Quaternions for computer graphics / / John Vince

• Record Nr.: UNINA9910497087203321
• Autore: Vince John (John A.)
• Pubbl/distr/stampa: London, England : , : Springer, , [2021]; ©2021
• ISBN: 9781447175094; 9781447175087
• Edizione: [Second edition.]
• Descrizione fisica: 1 online resource (188 pages)
• Disciplina: 512.5
• Soggetti: Quaternions
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- 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.