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Let \({\mathbb {D}}\) be the unit disc in \({\mathbb {C}}\) and let \(f:{\mathbb {D}} \rightarrow {\mathbb {C}}\) 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 \({\mathbb {D}}\).

A Characterization of Orthogonal Convergence in Simply Connected Domains