# Vector Parameters in Classical Hyperbolic Geometry

@inproceedings{Brezov2014VectorPI,
title={Vector Parameters in Classical Hyperbolic Geometry},
year={2014}
}
• Published 2014
• Mathematics
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…
14 Citations
• Mathematics
• 2015
In this paper we obtain the Lie derivatives of the scalar parameters in the generalized Euler decomposition with respect to arbitrary axes under left and right deck transformations. This problem can
• Physics
• 2013
We use a vector parameter technique to obtain the generalized Euler decompositions with respect to arbitrarily chosen axes for the three-dimensional special orthogonal group SO(3) and the
• Mathematics
• 2014
Here we present the generalized Euler decompositions of the six-dimensional Lie groups SO(4), SO*(4) and SO(2,2) using their (local) direct product structure [1] and a technique we have developed for
• Mathematics
• 2017
. The paper provides a geometric perspective on the inclusion map sl 2 ∼ = so 3 ⊂ so n realized via the Pl¨ucker embedding. It relates higher-dimensional representations of the complex Lie group SO 3
• Mathematics
• 2014
We propose a new method for decomposing SO(3) transformations into pairs of successive rotations (about orthogonal axes) and consider some applications in rigid body kinematics and navigation. The
• Physics
• 2018
We use both vector-parameter and quaternion techniques to provide a thorough description of several classes of rotations, starting with coaxial angular velocity of varying magnitude. Then, we fix the
• Mathematics
• 2013
We exploit a new technique for obtaining the generalized Euler decomposition of three-dimensional rotations [1], based on the vector parameter (also known as Rodrigues' or Gibbs's vector)
• D. Brezov
• Mathematics
• 2019
The paper provides a consistent study on the projective construction of low-dimensional motion groups starting with $$\mathrm {SO}(3)$$SO(3) and then gradually extending to the Galilean and
• Mathematics
• 2015
The Cayley maps for the Lie algebras su(1, 1) and so(2, 1) converting them into the corresponding Lie groups SU(1, 1) and SO(2, 1) along their natural vector-parameterizations are examined. Using the
• Mathematics
Royal Society Open Science
• 2013
This work gives the number Nm^,n^(U) as a function of U, the least value of a positive integer k such that U can be decomposed into a product of k rotations about either m^ or n^.

## References

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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
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
• Physics
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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
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
• Mathematics
Am. Math. Mon.
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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.
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
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
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