Quaternions for Computer Graphics

Preface
Contents
1 Introduction
	1.1 Rotation Transforms
	1.2 The Reader
	1.3 Aims and Objectives of This Book
	1.4 Mathematical Techniques
	1.5 Assumptions Made in This Book
	References
2 Number Sets and Algebra
	2.1 Introduction
	2.2 Number Sets
		2.2.1 Natural Numbers
		2.2.2 Real Numbers
		2.2.3 Integers
		2.2.4 Rational Numbers
	2.3 Arithmetic Operations
	2.4 Axioms
	2.5 Expressions
	2.6 Equations
	2.7 Ordered Pairs
	2.8 Groups, Rings and Fields
		2.8.1 Groups
		2.8.2 Abelian Group
		2.8.3 Rings
		2.8.4 Fields
		2.8.5 Division Ring
	2.9 Summary
		2.9.1 Summary of Definitions
	Reference
3 Complex Numbers
	3.1 Introduction
	3.2 Imaginary Numbers
	3.3 Powers of i
	3.4 Definition of a Complex Number
		3.4.1 Addition and Subtraction of Complex Numbers
		3.4.2 Multiplying a Complex Number by a Scalar
		3.4.3 Product of Complex Numbers
		3.4.4 Square of a Complex Number
		3.4.5 Norm of a Complex Number
		3.4.6 Complex Conjugate of a Complex Number
		3.4.7 Quotient of Complex Numbers
		3.4.8 Inverse of a Complex Number
		3.4.9 Square-Root of pmi
	3.5 Field Structure of Complex Numbers
	3.6 Ordered Pairs
		3.6.1 Addition and Subtraction of Ordered Pairs
		3.6.2 Multiplying an Ordered Pair by a Scalar
		3.6.3 Product of Ordered Pairs
		3.6.4 Square of an Ordered Pair
		3.6.5 Norm of an Ordered Pair
		3.6.6 Complex Conjugate of an Ordered Pair
		3.6.7 Quotient of an Ordered Pair
		3.6.8 Inverse of an Ordered Pair
		3.6.9 Square-Root of pmi
	3.7 Matrix Representation of a Complex Number
		3.7.1 Adding and Subtracting Complex Numbers
		3.7.2 Product of Two Complex Numbers
		3.7.3 Norm Squared of a Complex Number
		3.7.4 Complex Conjugate of a Complex Number
		3.7.5 Inverse of a Complex Number
		3.7.6 Quotient of a Complex Number
		3.7.7 Square-Root of pmi
	3.8 Summary
		3.8.1 Summary of Definitions
	3.9 Worked Examples
		3.9.1 Adding and Subtracting Complex Numbers
		3.9.2 Product of Complex Numbers
		3.9.3 Multiplying a Complex Number by i
		3.9.4 The Norm of a Complex Number
		3.9.5 The Complex Conjugate of a Complex Number
		3.9.6 The Quotient of Two Complex Numbers
		3.9.7 Divide a Complex Number by i
		3.9.8 Divide a Complex Number by -i
		3.9.9 The Inverse of a Complex Number
		3.9.10 The Inverse of i
		3.9.11 The Inverse of -i
	References
4 The Complex Plane
	4.1 Introduction
	4.2 Some History
	4.3 The Complex Plane
	4.4 Polar Representation
	4.5 Rotors
	4.6 Summary
		4.6.1 Summary of Definitions
	4.7 Worked Examples
		4.7.1 Rotate a Complex Number by i
		4.7.2 Product and Quotient Using Polar Form
		4.7.3 Design a Rotor to Rotate a Complex Number 30°
		4.7.4 Design a Rotor to Rotate a Complex Number -60°
	References
5 Triples and Quaternions
	5.1 Introduction
	5.2 Some History
	5.3 Triples
		5.3.1 Adding and Subtracting Triples
	5.4 The Birth of Quaternions
	References
6 Quaternion Algebra
	6.1 Introduction
	6.2 Some History
	6.3 Defining a Quaternion
		6.3.1 The Quaternion Units
		6.3.2 Example of Quaternion Products
	6.4 Algebraic Definition
	6.5 Adding and Subtracting Quaternions
	6.6 Real Quaternion
	6.7 Multiplying a Quaternion by a Scalar
	6.8 Pure Quaternion
	6.9 Unit Quaternion
	6.10 Additive Form of a Quaternion
	6.11 Binary Form of a Quaternion
	6.12 The Complex Conjugate of a Quaternion
	6.13 Norm of a Quaternion
	6.14 Normalised Quaternion
	6.15 Quaternion Products
		6.15.1 Product of Pure Quaternions
		6.15.2 Product of Unit-Norm Quaternions
		6.15.3 Square of a Quaternion
		6.15.4 Norm of the Quaternion Product
	6.16 Inverse Quaternion
	6.17 Matrices
		6.17.1 Orthogonal Matrix
	6.18 Quaternion Algebra
	6.19 Summary
		6.19.1 Summary of Definitions
	6.20 Worked Examples
		6.20.1 Adding and Subtracting Quaternions
		6.20.2 Norm of a Quaternion
		6.20.3 Unit-Norm Quaternions
		6.20.4 Quaternion Product
		6.20.5 Square of a Quaternion
		6.20.6 Inverse of a Quaternion
	References
7 3-D Rotation Transforms
	7.1 Introduction
	7.2 3-D Rotation Transforms
	7.3 Rotating About a Cartesian Axis
	7.4 Rotate About an Off-Set Axis
	7.5 Composite Rotations
	7.6 Rotating About an Arbitrary Axis
		7.6.1 Matrices
		7.6.2 Vectors
	7.7 Rodrigues\' Rotation Formula
	7.8 Summary
		7.8.1 Summary of Definitions
	7.9 Worked Examples
		7.9.1 Rotation Transform About an Off-Set Axis
		7.9.2 Test for Gimbal Lock
		7.9.3 The General Rotation Matrix
		7.9.4 Testing the Rotation Matrix
	References
8 Quaternions in Space
	8.1 Introduction
	8.2 Some History
		8.2.1 Composition Algebras
	8.3 Quaternion Products
		8.3.1 Special Case
		8.3.2 General Case
		8.3.3 Double Angle
	8.4 Quaternions in Matrix Form
		8.4.1 Vector Method
		8.4.2 Matrix Method
		8.4.3 Geometric Verification
	8.5 Multiple Rotations
	8.6 Rotating About an Off-Set Axis
	8.7 Frames of Reference
	8.8 Interpolating
		8.8.1 Linear Interpolation
	8.9 Interpolating Vectors
	8.10 Interpolating Quaternions
	8.11 Converting a Rotation Matrix to a Quaternion
	8.12 Euler Angles to Quaternion
	8.13 Summary
		8.13.1 Summary of Definitions
	8.14 Worked Examples
		8.14.1 Special Case Quaternion
		8.14.2 Rotating a Vector Using a Quaternion
		8.14.3 Evaluate qpq-1
		8.14.4 Evaluate qpq-1 Using a Matrix
		8.14.5 Slerp Interpolation
		8.14.6 Rotation Matrix into a Quaternion
	References
9 Conclusion
Index