A study of Jacobi-Perron boundary words for the generation of discrete planes
Résumé
The construction of a Sturmian word, and thus of a discrete line, from a continued fraction development generalizes to higher dimension. Given any vector v in R3 a list of 6-connected points approximating the, line defined by v may be obtained via a generalized continued fraction algorithm. By duality, a discrete plane with normal vector v can also be generated using a related technique. We focus on such discrete planes, more precisely on the finite patterns generated at each step of the process. We show that the choice of Jacobi-Perron algorithm as a higher dimension generalization of Euclid's algorithm together with the specific substitutions deduced from it allows us to guaranty the simple connectedness of those patterns.