Geometric interpretation of Euclid's algorithm and recognition of segments
Theoretical Computer Science
Digital straightness: a review
Discrete Applied Mathematics - The 2001 international workshop on combinatorial image analysis (IWCIA 2001)
Digital lines with irrational slopes
Theoretical Computer Science
Discrete rotations and symbolic dynamics
Theoretical Computer Science
Spirograph Theory: A Framework for Calculations on Digitized Straight Lines
IEEE Transactions on Pattern Analysis and Machine Intelligence
Revisiting digital straight segment recognition
DGCI'06 Proceedings of the 13th international conference on Discrete Geometry for Computer Imagery
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We study relations between digital lines and continued fractions. The main result is a parsimonious description of the construction of the digital line based only on the elements of the continued fraction 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 continued fraction expansions of their slopes. Our work is confined to irrational slopes since, to our knowledge, there exists no run-hierarchical and continued fraction based description for these, in contrast to rational slopes which have been extensively examined. The description is exact (it 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 continued fraction expansions, i.e. both quadratic surds and other irrationals. We also derive the connection between the Gauss map and the digitization parameters introduced by the author in 2007.