  Given a negatively curved geodesic metric space M, we study the asymptotic
penetration behaviour of geodesic lines of M in small neighbourhoods of closed
geodesics and of other compact convex subsets of M. We define a spiraling
spectrum which gives precise information on the asymptotic spiraling lengths of
geodesic lines around these objects. We prove analogs of the theorems of
Dirichlet, Hall and Cusick in this context. As a consequence, we obtain
Diophantine approximation results of real numbers, complex numbers, or elements
of the Heisenberg group by irrational quadratic ones.
