Further generalization of the continued fraction

In Introduction we discuss the history of the continued fraction and of its generalizations. Early authors proposed a new generalization of the continued fraction that gives periodicity for cubic irrationalities with positive discriminant. Here we propose a new generalization giving periodicity for cubic irrationalities with negative discriminant. We consider the simultaneous rational approximations of a number and its square. At first we describe the structure of the best integer approximations in the homogeneous coordinates when two real forms (linear and quadratic) are given. After that we propose an algorithm to compute the approximants. Examples of computations are given as well.