E-raamat: Dual Quaternions and Their Associated Clifford Algebras

Contents

2. Algebra

2.1 Quaternion Algebra

2.2 Conjugates

2.3 Dot Products and Norms: Lengths and Angles

3. Geometry

3.1 Points and Vectors in the Space of Dual Quaternions

3.2 Planes in the Space of Dual Quaternions

3.3 Lines in the Space of Dual Quaternions

3.3.1 Plucker Coordinates

3.3.2 Dual Plucker Coordinates

3.4 Duality in the Space of Dual Quaternions

4. Rigid Motions

4.1 Rotation and Translation

4.2 Rotations about Arbitrary Lines and Screw Transformations

4.3 Screw Transformations and Rigid Motions

4.4 Rotation and Translation on Planes

4.5 Rotation and Translation on Lines

4.6 Reflections

5. Rigid Motions as Rotations in 8-Dimensions

5.1 Rigid Motions as Linear Isometries in 8-Dimensions

5.2 Renormalization

6. Screw Linear Interpolation (ScLERP)

6.1 Spherical Linear Interpolation (SLERP) Revisited

6.2 The Trigonometric Form of the Screw Transformation

6.3 ScLERP

7. Perspective and Pseudo-Perspective

7.1 Perspective in the Quaternion Algebra

7.2 Rotation, Translation, and Duality

7.3 Perspective Projection

7.4 Pseudo-Perspective

8. Visualizing Quaternions and Dual Quaternions

9. Matrices vs. Dual Quaternions

9.1 Representations and Computations with Matrices and Dual Quaternions

9.2 Converting between Matrices and Dual Quaternions 9.2.1 Rigid Motions

9.2.2 Perspective and Pseudo-Perspective

10. Insights

11. Formulas

11.1 Algebra

11.2 Geometry

11.3 Duality

11.4 Transformations

11.5 Interpolation

11.6 Conversion Formulas

Appendix: Cross Products

1: A Brief Review of Clifford Algebra

1. Goals of Clifford Algebra

2. A Brief Introduction to Clifford Algebra

3. Basic Products: Clifford Product, Inner Product, and Outer Product

3.1 Exterior Algebra: The Outer (Wedge) Product for Arbitrary Grades

4. Duality

4.1 Duality in the Quaternion Algebra: Cross Products and Products of Pure Quaternions

2: The Plane Model of Clifford Algebra for Dual Quaternions

2.1. Algebra

2.2. Geometry

2.2.1 Planes

2.2.2 Points and Vectors

2.2.2.1 Incidence Relation: Point on Plane

2.2.3 Lines

2.2.3.1 Lines as the Intersection of Two Planes

2.2.3.2 Lines as Bivectors

2.2.3.3 Incidence Relations for Lines

2.2.4 Duality

2.2.4.1 Duality in the Quaternion Subalgebra

2.2.4.2 Duality in the Plane Model

2.2.4.3 Lines as the Join of Two Points

2.3. Transformations: Rotors and Versors

2.3.1 Translation

2.3.2 Rotation

2.3.3 Reflection

2.3.4 Perspective and Pseudo-Perspective

2.4. Insights

2.5. Formulas

2.5.1 Algebra

2.5.2 Geometry

2.5.3 Rotors and Versors

2.5.4 Perspective and Pseudo-Perspective

2.6. Comparisons Between Dual Quaternions and the Plane Model of Clifford Algebra

3. The Point Model of Clifford Algebra for Dual Quaternions

3.1. Algebra

3.2. Geometry

3.2.1 Points and Vectors

3.2.2 Planes

3.2.2.1 Incidence Relation: Point on Plane

3.2.3 Duality

3.2.3.1 Duality in the Point Model

3.2.4 Lines

3.2.4.1 Lines as the Join of Two Points

3.2.4.2 Lines as Bivectors

3.2.4.3 Incidence Relations for Lines

3.2.4.4 Lines as the Intersection of Two Planes

3.3. Transformations: Rotors and Versors

3.3.1 Translation

3.3.2 Rotation

3.3.3 Reflection

3.3.4 Perspective and Pseudo-Perspective

3.4. Insights

3.5. Formulas

3.5.1 Algebra

3.5.2 Geometry

3.5.3 Rotors and Versors

3.5.4 Perspective and Pseudo-Perspective

3.6. Comparisons Between the Point Model and the Plane Model of Clifford Algebra