Springs, Damping, and Smooth Following
• Ch 24 · Noise and Randomness for 3D
• Ch 25 · Floating-Point Precision and Numerical Pitfalls

Part VI — Projects and Case Study

• Ch 26 · Project: A Third-Person Camera Rig
• Ch 27 · Project: A Solar System with Orbits
• Ch 28 · Case Study: The Kestrel Drone Flight Simulator

Appendices

• App A · 3D Math Cheat Sheet
• App B · Formulas and Three.js Math API Reference
• App C · Debugging Transforms Checklist
• App D · Glossary

WHO IT IS FOR
JavaScript developers using Three.js who want to understand what their code is doing, front-end and creative developers moving from 2D to 3D, game and simulation hobbyists, and students who want an applied, code-first companion to linear algebra.

FORMAT
PDF (A4), 472 pages, English, clickable table of contents and PDF bookmarks. 1st Edition, 2026. Published by thenexbooks.

This book is independent and is not affiliated with or endorsed by the maintainers of Three.js, React, or any other project mentioned.

Book details