Lecture 1: Introduction to Clifford Algebras Pertti Lounesto 1.1 Introduction • 1.2 Clifford algebra of the Euclidean plane • 1.3 Quaternions • 1.4 Clifford algebra of the Euclidean space 3 • 1.5 The electron spin in a magnetic field • 1.6 From column spinors to spinor operators • 1.7 In 4D: Clifford algebra Cℓ 4 of 4 • 1.8 Clifford algebra of Minkowski spacetime • 1.9 The exterior algebra and contractions • 1.10 The Grassmann-Cayley algebra and shuffle products • 1.11 Alternative definitions of the Clifford algebra • 1.12 References Lecture 2: Mathematical Structure of Clifford Algebras Ian Porteous 2.1 Clifford algebras • 2.2 Conjugation • 2.3 References Lecture 3: Clifford Analysis John Ryan 3.1 Introduction • 3.2 Foundations of Clifford analysis • 3.3 Other types of Clifford holomorphic functions • 3.4 The equation D k ƒ = 0 • 3.5 Conformal groups and Clifford analysis • 3.6 Conformally flat spin manifolds • 3.7 Boundary behavior and Hardy spaces • 3.8 More on Clifford analysis on the sphere • 3.9 The Fourier transform and Clifford analysis • 3.10 Complex Clifford analysis • 3.11 References Lecture 4: Applications of Clifford Algebras in Physics William E. Baylis 4.
The Clifford algebra for the group of rigid body motions is described. Linear elements, that is points, lines and planes are identified as homogeneous elements in the algebra. In each case the action of the group of rigid motions on the linear elements is found. The relationships between these linear elements are found in terms of operations in the algebra. That is, incidence relations, the conditions for a point to lie on a line for example are found. Distance relations, like the distance between a point and a plane are found. Also the meet and join of linear elements, for example, the line determined by two planes and the plane defined by a line and a point, are found. Finally three examples of the use of the algebra are given: a computer graphics problem on the visibility of the apparent crossing of a pair of lines, an assembly problem concerning a double peg-in-hole assembly, and a problem from computer vision on finding epipolar lines in a stereo vision system.
The inertia matrix of any rigid body is the same as the inertia matrix of some system of four pointmasses. In this work, the possible disposition of these point-masses is investigated. It is found that every system of possible point-masses with the same inertia matrix can be parameterised by the elements of the orthogonal group in four-dimensional modulo-permutation of the points. It is shown that given a fixed inertia matrix, it is possible to find a system of point-masses with the same inertia matrix but where one of the points is located at some arbitrary point. It is also possible to place two point-masses on an arbitrary line or three of the points on an arbitrary plane. The possibility of placing some of the point-masses at infinity is also investigated. Applications of these ideas to rigid body dynamics are considered. The equation of motion for a rigid body is derived in terms of a system of four point-masses. These turn out to be very simple when written in a 6-vector notation.
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