Effective length-projection bounds for hyperbolic 3-manifolds diffeomorphic to $S\times \mathbb{R}$

We give a formula with explicit constants relating the subsurface projection $d_Y({\nu }^{-},{\nu }^{+})$ of the end invariants ${\nu }^{-},{\nu }^{+}$ of a hyperbolic 3-manifold $Q$ diffeomorphic to $S\times \mathbb{R}$ and the length of the geodesic representative in $Q$ of the multicurve $\mathrm{\partial }Y$. This makes effective and computable the large projections versus short curves relation proved by Minsky. We give an application to closed hyperbolic 3-manifolds fibering over the circle providing a geometric analog of the uniform projection bound in fibered faces of Minsky and Taylor.