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In this paper, we address non-Euclidean geometrical aspects of the Schur and Levinson-Szego algorithms. We first show that the Lobachevski geometry is, by construction, one natural geometrical environment of these algorithms, since they necessarily make use of automorphisms of the unit disk. We next consider the algorithms in the particular context of their application to linear prediction. Then the Schur (resp., Levinson-Szego) algorithm receives a direct (resp., polar) spherical trigonometry (ST) interpretation, which is a new feature of the classical duality of both algorithms.