Vince J. Quaternions for Computer Graphics

Springer, 2011. — 154 p. — ISBN: 0857297597, 9780857297594.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.Rotation Transforms.
The Reader.
Aims and Objectives of This Book.
Mathematical Techniques.
Assumptions Made in This Book.Number Sets and Algebra.Number Sets.
Arithmetic Operations.
Axioms.
Expressions.
Equations.
Ordered Pairs.
Groups, Rings and Fields.Complex Numbers.Imaginary Numbers.
Powers of i.
Complex Numbers.
Adding and Subtracting Complex Numbers.
Multiplying a Complex Number by a Scalar.
Complex Number Products.
Norm of a Complex Number.
Complex Conjugate.
Quotient of Two Complex Numbers.
Inverse of a Complex Number.
Square-Root of i.
Field Structure.
Ordered Pairs.
Matrix Representation of a Complex Number.Worked Examples.The Complex Plane.Some History.
The Complex Plane.
Polar Representation.
Rotors.Worked Examples.Quaternion Algebra.Some History.
Defining a Quaternion.
Algebraic Definition.
Adding and Subtracting Quaternions.
Real Quaternion.
Multiplying a Quaternion by a Scalar.
Pure Quaternion.
Unit Quaternion.
Additive Form of a Quaternion.
Binary Form of a Quaternion.
The Conjugate.
Norm of a Quaternion.
Normalised Quaternion.
Quaternion Products.
Inverse Quaternion.
Matrices.
Quaternion Algebra.Worked Examples.3D Rotation Transforms.3D Rotation Transforms.
Rotating About a Cartesian Axis.
Rotate About an Off-Set Axis.
Composite Rotations.
Rotating About an Arbitrary Axis.Worked Examples.Quaternions in Space.Some History.
Quaternion Products.
Quaternions in Matrix Form.
Multiple Rotations.
Eigenvalue and Eigenvector.
Rotating About an Off-Set Axis.
Frames of Reference.
Interpolating Quaternions.
Converting a Rotation Matrix to a Quaternion.
Euler Angles to Quaternion.Worked Examples.Appendix. Eigenvectors and Eigenvalues.