This article, continuation of previous works, presents the applications of geometric optimal control theory to the analysis of the Earth re-entry problem for a space shuttle where the control is the angle of bank, the cost is the total amount of thermal flux, and the system is subject to state constraints on the thermal flux, the normal acceleration and the dynamic pressure. Our analysis is based on the evaluation of the reachable set using the maximum principle and direct computations with the boundary conditions according to the CNES research project\footnote{The project is partially supported by the Centre National d'Etude Spatiales.}. The optimal solution is approximated by a concatenation of bang and boundary arcs, and is numerically computed with a multiple-shooting method.