  We consider the question of how well points in a quadric hypersurface
$M\subset\mathbb R^d$ can be approximated by rational points of $\mathbb
Q^d\cap M$. This contrasts with the more common setup of approximating points
in a manifold by all rational points in $\mathbb Q^d$. We provide complete
answers to major questions of Diophantine approximation in this context. Of
particular interest are the impact of the real and rational ranks of the
defining quadratic form, quantities whose roles in Diophantine approximation
have never been previously elucidated. Our methods include a correspondence
between the intrinsic Diophantine approximation theory on a rational quadric
hypersurface and the dynamics of the group of projective transformations which
preserve that hypersurface, similar to earlier results in the non-intrinsic
setting due to Dani ('86) and Kleinbock--Margulis ('99).
