The continuity of the metric projection onto an approximately compact set in a uniformly convex and uniformly smooth Banach space is investigated. The concept of directional radius of curvature at a point is defined: this allows the author to obtain an explicit modulus of continuity for the metric projection onto some sets (which are not necessarily convex). For the case of subspaces, these estimates improve upon some of the results given in a paper by \textit{B. O. Bjornestal} [quoted here as a ''preprint'', but published in the book ''Approximation theory'', Banach Center Publ., 4, PWN, Warsaw, 43-53 (1979)].