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.
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References
Beskin, N.M.: Fascinating Fractions. Mir Publishers, Moscow (1986) (Revised from the 1980 Russian edition)
Brezinski, C.: History of Continued Fractions and Padé Approximants. Springer, Heidelberg (1991) (Printed in USA)
Brons, R.: Linguistic Methods for the Description of a Straight Line on a Grid. Comput. Graphics Image Processing 3, 48–62 (1974)
Bruckstein, A.M.: Self-Similarity Properties of Digitized Straight Lines. Contemp. Math. 119, 1–20 (1991)
Debled, I.: Étude et reconnaissance des droites et plans discrets. Ph.D. Thesis, Strasbourg: Université Louis Pasteur, pp. 209 (1995)
Dorst, L., Duin, R.P.W.: Spirograph Theory: A Framework for Calculations on Digitized Straight Lines. IEEE Transactions on Pattern Analysis and Machine Intelligence PAMI 6(5), 632–639 (1984)
Khinchin, A.Ya.: Continued Fractions, 3rd edn. Dover Publications (1997)
Klette, R., Rosenfeld, A.: Digital straightness – a review. Discrete Appl. Math. 139(1–3), 197–230 (2004)
Lothaire, M.: Algebraic Combinatorics on Words. Cambridge Univ. Press, Cambridge (2002)
Perron, O.: Die Lehre von den Kettenbrüchen. Band I: Elementare Kettenbrüche, 3rd edn (1954)
Réveillès, J.-P.: Géométrie discrète, calculus en nombres entiers et algorithmique, 251 pages, Strasbourg: Université Louis Pasteur, Thèse d’État (1991)
Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences, http://www.research.att.com/~njas/sequences/A003285
Stephenson, P.D.: The Structure of the Digitised Line: With Applications to Line Drawing and Ray Tracing in Computer Graphics. North Queensland, Australia, James Cook University. Ph.D. Thesis (1998)
Stolarsky, K.B.: Beatty sequences, continued fractions, and certain shift operators. Canad. Math. Bull. 19, 473–482 (1976)
Uscka-Wehlou, H.: Digital lines with irrational slopes. Theoret. Comput. Sci. 377, 157–169 (2007)
Venkov, B.A.: Elementary Number Theory. Translated and edited by Helen Alderson. Wolters-Noordhoff, Groningen (1970)
de Vieilleville, F., Lachaud, J.-O.: Revisiting Digital Straight Segment Recognition. In: Kuba, A., Nyúl, L.G., Palágyi, K. (eds.) DGCI 2006. LNCS, vol. 4245, pp. 355–366. Springer, Heidelberg (2006)
Voss, K.: Discrete Images, Objects, and Functions in ZZn. Springer-Verlag (1993)
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Uscka-Wehlou, H. (2008). Continued Fractions and Digital Lines with Irrational Slopes. In: Coeurjolly, D., Sivignon, I., Tougne, L., Dupont, F. (eds) Discrete Geometry for Computer Imagery. DGCI 2008. Lecture Notes in Computer Science, vol 4992. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79126-3_10
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DOI: https://doi.org/10.1007/978-3-540-79126-3_10
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