Here we derive analytic expressions for the scalar parameters which appear in the generalized Euler decomposition of the rotational matrices in R 3. The axes of rotations in the decomposition are almost arbitrary and they need only to obey a simple condition to guarantee that the problem is well posed. A special attention is given to the case when the rotation is decomposable using only two rotations and for this case quite elegant expressions for the parameters were derived. In certain cases one encounters infinite parameters due to the rotations by an angle π (the so called half turns). We utilize both geometric and algebraic methods to obtain those conditions that can be used to predict and deal with various configurations of that kind and then, applying l'Hôpital's rule, we easily obtain the solutions in terms of linear fractional functions. The results are summarized in two Tables and a flowchart presenting in full details the procedure. Contents 1 Introduction 60 2 The Generic Case 64 3 The Symmetric Case 69 4 Decomposition Into Two Rotations 71

The paper discusses manipulator modelling and control through a nonstandard parametrization of rotation motions. The advantage of the method is the computational facilities arising at the kinematical level, not from efficient presentation, but from fundamental topological considerations of the configurational manifold provided with a Lie group structure.

We use a vector parameter description of the Lorentz groups in R 2,1 and R 3,1 to obtain an exact expression for the Thomas factor as a geometric phase. The effect of phase accumulation in Thomas-Wigner precession phenomena is seen as a manifestation of the hyperbolic solid angle theorem. On the infinitesimal level, our description involves affine connections on the noncompact Hopf fibrations U(1) → SU(1, 1) → Δ and SU(2) → PSL(2, C) → H 3 . The associated gauge field is a restriction of the familiar Yang-Mills anti-instanton. We also consider the dual compact case, and we discuss generalizations to arbitrary dimensions and applications in various branches of theoretical physics.