The сorrect generalization of the continued fraction

In a three-dimensional space we consider three gomogeneous linear forms. In another three-dimensional space, where coordinates are modulus of values of these forms, we consider the convex hull of points corresponding to all integers of the first space, exept the origin. The proposed generalization of the continued fraction is a motion along faces of the convex hull. Two examples are given. Thus that gives a solution of an old problem, which was attacked without a success by a lot of mathematicians (Euler, Hermit, Jacobi, Poincare, Minkowski, Klein, Arnold etc.).