  We study geometry of confocal quadrics in pseudo-Euclidean spaces of an
arbitrary dimension $d$ and any signature, and related billiard dynamics. The
goal is to give a complete description of periodic billiard trajectories within
ellipsoids. The novelty of our approach is based on introduction of a new
discrete combinatorial-geometric structure associated to a confocal pencil of
quadrics, a colouring in $d$ colours, by which we decompose quadrics of $d+1$
geometric types of a pencil into new relativistic quadrics of $d$ relativistic
types. Deep insight of related geometry and combinatorics comes from our study
of what we call discriminat sets of tropical lines $\Sigma^+$ and $\Sigma^-$
and their singularities. All of that enable usto get an analytic criterion
describing all periodic billiard trajectories, including the light-like ones as
those of a special interest.
