Abstract:
There is a very close relationship between digital straight lines and binary words with the property of balance. During my Ph.D. studies I described, by means of continued fractions, the run hierarchical structure of digital straight lines (equivalently, of upper mechanical words) with irrational positive slopes less than one. I defined two equivalence relations on the set of slopes. Both relations are defined on the continued fraction elements of the slopes and they have their interpretation in the run hierarchical structure. This causes some questions with combinatorial nature, for example, about the description of the continued fraction expansion of slopes in each equivalence class. Another combinatorial piece is the fixed point theorem for the lines and words. The fixed points are the lines (words) with self-balanced construction in terms of long and short runs on all digitization levels.