  We use the results of Moser and Kn\"orrer on relations between geodesics on
quadrics and solutions of the classical Neumann system to describe explicitly
the geodesic scattering on hyperboloids.
  We explain the relation of Kn\"orrer's reparametrisation with projectively
equivalent metrics on quadrics introduced by Tabachnikov and independently by
Matveev and Topalov, giving a new proof of their result. We show that the
projectively equivalent metric is regular on the projective closure of
hyperboloids and extend Kn\"orrer's map to this closure.
