Abstract
We study patterns that appear in discrete circles with integer center and radius. As the radius goes to infinity, the patterns get closer to digital straight segments: the notion of tangent words (described in Monteil DGCI 2011) allows to grasp their shape. Unexpectedly, some tangent convex words do not appear infinitely often due to deep arithmetical reasons related to an underlying Pell-Fermat equation. The aim of this paper is to provide a complete characterization of the patterns that appear in integer discrete circles for infinitely many radii.
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Andres, E.: Discrete circles, rings and spheres. Computers & Graphics 18(5), 695–706 (1994)
Andres, E.: Defining Discrete Objects for Polygonalization: The Standard Model. In: Braquelaire, A., Lachaud, J.-O., Vialard, A. (eds.) DGCI 2002. LNCS, vol. 2301, pp. 313–325. Springer, Heidelberg (2002)
Bresenham, J.: A linear algorithm for incremental digital display of circular arcs. Commun. ACM 20(2), 100–106 (1977)
Cohen-Or, D., Kaufman, A.: Fundamentals of surface voxelization. Graphical Models and Image Processing 57(6), 453–461 (1995)
Fiorio, C., Toutant, J.-L., Jamet, D.: Discrete Circles: an arithmetical approach with non-constant thickness. In: Vision Geometry XIV, IS&T/SPIE 18th Annual Symposium Electronic Imaging, pp. 60660C01–60660C12. SPIE (January 2006)
Freeman, H.: Computer processing of line-drawing images. Computing Surveys 6, 57–97 (1974)
Kulpa, Z.: On the properties of discrete circles, rings, and disks. Computer Graphics and Image Processing 10(4), 348–365 (1979)
Lothaire, M.: Algebraic combinatorics on words. Encyclopedia of Mathematics and its Applications, vol. 90. Cambridge University Press, Cambridge (2002)
Monteil, T.: Another Definition for Digital Tangents. In: Debled-Rennesson, I., Domenjoud, E., Kerautret, B., Even, P. (eds.) DGCI 2011. LNCS, vol. 6607, pp. 95–103. Springer, Heidelberg (2011)
Monteil, T.: The complexity of tangent words. In: Ambroz, P., Holub, S., Masáková, Z. (eds.) WORDS. EPTCS, vol. 63, pp. 152–157 (2011)
Pytheas Fogg, N.: Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics, vol. 1794. Springer, Berlin (2002)
Robertson, J.P.: Solving the generalized Pell equation x 2 − Dy 2 = N (2004), Preprint available at http://www.jpr2718.org/pell.pdf
Tougne, L.: Circle Digitization and Cellular Automata. In: Miguet, S., Ubéda, S., Montanvert, A. (eds.) DGCI 1996. LNCS, vol. 1176, pp. 283–294. Springer, Heidelberg (1996)
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Hoarau, A., Monteil, T. (2013). Persistent Patterns in Integer Discrete Circles. In: Gonzalez-Diaz, R., Jimenez, MJ., Medrano, B. (eds) Discrete Geometry for Computer Imagery. DGCI 2013. Lecture Notes in Computer Science, vol 7749. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37067-0_4
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DOI: https://doi.org/10.1007/978-3-642-37067-0_4
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