Visualizing Quaternions - 1st Edition

Visualizing Quaternions

1st Edition

Authors: Andrew Hanson
Imprint: Morgan Kaufmann
95.95 + applicable tax
58.99 + applicable tax
73.95 + applicable tax
89.95 + applicable tax
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Table of Contents

  • FOREWORD by Steve Cunningham
      • 1.1 Hamilton's Walk
      • 1.2 Then Came Octonions
      • 1.3 The Quaternion Revival
      • 2.1 The Belt Trick
      • 2.2 The Rolling Ball
      • 2.3 The Apollo 10 Gimbal-lock Incident
      • 2.4 3D Game Developer's Nightmare
      • 2.5 The Urban Legend of the Upside-down F16
      • 2.6 Quaternions to the Rescue
      • 3.1 Vectors
      • 3.2 Length of a Vector
      • 3.3 3D Dot Product
      • 3.4 3D Cross Product
      • 3.5 Unit Vectors
      • 3.6 Spheres
      • 3.7 Matrices
      • 3.8 Complex Numbers
      • 5.1 The Complex Number Connection
      • 5.2 The Cornerstones of Quaternion Visualization
      • 6.1 2D Rotations
        • 6.1.1 Relation to Complex Numbers
        • 6.1.2 The Half-angle Form
        • 6.1.3 Complex Exponential Version
      • 6.2 Quaternions and 3D Rotations
        • 6.2.1 Construction
        • 6.2.2 Quaternions and Half Angles
        • 6.2.3 Double Values
      • 6.3 Recovering Θ and n
      • 6.4 Euler Angles and Quaternions
      • 6.5 † Optional Remarks
        • 6.5.1 † Connections to Group Theory
        • 6.5.2 † "Pure" Quaternion Derivation
        • 6.5.3 † Quaternion Exponential Version
      • 6.6 Conclusion
      • 7.1 Algebra of Complex Numbers
        • 7.1.1 Complex Numbers
        • 7.1.2 Abstract View of Complex Multiplication
        • 7.1.3 Restriction to Unit-length Case
      • 7.2 Quaternion Algebra
        • 7.2.1 The Multiplication Rule
        • 7.2.2 Scalar Product
        • 7.2.3 Modulus


Introduced 160 years ago as an attempt to generalize complex numbers to higher dimensions, quaternions are now recognized as one of the most important concepts in modern computer graphics. They offer a powerful way to represent rotations and compared to rotation matrices they use less memory, compose faster, and are naturally suited for efficient interpolation of rotations. Despite this, many practitioners have avoided quaternions because of the mathematics used to understand them, hoping that some day a more intuitive description will be available.
The wait is over. Andrew Hanson's new book is a fresh perspective on quaternions. The first part of the book focuses on visualizing quaternions to provide the intuition necessary to use them, and includes many illustrative examples to motivate why they are important—a beautiful introduction to those wanting to explore quaternions unencumbered by their mathematical aspects. The second part covers the all-important advanced applications, including quaternion curves, surfaces, and volumes. Finally, for those wanting the full story of the mathematics behind quaternions, there is a gentle introduction to their four-dimensional nature and to Clifford Algebras, the all-encompassing framework for vectors and quaternions.

Key Features

  • Richly illustrated introduction for the developer, scientist, engineer, or student in computer graphics, visualization, or entertainment computing.
    Covers both non-mathematical and mathematical approaches to quaternions.
    Companion website with an assortment of quaternion utilities and sample code, data sets for the book's illustrations, and Mathematica notebooks with essential algebraic utilities.


Programmers and developers in computer graphics and the game industry, scientists and engineers working in aerospace and scientific visualization, students of game development and computer graphics, and those interested in quaternions but who have limited math background.


Morgan Kaufmann
eBook ISBN:


“Almost all computer graphics practitioners have a good grasp of the 3D Cartesian space. However, in many graphics applications, orientations and rotations are equally important, and the concepts and tools related to rotations are less well-known.
Quaternions are the key tool for understanding and manipulating orientations and rotations, and this book does a masterful job of making quaternions accessible. It excels not only in its scholarship, but also provides enough detailed figures and examples to expose the subtleties encountered when using quaternions. This is a book our field has needed for twenty years and I’m thrilled it is finally here.”
—Peter Shirley, Professor, University of Utah

“This book contains all that you would want to know about quaternions, including a great many things that you don’t yet realize that you want to know!”
—Alyn Rockwood, Vice President, ACM SIGGRAPH

“We need to use quaternions any time we have to interpolate orientations, for animating a camera move, simulating a rollercoaster ride, indicating fluid vorticity or displaying a folded protein, and it’s all too easy to do it wrong. This book presents gently but deeply the relationship between orientations in 3D and the differential geometry of the three-sphere in 4D that we all need to understand to be proficient in modern science and engineering, and especially computer graphics.”
—John C. Hart, Associate Professor, Department of Computer Science, University of Illinois Urbana-Champaign, and Editor-in-Chief, ACM Transactions on Graphics

Visualizing Quaternions is a comprehensive, yet superbly readable introduction to the concepts, mechanics, geometry, and graphical applications of Hamilton’s lasting contribution to the mathematical description of the real world. To write effectively on this subject, an author has to be a mathematician, physicist and computer scientist; Hanson is all three.
Still, the reader

About the Authors

Andrew Hanson Author

Andrew J. Hanson is a professor of computer science at Indiana University in Bloomington, Indiana, and has taught courses in computer graphics, computer vision, programming languages, and scientific visualization. He received a BA in chemistry and physics from Harvard College and a PhD in theoretical physics from MIT. Before coming to Indiana University, he did research in theoretical physics at the Institute for Advanced Study, Cornell University, the Stanford Linear Accelerator Center, and the Lawrence-Berkeley Laboratory, and then in computer vision at the SRI Artificial Intelligence Center in Menlo Park, CA. He has published a wide variety of technical articles concerning problems in theoretical physics, machine vision, computer graphics, and scientific visualization methods. His current research interests include scientific visualization (with applications in mathematics, cosmology and astrophysics, special and general relativity, and string theory), optimal model selection, machine vision, computer graphics, perception, collaborative methods in virtual reality, and the design of interactive user interfaces for virtual reality and visualization applications.

Affiliations and Expertise

Indiana University, Bloomington, U.S.A.