Vector Parameters in Classical Hyperbolic Geometry

@inproceedings{Brezov2014VectorPI,
  title={Vector Parameters in Classical Hyperbolic Geometry},
  author={Danail S. Brezov and Clementina D. Mladenova and Iva{\"i}lo M. Mladenov},
  year={2014}
}
Presented by Ivailo 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… 

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References

SHOWING 1-10 OF 35 REFERENCES

Vector Decompositions of Rotations

Presented by Ivaïlo M. Mladenov Abstract. Here we derive analytic expressions for the scalar parameters which appear in the generalized Euler decomposition of the rotational matrices in R. The axes

Relations Among Low-Dimensional Simple Lie Groups

Communicated by Gregory L. Naber Abstract. The compact classical Lie groups can be regarded as groups of n × n matrices over the real, complex, and quaternion fields R, C, and Q that satisfy

Wigner rotations, Bargmann invariants and geometric phases

The concept of the 'Wigner rotation', familiar from the composition law of (pure) Lorentz transformations, is described in the general setting of Lie group coset spaces and the properties of coset

On the possibility of application of one system of hypercomplex numbers in inertial navigation

A hypercomplex number system (generalized quaternions) is defined to extend the concept of Hamilton's and Clifford's biquaternions. Characteristic properties of numbers of this system, as well as

A Disorienting Look at Euler's Theorem on the Axis of a Rotation

In modern terms, Euler's result stating that in three dimensions every rotation of a sphere about its center has an axis is formulated in terms of rotation matrices as follows.

VECTOR DECOMPOSITION OF FINITE ROTATIONS

Geometry of the Saxon-Hutner theorem

The Wigner angle as an anholonomy in rapidity space

A product of two non-collinear boosts (i.e., pure Lorentz transformations) can be written as the product of a boost and a rotation, the angle of rotation being known as Wigner’s angle. This paper

Analytic Hyperbolic Geometry: Mathematical Foundations And Applications

This is the first book on analytic hyperbolic geometry, fully analogous to analytic Euclidean geometry. Analytic hyperbolic geometry regulates relativistic mechanics just as analytic Euclidean

The Proper-Time Lorentz Group Demystified

Abstract. Proper velocities are measured by proper time as opposed to coordinate velocities, which are measured by coordinate time. The standard Lorentz transformation group, in which each