Visualizing More Quaternions, Volume Two updates on proteomics-related material that will be useful for biochemists and biophysicists, including material related to electron microscopy (and specifically cryo-EVisualizing. Dr. Andrew J. Hanson’s groundbreaking book updates and extends concepts that have evolved since the first book published in 2005, adding entirely new insights that Dr. Hanson’s research has recently developed. This includes the applications of quaternion methods to proteomics and molecular crystallography problems, which are domains with significant current research and application activity.
In addition to readers interested in quaternions for their own sake, scientists involved in computer graphics, animation, shape modeling, and scientific visualization, and readers from several other disciplines will benefit from this new volume. Foremost among these, and the target of the first several chapters, are scientists involved in molecular chemistry where techniques based on quaternion eigensystems have become a standard tool for evaluating the quality of shape matching.
Explores quaternion based approaches to the matching of point cloud pairs, including approaches to data from orthographic and perspective projections.
Develops extensive applications of quaternion frames to protein orientation analysis.
Analyzes the application of quaternion methods to physics problems ranging from quantum computing to special relativity and gravitational instantons.
1. Chapter 1 Introduction - p. 3 2. Chapter 2 Basic quaternion formulas - p. 9 3. Chapter 3 What are quaternions? - p. 15 4. Chapter 4 What is quaternion visualization? - p. 31 5. Chapter 5 Interacting with quaternions as hyperspheres - p. 49 6. Chapter 6 Quaternions and uniform rotation distributions - p. 57 7. Part 2 Orientation matching - p. 69 8. References - p. 70 9. Chapter 7 Introduction: 2D cloud alignment problem - p. 71 10. Chapter 8 The 3D quaternion-based alignment problem - p. 77 11. Chapter 9 Quaternion of a 3D rotation from the Bar-Itzhack method - p. 95 12. Chapter 10 The 3D quaternion-based frame alignment problem - p. 103 13. Chapter 11 The combined point-frame alignment problem - p. 125 14. Part 3 Quaternion adjugate methods for pose estimation - p. 133 15. References - p. 133 16. Chapter 12 Exploring the quaternion adjugate variables - p. 135 17. Keywords - p. 135 18. 12.1 Overview of the quaternion extraction problem - p. 135 19. 12.2 Fundamental background - p. 136 20. 12.3 2D rotations and the quaternion map - p. 137 21. 12.4 3D rotations and the quaternion map - p. 147 22. References - p. 157 23. Chapter 13 Quaternion-related machine learning - p. 159 24. Chapter 14 Exact 2D rotations for cloud matching and pose estimation tasks - p. 175 25. Chapter 15 Exact 3D rotations for cloud matching and pose estimation tasks - p. 197 26. Keywords - p. 197 27. 15.1 Basics - p. 199 28. 15.2 3D point-cloud matching - p. 199 29. 15.3 Direct solution of the error-free 3D match problem - p. 202 30. 15.4 3D to 2D orthographic pose estimation - p. 208 31. 15.5 3D pose estimation with perspective projection - p. 217 32. 15.6 Remarks on handling data with measurement errors - p. 228 33. References - p. 231 34. Part 4 Quaternion proteomics - p. 233 35. References - p. 233 36. Chapter 16 Quaternion protein frame maps - p. 235 37. Part 5 Spherical geometry and quaternions - p. 271 38. Chapter 17 Euclidean center of mass and barycentric coordinates - p. 273 39. Chapter 18 Quaternion averaging and the barycenter - p. 281 40. Chapter 19 The quaternion barycentric coordinate problem - p. 289 41. Keywords - p. 289 42. 19.1 Introduction - p. 289 43. 19.2 Review of spherical trigonometry - p. 291 44. 19.3 Spherical triangles as coordinates - p. 296 45. 19.4 Brown–Worsey no-go theorem - p. 299 46. 19.5 Möbius dual coordinates for points in a spherical simplex - p. 301 47. 19.6 Quaternion barycentric coordinates - p. 309 48. 19.7 Remark on intractability of spherical volume ratios for N>2 - p. 310 49. References - p. 311 50. Part 6 Extending quaternions to 4D and SE(3) - p. 313 51. References - p. 314 52. Chapter 20 The 4D quaternion-based coordinate and orientation frame alignment problems - p. 315 53. Chapter 21 Dual quaternions and SE(3) - p. 331 54. Part 7 Quaternion applications in physics - p. 345 55. Chapter 22 Quantum computing and quaternion qubits - p. 347 56. Chapter 23 Introduction to quaternions and special relativity - p. 357 57. Chapter 24 Complex quaternions and special relativity - p. 371 58. Keywords - p. 371 59. 24.1 Introducing 3 space and 1 time - p. 371 60. 24.2 Exploring complexified quaternions - p. 372 61. 24.3 Relativity from SL(2,C) and complexified quaternions - p. 375 62. 24.4 Four-vector Pauli matrices and the spacetime Lorentz group - p. 379 63. 24.5 Miscellaneous properties - p. 384 64. 24.6 Summary of (3 + 1)-dimensional relativity and complexified quaternions - p. 387 65. References - p. 387 66. Part 8 Quaternion discrete symmetries and gravitons - p. 389 67. References - p. 390 68. Chapter 25 Geometry of the ADE symmetries of the 2-sphere - p. 391 69. Chapter 26 The discrete ADE rotation groups and their quaternions - p. 403 70. Chapter 27 The quaternion presentation form of the discrete ADE groups - p. 429 71. Chapter 28 The Klein invariants of the ADE discrete quaternion groups - p. 447 72. Chapter 29 Quaternion-based embedding and visualization of the A1 gravitational instanton - p. 475 73. Chapter 30 Conclusion - p. 487 74. Appendix A Notation - p. 489 75. Appendix B Quaternion methods - p. 497 76. Appendix C Eigensystem of the rotation matrix - p. 503 77. Appendix D Lie algebra origins of the rotation matrix - p. 505 78. Appendix E Derivation of rotation group representations - p. 507 79. Appendix F Quaternion from double 3D Clifford reflection - p. 511 80. Appendix G Example code for rotation to quaternion algorithm - p. 515 81. Appendix H Selected details of the 3D perspective solution - p. 519 82. Appendix I Exact cloud matching rotations in any dimension - p. 525 83. Appendix J Details of the Cardano fourth-order eigenvalue solution - p. 529 84. Appendix K A quaternion context for traditional Ramachandran protein plots - p. 539 85. Appendix L Determinants and the Levi-Civita symbol - p. 547 86. Appendix M The dual basis - p. 553 87. Appendix N Equivalence of SVD and quaternion eigensystem methods - p. 559 88. References - p. 565 89. Index - p. 575
Visualizing More Quaternions, Volume Two updates on proteomics-related material that will be useful for biochemists and biophysicists, including material related to electron microscopy (and specifically cryo-EVisualizing. Dr. Andrew J. Hanson’s groundbreaking book updates and extends concepts that have evolved since the first book published in 2005, adding entirely new insights that Dr. Hanson’s research has recently developed. This includes the applications of quaternion methods to proteomics and molecular crystallography problems, which are domains with significant current research and application activity.
In addition to readers interested in quaternions for their own sake, scientists involved in computer graphics, animation, shape modeling, and scientific visualization, and readers from several other disciplines will benefit from this new volume. Foremost among these, and the target of the first several chapters, are scientists involved in molecular chemistry where techniques based on quaternion eigensystems have become a standard tool for evaluating the quality of shape matching.
Explores quaternion based approaches to the matching of point cloud pairs, including approaches to data from orthographic and perspective projections.
Develops extensive applications of quaternion frames to protein orientation analysis.
Analyzes the application of quaternion methods to physics problems ranging from quantum computing to special relativity and gravitational instantons.
1. Chapter 1 Introduction - p. 3 2. Chapter 2 Basic quaternion formulas - p. 9 3. Chapter 3 What are quaternions? - p. 15 4. Chapter 4 What is quaternion visualization? - p. 31 5. Chapter 5 Interacting with quaternions as hyperspheres - p. 49 6. Chapter 6 Quaternions and uniform rotation distributions - p. 57 7. Part 2 Orientation matching - p. 69 8. References - p. 70 9. Chapter 7 Introduction: 2D cloud alignment problem - p. 71 10. Chapter 8 The 3D quaternion-based alignment problem - p. 77 11. Chapter 9 Quaternion of a 3D rotation from the Bar-Itzhack method - p. 95 12. Chapter 10 The 3D quaternion-based frame alignment problem - p. 103 13. Chapter 11 The combined point-frame alignment problem - p. 125 14. Part 3 Quaternion adjugate methods for pose estimation - p. 133 15. References - p. 133 16. Chapter 12 Exploring the quaternion adjugate variables - p. 135 17. Keywords - p. 135 18. 12.1 Overview of the quaternion extraction problem - p. 135 19. 12.2 Fundamental background - p. 136 20. 12.3 2D rotations and the quaternion map - p. 137 21. 12.4 3D rotations and the quaternion map - p. 147 22. References - p. 157 23. Chapter 13 Quaternion-related machine learning - p. 159 24. Chapter 14 Exact 2D rotations for cloud matching and pose estimation tasks - p. 175 25. Chapter 15 Exact 3D rotations for cloud matching and pose estimation tasks - p. 197 26. Keywords - p. 197 27. 15.1 Basics - p. 199 28. 15.2 3D point-cloud matching - p. 199 29. 15.3 Direct solution of the error-free 3D match problem - p. 202 30. 15.4 3D to 2D orthographic pose estimation - p. 208 31. 15.5 3D pose estimation with perspective projection - p. 217 32. 15.6 Remarks on handling data with measurement errors - p. 228 33. References - p. 231 34. Part 4 Quaternion proteomics - p. 233 35. References - p. 233 36. Chapter 16 Quaternion protein frame maps - p. 235 37. Part 5 Spherical geometry and quaternions - p. 271 38. Chapter 17 Euclidean center of mass and barycentric coordinates - p. 273 39. Chapter 18 Quaternion averaging and the barycenter - p. 281 40. Chapter 19 The quaternion barycentric coordinate problem - p. 289 41. Keywords - p. 289 42. 19.1 Introduction - p. 289 43. 19.2 Review of spherical trigonometry - p. 291 44. 19.3 Spherical triangles as coordinates - p. 296 45. 19.4 Brown–Worsey no-go theorem - p. 299 46. 19.5 Möbius dual coordinates for points in a spherical simplex - p. 301 47. 19.6 Quaternion barycentric coordinates - p. 309 48. 19.7 Remark on intractability of spherical volume ratios for N>2 - p. 310 49. References - p. 311 50. Part 6 Extending quaternions to 4D and SE(3) - p. 313 51. References - p. 314 52. Chapter 20 The 4D quaternion-based coordinate and orientation frame alignment problems - p. 315 53. Chapter 21 Dual quaternions and SE(3) - p. 331 54. Part 7 Quaternion applications in physics - p. 345 55. Chapter 22 Quantum computing and quaternion qubits - p. 347 56. Chapter 23 Introduction to quaternions and special relativity - p. 357 57. Chapter 24 Complex quaternions and special relativity - p. 371 58. Keywords - p. 371 59. 24.1 Introducing 3 space and 1 time - p. 371 60. 24.2 Exploring complexified quaternions - p. 372 61. 24.3 Relativity from SL(2,C) and complexified quaternions - p. 375 62. 24.4 Four-vector Pauli matrices and the spacetime Lorentz group - p. 379 63. 24.5 Miscellaneous properties - p. 384 64. 24.6 Summary of (3 + 1)-dimensional relativity and complexified quaternions - p. 387 65. References - p. 387 66. Part 8 Quaternion discrete symmetries and gravitons - p. 389 67. References - p. 390 68. Chapter 25 Geometry of the ADE symmetries of the 2-sphere - p. 391 69. Chapter 26 The discrete ADE rotation groups and their quaternions - p. 403 70. Chapter 27 The quaternion presentation form of the discrete ADE groups - p. 429 71. Chapter 28 The Klein invariants of the ADE discrete quaternion groups - p. 447 72. Chapter 29 Quaternion-based embedding and visualization of the A1 gravitational instanton - p. 475 73. Chapter 30 Conclusion - p. 487 74. Appendix A Notation - p. 489 75. Appendix B Quaternion methods - p. 497 76. Appendix C Eigensystem of the rotation matrix - p. 503 77. Appendix D Lie algebra origins of the rotation matrix - p. 505 78. Appendix E Derivation of rotation group representations - p. 507 79. Appendix F Quaternion from double 3D Clifford reflection - p. 511 80. Appendix G Example code for rotation to quaternion algorithm - p. 515 81. Appendix H Selected details of the 3D perspective solution - p. 519 82. Appendix I Exact cloud matching rotations in any dimension - p. 525 83. Appendix J Details of the Cardano fourth-order eigenvalue solution - p. 529 84. Appendix K A quaternion context for traditional Ramachandran protein plots - p. 539 85. Appendix L Determinants and the Levi-Civita symbol - p. 547 86. Appendix M The dual basis - p. 553 87. Appendix N Equivalence of SVD and quaternion eigensystem methods - p. 559 88. References - p. 565 89. Index - p. 575
