Orientations and Rotations

by Morawiec, Adam

• ISBN13: 9783540407348
• ISBN10: 3540407340
• Format: Hardcover
• Copyright: 2004-02-01
• Publisher: Springer Verlag

Summary

The book is a comprehensive account of orientations and rotations of three-dimensional objects. The author's interest in orientations stems from the field of crystallographic texture analysis ' a branch of knowledge concerned with orientations of crystallites constituting polycrystalline materials. Essentially, Orientations and Rotations treats the mathematical and computational foundations of texture analysis. It contains an extensive and thorough introduction to parameterizations and geometry of the rotation space. Since the notions of orientations and rotations are of primary importance for science and engineering, the book can be useful for a very broad audience using rotations in other fields (robotic manipulators, spacecraft control, computer animation, limb or eye kinematics, etc.).

Table of Contents

Preliminaries	1	(18)
Rotations as Displacements	1	(4)
Composition of Rotations	5	(2)
Improper Rotations	7	(1)
Matrix Representation	8	(2)
Formal Approach to Orthogonal Transformations	10	(9)
Parameterizations	19	(32)
Half-turns	19	(1)
Cayley Transformation and Rodrigues Parameters	20	(3)
Axis and Angle Parameters	23	(4)
Euler Angles	27	(3)
Cayley--Klein Parameters	30	(5)
Quaternions	35	(6)
Rotation Vector	41	(3)
Rotation Matrix in Non--Cartesian Coordinate Systems	44	(2)
Miller Indices	46	(3)
Computational Properties of Parameterizations	49	(2)
Geometry of the Rotation Space	51	(22)
SO(3) as a Riemannian Manifold	52	(9)
Exponential Mapping	61	(3)
SO(3) as a Lie Group	64	(6)
Integration on SO(3)	70	(3)
More on Small Orientation Changes	73	(8)
Vector of Infinitesimal Rotation	73	(3)
Rotation Rate Field and Continuity Equation	76	(1)
Short Excursion into Mechanics	77	(4)
Some Statistical Issues	81	(12)
Mean Orientation	81	(5)
Distributions on the Rotation Manifold	86	(4)
Generation of Orientations	90	(2)
Comparing Smooth Orientation Functions	92	(1)
Symmetry	93	(22)
Finite Point Groups	93	(2)
Crystallographic Point Groups	95	(2)
Asymmetric Domains	97	(7)
Asymmetric Domains in Rodrigues Space	104	(11)
Misorientation Angle and Axis Distributions	115	(14)
Misorientation Angle Distributions	116	(8)
Distributions of Rotation Axes	124	(5)
Crystalline Interfaces and Symmetry	129	(6)
Symmetrically Equivalent Boundaries	129	(2)
Boundary Distributions	131	(4)
Crystallographic Textures	135	(16)
Texture Components in Cubic--Orthorhombic Case	136	(2)
Rational Orientation Relationships	138	(4)
Coincident Sublattices	142	(9)
Diffraction Geometry	151	(14)
Elementary Relations	152	(1)
Orientations of Individual Crystallites	152	(7)
Orientation Distributions from Pole Figures	159	(6)
Effective Elastic Properties of Polycrystals	165	(24)
Definitions and Simplest Principles	165	(8)
Perturbation Methods	173	(14)
Related Issues	187	(2)
References	189	(8)
Index	197