Rotations and Lorentz transformations are compared geometrically and algebraically for two-, three-, and four-dimensional space and space-time. The close analogy between the two types of transformation is revealed by means of orthogonal and pseudo-orthogonal matrices, complex numbers and double numbers, and quaternions and tetrons. The most natural and elegant forms are shown to be the complex and hypercomplex number representations. The relationship is essentially based on the direct correspondence between the circular functions sinθ and cosθ, and the hyperbolic functions sinhφ and coshφ.
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References
1.
BallR S, 1900A Treatise on the Theory of Screws (Cambridge University Press, London)
2.
BlumenthalL MMengerK, 1970Studies in Geometry (Freeman, San Francisco)
ChaslesM, 1830“Note sur les propriétés générales du système de deux corps semblables entr'eux et placés d'une manière quelconque dans l'espace; et sur le déplacement fini ou infiniment petit d'un corps solide libre”Férussac, Bulletin des Sciences Mathématiques14321–326
5.
CliffordW K, 1876“On the classification of geometric algebras” published in 1882 as paper 43 of Mathematical Papers Ed. TuckerR, (Macmillan, London)
6.
Du ValP, 1964Homographies, Quaternions and Rotations (Clarendon Press, London)
7.
FeynmanR P, 1962The Theory of Fundamental Processes (Benjamin, New York)
8.
ForderH G, 1960The Calculus of Extension (Chelsea, New York)
RooneyJ, 1977“A survey of representations of spatial rotation about a fixed point”Environment and Planning B4185–210
19.
RooneyJ, 1978a“A comparison of representations of general spatial screw displacement”Environment and Planning B545–88
20.
RooneyJ, 1978b“On the three types of complex number and planar transformations”Environment and Planning B589–99
21.
SchaferR D, 1966An Introduction to Nonassociative Algebras (Academic Press, London)
22.
SempleJ GRothL, 1949An Introduction to Algebraic Geometry (Clarendon Press, London)
23.
SyngeJ L, 1956Relativity: The Special Theory (North-Holland, Amsterdam)
24.
SyngeJ L, 1972“Quaternions, Lorentz transformations, and the Conway-Dirac-Eddington matrices”Communications of the Dublin Institute for Advanced Studies, Series Anumber 21
25.
SyngeJ LGriffithB A, 1959Principles of Mechanics (McGraw-Hill, New York)
26.
TinkhamM, 1964Group Theory and Quantum Mechanics (McGraw-Hill, New York)
27.
YaglomI M, 1968Complex Numbers in Geometry (Academic Press, New York)
