A characterization of orthogonal convergence in simply connected domains

Let $mathbb D$ be the unit disc in $mathbb C$ and let $f:mathbb D to mathbb C$ be a Riemann map, $Delta=f(mathbb D)$. We give a necessary and sufficient condition in terms of hyperbolic distance and horocycles which assures that a compactly divergent sequence ${z_n}subset Delta$ has the property that ${f^{-1}(z_n)}$ converges orthogonally to a point of $partial mathbb D$. We also give some applications of this to the slope problem for continuous semigroups of holomorphic self-maps of $mathbb D$.