Numerical Calculations in Clifford Algebra A Practical Guide for Engineers and Scientists

An intuitive combination of the theory of Clifford algebra with numerous worked and computed examples and calculations

Numerical Calculations in Clifford Algebra: A Practical Guide for Engineers and Scientists is an accessible and practical introduction to Clifford algebra, with comprehensive coverage of the theory and calculations. The book offers many worked and computed examples at a variety of levels of complexity and over a range of different applications making extensive use of diagrams to maintain clarity. The author introduces and documents the Clifford Numerical Suite, developed to overcome the limitations of existing computational packages and to enable the rapid creation and deployment of sophisticated and efficient code.

Applications of the suite include Fourier transforms for arrays of any types of Clifford numbers and the solution of linear systems in which the coefficients are Clifford numbers of particular types, including scalars, bicomplex numbers, quaternions, Pauli matrices, and extended electromagnetic fields. Readers will find:

• A thorough introduction to Clifford algebra, with a combination of theory and practical implementation in a range of engineering problems
• Comprehensive explorations of a variety of worked and computed examples at various levels of complexity
• Practical discussions of the conceptual and computational tools for solving common engineering problems
• Detailed documentation on the deployment and application of the Clifford Numerical Suite

Perfect for engineers, researchers, and academics with an interest in Clifford algebra, Numerical Calculations in Clifford Algebra: A Practical Guide for Engineers and Scientists will particularly benefit professionals in the areas of antenna design, digital image processing, theoretical physics, and geometry.

Andrew Seagar, PhD, is Director for the Bachelor of Engineering Programs at the Gold Coast Campus of the School of Engineering at Griffith University in Australia. He has experience in a variety of research, commercial development, and academic positions around the world, primarily in the areas of electrical or biomedical engineering.

Preface xix

I Entities and Operations

1 Introduction 3

1.1 Operations 3

1.2 History 4

1.3 Alternative Forms 4

1.4 Naming 5

1.5 Structure 6

1.5.1 Algebraic 6

1.5.2 Numeric 6

1.6 Entities 9

2 Input 11

2.1 Syntax 11

2.2 Constants 12

2.2.1 Specific Types 12

2.2.2 General 13

2.3 Variables 15

2.3.1 Checking and Converting 15

3 Output 21

3.1 Tree Format 21

3.2 Numeric Formats 24

3.2.1 Default Format 24

3.2.2 Defined Format 25

3.3 Extended Formats 26

3.3.1 Rounding 26

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3.3.2 Parts of Coefficients 27

3.4 Selected Components 28

3.5 Primitive Formats 29

3.6 Recovered Values 30

4 Unary Operations 33

4.1 Theory 33

4.1.1 Negation 33

4.1.2 Involution 34

4.1.3 Pair Exchange 34

4.1.4 Reversion 35

4.1.5 Clifford Conjugation 36

4.1.6 Supplementation and Pseudoscalar 36

4.2 Practice 37

4.2.1 Example Code 37

4.2.2 Example Output 38

5 Binary Operations 41

5.1 Geometric Origins 41

5.1.1 Outer Multiplication 41

5.1.2 Orthogonal Components 43

5.1.3 Inner Multiplication 44

5.1.4 Names 45

5.2 Multiplication of Units 46

5.2.1 Progressive and Regressive Multiplication 46

5.2.2 Outer, Inner and Central Multiplication 47

5.2.3 Multiplication by Scalars 49

5.3 Central Multiplication 49

5.3.1 Primal Units 50

5.3.2 Evolved and Other Units 51

5.3.3 Numbers 52

5.4 Practice 53

5.4.1 Example Code 53

5.4.2 Example Output 54

5.4.3 Multiplication Tables 54

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6 Vectors and Geometry 59

6.1 Theory 59

6.1.1 Magnitude 59

6.1.2 Inverse 60

6.1.3 Reflection 60

6.1.4 Projection 60

6.1.5 Rotation 61

6.2 Practice 61

6.2.1 Example Code 61

6.2.2 Example Output 63

7 Quaternions 65

7.1 Theory 65

7.1.1 Magnitude 65

7.1.2 Inverse 66

7.1.3 Reflection and Projection 66

7.1.4 Rotation 66

7.1.5 Intersection 67

7.1.6 Factorisation 67

7.2 Practice 68

7.2.1 Example Code 68

7.2.2 Example Output 70

8 Pauli Matrices 73

8.1 Theory 73

8.1.1 Recovery of Components 73

8.1.2 Magnitude 74

8.1.3 Inverse 74

8.1.4 Reflection, Projection and Rotation 74

8.2 Practice 74

8.2.1 Example Code 74

8.2.2 Example Output 76

9 Bicomplex Numbers 79

9.1 Theory 79

9.1.1 Conjugate 79

9.1.2 Magnitude 80

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9.1.3 Inverse 80

9.1.4 Reflection, Projection, and Rotation 80

9.2 Practice 81

9.2.1 Example Code 81

9.2.2 Example Output 82

10 Electromagnetic Fields 85

10.1 Theory 85

10.1.1 Time and Frequency 85

10.1.2 Electromagnetic Entities 85

10.1.3 Dirac Operators 86

10.1.4 Maxwell’s Equations 86

10.1.5 Simplified Notation 87

10.1.6 Magnitude 87

10.1.7 Inverse 87

10.1.8 Reflection 88

10.1.9 Projection 88

10.1.10 Rotation 88

10.2 Practice 88

10.2.1 Example Code 88

10.2.2 Example Output 90

10.3 Field Arithmetic 92

10.3.1 Extensions based on Quaternions 92

10.3.2 Inverses 92

10.3.3 Example Code 94

10.3.4 Example Output 95

11 Arrays of Clifford Numbers 97

11.1 Theory 97

11.2 Practice 98

11.2.1 Example Code 98

11.2.2 Example Output 100

12 Power Series 103

12.1 Theory 103

12.1.1 User Defined 103

12.1.2 Predefined 104

Contents c Seagar/Wiley, September 16, 2022 ix

12.1.3 Convergence 105

12.1.4 Factorisation 105

12.1.5 Squaring 106

12.2 Practice 106

12.2.1 User Defined 106

12.2.2 Predefined 109

13 Matrices of Clifford Numbers 121

13.1 Background 121

13.2 Inversion 122

13.3 Practice 123

13.3.1 Example Code 123

13.3.2 Example Output 126

II Customisation

14 Memory 133

14.1 Memory Usage 133

14.2 Examples 134

14.2.1 Memory Tree Sparsity 134

14.2.2 Memory Expansion 138

14.2.3 Memory Recycling 139

15 Errors 143

15.1 User Errors 143

15.1.1 Syntax Errors and Messages 146

15.2 System Errors 146

15.3 Recovery 146

15.4 Beneficial Usage 150

16 Extension 155

16.1 Accumulation 155

16.2 Multiplication 156

16.3 Transformation 158

16.4 Filtration 159

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III Application

17 Verification 165

17.1 Identities 165

17.2 Tests 165

17.2.1 Example Code 165

17.2.2 Example Output 167

18 Lines Not Parallel 173

18.1 Theory 173

18.1.1 Common Plane 173

18.1.2 No Plane in Common 175

18.2 Practice 177

18.2.1 Example Code 177

18.2.2 Example Output 180

19 Perspective Projection 181

19.1 Theory 181

19.2 Practice 182

19.2.1 Example Code 182

19.2.2 Example Output 185

20 Linear Systems 187

20.1 Theory 187

20.2 Practice 188

20.2.1 Example Code 188

20.2.2 Example Output 190

21 Fast Fourier Transform 191

21.1 Theory 191

21.2 Practice 192

21.2.1 Example Code 192

21.2.2 Example Output 196

22 Hertzian Dipole 199

22.1 Theory 199

22.2 Practice 200

22.2.1 Example Code 200

22.2.2 Example Output 203

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23 Finite Difference Time Domain 207

23.1 Theory 207

23.1.1 Analytical Solution 207

23.1.2 Series Solution 207

23.1.3 Analytical Example 208

23.1.4 Numerical Derivatives 209

23.2 Practice 210

23.2.1 Example Code 210

23.2.2 Example Output 215

24 Cauchy Extension 221

24.1 Background 221

24.2 Theory 221

24.2.1 Two Dimensions 221

24.2.2 Three Dimensions 222

24.2.3 Singularity 222

24.2.4 The Taming Function 223

24.2.5 Construction 223

24.3 Practice 225

24.3.1 Example Code 225

24.3.2 Example Output 228

25 Electromagnetic Scattering 233

25.1 Background 233

25.2 Theory 234

25.3 Practice 235

25.3.1 Example Code 235

25.3.2 Example Output 236

IV Programming

26 Interfaces 243

26.1 Configuration and Observation 243

26.1.1 Management 243

26.1.2 Printing 244

26.2 Simple Entities 245

26.2.1 Units 245

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26.2.2 Components 245

26.2.3 Numbers 247

26.3 Higher Entities 249

26.3.1 Vectors 249

26.3.2 Bicomplex Numbers 250

26.3.3 Quaternions 250

26.3.4 Pauli Matrices 251

26.3.5 Electromagnetic Fields 251

26.4 Multiple Entities 251

26.4.1 Arrays 251

26.4.2 Fast Fourier Transforms 252

26.4.3 Series 252

26.4.4 Matrices 253

27 Descriptions 255

27.1 Arguments 255

27.2 Datatypes 255

27.3 Formats 257

27.4 Manual Pages 259

27.4.1 A–E 259

27.4.2 F–J 276

27.4.3 K–O 294

27.4.4 P–T 306

27.4.5 U–Z 360

27.5 Quick Reference 367

Appendices 375

A Key to Example Code and Results 375

Index 377

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