Visualizing More Quaternions

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 h...

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Detalles Bibliográficos
Autor principal:	Hanson, Andrew (-)
Formato:	Libro electrónico
Idioma:	Inglés
Publicado:	San Diego : Elsevier Science & Technology 2024.
Edición:	1st ed
Materias:	Quaternions.
Ver en Biblioteca Universitat Ramon Llull:	https://discovery.url.edu/permalink/34CSUC_URL/1im36ta/alma991009837635206719

Tabla de Contenidos:

Front Cover
Visualizing More Quaternions
Copyright
Contents
Preface
Biography
1 Review
1 Introduction
1.1 Introductory remarks
1.2 Hamilton's walk
1.3 Philosophical thoughts
2 Basic quaternion formulas
3 What are quaternions?
3.1 Quaternions are extended complex numbers
3.2 Quaternions are two-by-two matrices
3.3 Quaternions are a complexified pair of complex numbers
3.4 Quaternions are the group SU(2)
3.5 Quaternion actions can be realized in complex projective space
3.6 Quaternions are one of the four normed division algebras
3.7 Quaternions are a special case of Clifford algebras
3.8 Quaternions are points on the 3-sphere S3
3.9 Synopsis
4 What is quaternion visualization?
4.1 Fundamental idea: the circle S1
4.2 A richer example: the 2-sphere S2
4.3 Seeing quaternions as points on the 3-sphere S3
4.4 Interactively following a quaternion path of incremental rotations
4.5 Displaying every point of the quaternion 3-sphere S3
4.6 Examples of quaternion visualization strategies
5 Interacting with quaternions as hyperspheres
5.1 The 3D rolling ball
5.2 The 4D rolling ball
5.3 The ND rolling ball
6 Quaternions and uniform rotation distributions
6.1 Points on spheres
6.2 Theoretical basis for uniformity of spherical distributions
6.3 Elements of the uniform random sphere
6.4 Attractive incorrect quaternion sampling
6.5 More uniform quaternions: Gaussian and Shoemake/Diaconis
2 Orientation matching
7 Introduction: 2D cloud alignment problem
7.1 Context of the alignment task
7.2 Remark: some alignment problems are very simple
8 The 3D quaternion-based alignment problem
8.1 Introduction
8.2 Spatial alignment of matched 3D point sets
8.3 Reviewing the 3D spatial alignment RMSD problem.
8.4 Algebraic solution of the eigensystem for 3D spatial alignment
8.5 Conclusion
9 Quaternion of a 3D rotation from the Bar-Itzhack method
9.1 Classic methods for finding the quaternion of a 3D rotation
9.2 The Bar-Itzhack approach
9.3 Optimal quaternions from projection matrix data
9.4 Summary of finding quaternions from rotation data
10 The 3D quaternion-based frame alignment problem
10.1 Introduction to quaternion orientation frames
10.2 Overview of 3D quaternion frame alignment
10.3 Alternative matrix forms of the linear vector chord distance
10.4 Evaluating the 3D orientation frame solution
11 The combined point-frame alignment problem
11.1 Combined optimization measures
11.2 Practical simplification of the composite measure
3 Quaternion adjugate methods for pose estimation
12 Exploring the quaternion adjugate variables
12.1 Overview of the quaternion extraction problem
12.2 Fundamental background
12.3 2D rotations and the quaternion map
12.3.1 Direct solution of the 2D problem
12.3.2 Graphical illustration of the 2D rotation case
12.3.3 Variational approach: the Bar-Itzhack method in 2D
12.3.4 Bar-Itzhack errorful measurement strategy
12.3.5 Summary
12.4 3D rotations and the quaternion map
12.4.1 Direct solution of the 3D problem
12.4.2 Variational approach: Bar-Itzhack in 3D
12.4.3 Bar-Itzhack variational approach to 3D noisy data
13 Quaternion-related machine learning
13.1 Overview
13.2 Reports of quaternion deficiencies for machine learning
13.3 Exercises in machine learning with quaternions
13.3.1 Loss layer data encoding
13.3.2 Network setup and training
13.4 2D rotation training and results
13.5 3D rotation training and results
13.6 Basic elements of applying a network to the cloud matching problem.
13.7 Experiments with training to the cross-covariance matrix E
14 Exact 2D rotations for cloud matching and pose estimation tasks
14.1 2D point-cloud orientation matching
14.2 2D parallel projection pose estimation
14.3 2D perspective projection
15 Exact 3D rotations for cloud matching and pose estimation tasks
15.1 Basics
15.2 3D point-cloud matching
15.3 Direct solution of the error-free 3D match problem
15.3.1 Solving the 3D match least-squares loss function algebraically
15.4 3D to 2D orthographic pose estimation
15.4.1 Least-squares solution to error-free orthographic pose problem
15.4.2 Remarks on variational approaches to solving matching problems
15.5 3D pose estimation with perspective projection
15.5.1 The exact perspective solution
15.6 Remarks on handling data with measurement errors
4 Quaternion proteomics
16 Quaternion protein frame maps
16.1 Quaternion maps of global protein structure
16.2 Related work
16.3 Introduction to quaternion maps
16.3.1 Quaternion orientation frames
16.4 Quaternion distance
16.5 Protein frame geometry
16.5.1 Visualizing quaternions as geometry
16.5.2 Visualizing quaternions as coordinates
16.5.3 Collections of frames via quaternion maps
16.6 Studies in quaternion frame maps
16.6.1 Alpha-helix model: quaternion frames of idealized curves
16.6.2 Beta-sheet model: extreme quaternion frames
16.6.3 Quaternion frames from spline curves of PDB backbones
16.7 Quaternion frames from discrete PDB data
16.8 Example applications of discrete global quaternion frames
16.8.1 Quaternion frames of rigid proteins
16.8.2 Statistical quaternion groupings of NMR data
16.8.3 Enzyme functional structures
16.9 Tools for exploring and comparing quaternion maps
16.9.1 Aids for exploring quaternion maps.
16.9.2 Metric for comparing quaternion maps
16.9.3 Quaternion rings: orientation freedom spaces
16.10 Conclusion
5 Spherical geometry and quaternions
17 Euclidean center of mass and barycentric coordinates
17.1 The 1D simplex: N=1
17.2 Triangular simplex: N=2
17.3 Tetrahedral simplex barycenter: N=3
17.4 The N-simplex barycenter
17.5 Variational approach to the Euclidean center of mass
18 Quaternion averaging and the barycenter
18.1 Introduction
18.2 The geometry of inter-quaternion distances
18.3 Analysis of the quaternion averaging problem
19 The quaternion barycentric coordinate problem
19.1 Introduction
19.2 Review of spherical trigonometry
19.3 Spherical triangles as coordinates
19.4 Brown-Worsey no-go theorem
19.5 Möbius dual coordinates for points in a spherical simplex
19.6 Quaternion barycentric coordinates
19.7 Remark on intractability of spherical volume ratios for N&gt
2
6 Extending quaternions to 4D and SE(3)
20 The 4D quaternion-based coordinate and orientation frame alignment problems
20.1 Foundations of quaternions for 4D problems
20.2 Double-quaternion approach to the 4D RMSD problem
20.2.1 Starting point for the 4D RMSD problem
20.2.2 A tentative 4D eigensystem
20.2.3 Issues with the naive 4D approach
20.2.4 Insights from the singular value decomposition
20.3 4D orientation frame alignment
20.3.1 The 4D orientation frame alignment problem
20.4 Extracting quaternion pairs from 4D rotation matrices
21 Dual quaternions and SE(3)
21.1 Background
21.2 Introduction to dual quaternions
21.3 The fundamental idea of quaternion-like translations
21.4 Refining the dual quaternion to use dual trigonometry
21.5 The motion of a dual quaternion
21.6 Brief discussion of applications
7 Quaternion applications in physics.
22 Quantum computing and quaternion qubits
22.1 Context of quantum computing
22.2 The qubit
22.3 The Bloch sphere
22.4 Quaternions as quantum computing operators
22.5 Higher-dimensional qubit geometry
22.6 Conclusion
23 Introduction to quaternions and special relativity
23.1 Rotations in 2D tell us a lot about 2D relativity
23.2 Relativity with one time and two space dimensions
23.3 Sequential noncommuting transformations
23.4 Quaternion framework for 2 + 1 relativity
23.5 Features of light in lower dimensions of spacetime
23.6 Some simple 2 + 1 relativistic camera effects
23.7 What is next?
24 Complex quaternions and special relativity
24.1 Introducing 3 space and 1 time
24.2 Exploring complexified quaternions
24.3 Relativity from SL(2,C) and complexified quaternions
24.4 Four-vector Pauli matrices and the spacetime Lorentz group
24.5 Miscellaneous properties
24.6 Summary of (3 + 1)-dimensional relativity and complexified quaternions
8 Quaternion discrete symmetries and gravitons
25 Geometry of the ADE symmetries of the 2-sphere
25.1 Introduction to discrete groups on the sphere
25.2 Coordinates of the Ak cyclic geometry
25.3 Coordinates of the Dk dihedral geometry
25.4 Coordinates of the E6 tetrahedral geometry
25.5 Coordinates of the E7 octahedral geometry
25.6 Coordinates of the E8 icosahedral geometry
26 The discrete ADE rotation groups and their quaternions
26.1 Introduction
26.2 Actions of the SO(3) ADE rotations in Euclidean 3D space
26.3 Identifying the 3D symmetric ADE transformations
26.4 The quaternion actions of the Klein groups
26.4.1 The Ak cyclic quaternion group
26.4.2 The Dk dihedral quaternion group
26.4.3 The E6 tetrahedral quaternion group
26.4.4 The E7 octahedral quaternion group.
26.4.5 The E8 icosahedral quaternion group.

Visualizing More Quaternions

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