Continued Fractions and Digital Lines with Irrational Slopes

Abstract

This paper expands on previous work on relationships between digital lines and continued fractions (CF). The main result is a parsimonious description of the construction of the digital line based only on the elements of the CF representing its slope and containing only simple integer computations. The description reflects the hierarchy of digitization runs, which raises the possibility of dividing digital lines into equivalence classes depending on the CF expansions of their slopes. Our work is confined to irrational slopes since, to our knowledge, there exists no such description for these, in contrast to rational slopes which have been extensively examined. The description is exact and does not use approximations by rationals. Examples of lines with irrational slopes and with very simple digitization patterns are presented. These include both slopes with periodic and non-periodic CF expansions, i.e. both quadratic surds and other irrationals.Keywordsdigital geometrytheory of digital linesirrational slopecontinued fractionquadratic surd