JGSP 30 (2013) 19–48
VECTOR PARAMETERS IN CLASSICAL HYPERBOLIC GEOMETRY
DANAIL BREZOV, CLEMENTINA MLADENOVA AND IVAÏLO MLADENOV
Presented by Ivaïlo M. Mladenov
Abstract. Here we use an extension of Rodrigues’ vector parameter construction
for pseudo-rotations in order to obtain explicit formulae for the generalized Euler
decomposition with arbitrary axes for the structure groups in the classical models
of hyperbolic geometry. Although the construction is projected from the universal
cover SU(1, 1) ' SL(2, R), most attention is paid to the 2 + 1 Minkowski space
model, following the close analogy with the Euclidean case, and various decompositions of the restricted Lorentz group SO+ (2, 1) are investigated in detail. At
the end we propose some possible applications in special relativity and scattering
theory.
Contents
1
Introduction
21
2
Projective Construction
2.1 The Poincaré Disk and Pseudo-quaternions . . . . . . . . . . . . . . . .
22
22
2.2
2.3
2.4
The Upper Half Plane Model . . . . . . . . . . . . . . . . . . . . . . . .
Rodrigues’ Formula and Vector Parametrization . . . . . . . . . . . . . .
Composition Law for the Vector Parameters . . . . . . . . . . . . . . . .
23
25
27
Generalized Euler Decomposition
3.1 The Case of Three Axes . . . . . . . . . . . . . . . . . . . . . . . . . .
29
29
3.2
3.3
The Case of Two Axes . . . . . . . . . . . . . . . . . . . . . . . . . . .
Degenerate Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
34
3.4
3.5
The Trivial Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Divergencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
35
3
4
Null Vectors and Invariant Axes
37
4.1
4.2
Euler’s Fixed Axis Theorem . . . . . . . . . . . . . . . . . . . . . . . .
The Parabolic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
39
4.3
The Embedding SO(2) ⊂ SO+ (2, 1) . . . . . . . . . . . . . . . . . . . .
41
doi: 10.7546/jgsp-30-2013-19-48
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