Permutation asymmetry of the relativistic velocity addition law and non-Euclidean geometry

The asymmetry of the relativistic addition law for noncollinear velocities under the velocity permutation leads to two modified triangles on a Euclidean plane depicting the addition of unpermuted and permuted velocities and the appearance of a nonzero angle $\omega$ between two resulting velocities. A particle spin rotates through the same angle $\omega$ under a Lorentz boost with a velocity noncollinear to the particle velocity. Three mutually connected three-parameter representations of the angle $\omega$, obtained by the author earlier, express the three-parameter symmetry of the sides and angles of two Euclidean triangles identical to the sine and cosine theorems for the sides and angles of a single geodesic triangle on the surface of a pseudosphere. Namely, all three representations of the angle $\omega$, after a transformation of one of them, coincide with the representations of the area of a pseudospherical triangle expressed in terms of any two of its sides and the angle between them. The angle $\omega$ is also symmetrically expressed in terms of three angles or three sides of a geodesic triangle, and therefore it is an invariant of the group of triangle motions over the pseudo-sphere surface, the group that includes the Lorentz group. Although the pseudospheres in Euclidean and pseudo-Euclidean spaces are locally isometric, only the latter is isometric to the entire Lobachevsky plane and forms a homogeneous isotropic curved 4-velocity space in the flat Minkowski space. In this connection, relativistic physical processes that may be related to the pseudosphere in Euclidean space are especially interesting.