Quaternions for Computer Graphics

verfasst von: John Vince

Verlag: Springer London

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

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

John Vince

Chapter 2. Number Sets and Algebra

Abstract

Chapter 2 reviews basic ideas of natural, real, integer and rational number sets, and how they are manipulated arithmetically and algebraically. The chapter contains sections on axioms, expressions, equations and ordered pairs, and concludes with an introductory description of groups, abelian groups, rings and fields. The chapter summarises key formulae and contains some useful worked examples.

John Vince

Chapter 3. Complex Numbers

Abstract

Chapter 3 shows how equations that have no real roots give rise to imaginary numbers that square to −1, which, in turn, lead to complex numbers. Definitions and examples are given for adding, subtracting, multiplying and dividing complex numbers. Further sections introduce concepts of the norm, complex conjugate, inverse and square-root of a complex number. Finally, it is shown how a complex number is represented as an ordered pair and as a matrix. The chapter summarises key formulae and contains some useful worked examples.

John Vince

Chapter 4. The Complex Plane

Abstract

Chapter 4 describes the complex plane which provides a graphical representation for complex numbers. The chapter also contains historical information about the complex plane’s invention, and complements similar historical events associated with quaternions. Polar representation of a complex number is described and how it provides a useful mechanism to visualize rotations in the plane. The chapter summarises key formulae and contains some useful worked examples.

John Vince

Chapter 5. Quaternion Algebra

Abstract

Chapter 5 defines a quaternion and its associated algebra. Definitions and examples are given for adding, subtracting and multiplying quaternions. Further sections introduce pure, real and unit quaternions and how to conjugate, normalise and invert them. The matrix form of a quaternion is described in some detail, as this is useful for implementing rotations in space. The chapter summarises key formulae and contains some useful worked examples.

John Vince

Chapter 6. 3D Rotation Transforms

Abstract

Chapter 6 revises 3D rotation transforms and how they are used to rotate points about a Cartesian axis and an off-set axis. In particular, gimbal lock is described and the need to rotate points about an arbitrary 3D axis. To this end, a matrix transform is developed that achieves such a rotation, which provides a basis for understanding a similar transform using quaternions. The chapter summarises key formulae and contains some useful worked examples.

John Vince

Chapter 7. Quaternions in Space

Abstract

Chapter 7 is the focal point of the book and shows how quaternions are used to rotate vectors about an arbitrary 3D axis. The chapter begins by reviewing some of the history associated with quaternions, in particular, the role of Benjamin Olinde Rodrigues, who discovered the importance of half-angles in rotation transforms. Quaternion products are described and how they are employed to rotate points. Further sections explain how to compute eigenvalues and eigenvectors of a quaternion. Finally, the chapter contains details on frames of reference, interpolating quaternions, matrix to quaternion conversion, and Euler angles to quaternion conversion. The chapter summarises key formulae and contains some useful worked examples.

John Vince

Chapter 8. Conclusion

Abstract

Chapter 8 draws the book to a close by reviewing the original aims and objectives.

John Vince

Backmatter

Titel: Quaternions for Computer Graphics
verfasst von: John Vince
Verlag: Springer London
Electronic ISBN: 978-0-85729-760-0
Print ISBN: 978-0-85729-759-4
DOI: https://doi.org/10.1007/978-0-85729-760-0