# 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…

## 14 Citations

### Variations of (pseudo-)rotations and the Laplace-Beltrami operator on homogeneous spaces

- 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…

### Some new results on three-dimensional rotations and pseudo-rotations

- 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…

### Factorizations in Special Relativity and Quantum Scattering on the Line

- Mathematics
- 2016

We extend an old result due to Pina on three-dimensional rotations to the hyperbolic case and utilize it to construct a specific factorization scheme for the isometries in \(\mathbb {R}^{2,1}\!\).…

### Generalized Euler decompositions of some six-dimensional Lie groups

- 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…

### Higher-Dimensional Representations of SL 2 and its Real Forms Via Pl¨ucker Embedding

- 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…

### Two-axes Decompositions of (Pseudo-)Rotations and Some of Their Applications

- 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…

### From the Kinematics of Precession Motion to Generalized Rabi Cycles

- 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…

### New perspective on the gimbal lock problem

- 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)…

### VECTOR-PARAMETER FORMS OF SU ( 1 , 1 ) , SL ( 2 , R ) AND THEIR CONNECTION TO SO

- 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…

### The minimum number of rotations about two axes for constructing an arbitrarily fixed rotation

- MathematicsRoyal 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

SHOWING 1-10 OF 35 REFERENCES

### Vector Decompositions of Rotations

- Mathematics
- 2012

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

- Mathematics
- 2012

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

- Physics
- 2003

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

- Physics
- 2002

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

- MathematicsAm. Math. Mon.
- 2009

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.

### The Wigner angle as an anholonomy in rapidity space

- Physics
- 1997

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

- Mathematics
- 2005

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

- Physics
- 2005

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…