  The effect of the linear-fractional transformations on the parallel lines in
the spacetime has been studied. Fock-Lorentz transformations maps a line to a
line, from which one can obtain the combinations rule for the velocities in the
Fock-Lorentz transformations. Rigidity is defined as a consequences of holding
parallelism under the transformations. The Fock-Lorentz transformations do not
preserve rigidity, which leads to some novel results such as growing distances
alongside with advancing time. Also, it is shown that the time coordinates of
events will come closer to each other in the transformed coordinates by going
back in time
