  Through the use of sub-Riemannian metrics we provide quantitative estimates
for the maximal tight neighbourhood of a Reeb orbit on a three-dimensional
contact manifold. Under appropriate geometric conditions we show how to
construct closed curves which are boundaries of overtwisted disks. We introduce
the concept of contact Jacobi curve, and prove sharp lower bounds of the
so-called tightness radius (from a Reeb orbit) in terms of Schwarzian
derivative bounds. We also prove similar, but non-sharp, comparison theorems in
terms of sub-Riemannian canonical curvature bounds. We apply our results to
K-contact sub-Riemannian manifolds. In this setting, we prove a contact
analogue of the celebrated Cartan-Hadamard theorem.
