  Given a group of kinematical symmetry generators, one can construct a
compatible noncommutative spacetime and deformed phase space by means of
projective geometry. This was the main idea behind the very first model of
noncommutative spacetime, proposed by H.S. Snyder in 1947. In this framework,
spacetime coordinates are the translation generators over a manifold that is
symmetric under the required generators, while momenta are projective
coordinates on such a manifold. In these proceedings we review the construction
of Euclidean and Lorentzian noncommutative Snyder spaces and investigate the
freedom left by this construction in the choice of the physical momenta,
because of different available choices of projective coordinates. In
particular, we derive a quasi-canonical structure for both the Euclidean and
Lorentzian Snyder noncommutative models such that their phase space algebra is
diagonal although no longer quadratic.
