Two-sided generalization of the continued fraction

Let in a three-dimensional real space two forms be given: a linear form and a quadratic one which is a product of two complex conjugate linear forms. Their root sets are a plane and a straight line correspondingly. We assume that the line does not lie in the plane. Voronoi (1896) and authors (2005) proposed two different algorithms for computation of integer points giving the best approximations to roots of these two forms. The both algorithms are one-side: the Voronoi algorithms is directed to the plane and the authors algorithm is directed to the line.
Here we propose an algorithm, which works in both directions. We give also examples of two-sided computations by means of the new algorithm.