Quaternions for Computer Graphics

By John Vince

Sir William Rowan Hamilton was a genius, and will be remembered for his significant contributions to physics and mathematics. The Hamiltonian, which is used in quantum physics to describe the total energy of a system, would have been a major achievement for anyone, but Hamilton also invented quaternions, which paved the way for modern vector analysis.

Quaternions are one of the most documented inventions in the history of mathematics, and this book is about their invention, and how they are used to rotate vectors about an arbitrary axis. Apart from introducing the reader to the features of quaternions and their associated algebra, the book provides valuable historical facts that bring the subject alive.

Quaternions for Computer Graphics introduces the reader to quaternion algebra by describing concepts of sets, groups, fields and rings. It also includes chapters on imaginary quantities, complex numbers and the complex plane, which are essential to understanding quaternions. The book contains many illustrations and worked examples, which make it essential reading for students, academics, researchers and professional practitioners.

• Language: English
• Publisher: Springer
• Release date: Jun 11, 2011
• ISBN: 9780857297600

Book preview

John VinceQuaternions for Computer Graphics10.1007/978-0-85729-760-0_1© Springer-Verlag London Limited 2011

1. Introduction

John Vince¹  

(1)

Bournemouth University, Bournemouth, UK

John Vince

URL: www.johnvince.co.uk

Abstract

Chapter 1 covers the book’s aims and objectives and the reader’s technical profile.

1.1 Rotation Transforms

In computer graphics we use transforms to modify the position and orientation of an object or a virtual camera. Such transforms generally comprise: scale, translation and rotation. The first two transforms are straight forward, but rotations do cause problems. This is because we normally construct a rotation transform from individual rotations about the x-, y- and z-axes. Although such transforms work, they are far from perfect. What really is required, is a technique that is intuitive, simple and accurate.

Over the years, rotation transforms have embraced direction cosines, Euler angles, Euler–Rodrigues parameterisation, quaternions and multivectors. The last two techniques are the most recent, and are historically related. However, the subject of this book is quaternions, and how they can be used within computer graphics.

1.2 The Reader

This book is aimed at readers studying or working in computer graphics and require an overview of quaternions. They are probably the same people I have encountered asking questions on Internet forums about quaternions, how they work, and how they can be coded. Hopefully, this book will answer most of these questions.

1.3 Aims and Objectives of This Book

The primary aim of this book is to introduce the reader to the subject of quaternions and how they can be used to rotate points about an arbitrary axis. A secondary aim is to make the reader aware of the human dimension behind all mathematical discovery. Personally, I believe that we must never lose sight of the fact that mathematicians are human beings. And although they may be endowed with extraordinary mathematical skills, they fall in love, marry, raise families, die, and leave behind an amazing edifice of knowledge, from which we all