On generalisations of the continued fraction

In Introduction we discuss the history of the continued fraction and of its generalisations. In § 1 we compare the geometric interpretations of the continued fraction given by Klein, by Voronoi and by author, and define the convex continued fraction. In § 2 we propose an algorithm of computation of the convex continued fraction. In § 3 we compare the geometric interpretations of the multidimensional generalisations of the continued fraction given by Klein, by Voronoi and by author (see preprint no. 86/2003). In § 4 we propose an algorithm of computation of a generalisation of the covex continued fraction. In § 5 we compare points of Klein, of Voronoi and of the author in two-dimensional and three-dimensional cases.