Abstract
Our concern is the digitalization of line segments in ℤ2 as considered by Chun et al. [5] and Christ et al. [4]. The key property that differentiates the research of Chun et al. and Christ et al. from other research in digital line segment construction is that the intersection of any two segments must be connected. Such a system of segments is called a consistent digital line segments system (CDS). Chun et al. give a construction for all segments in ℤd that share a common endpoint (called consistent digital rays (CDR)) that has asymptotically optimal Hausdorff distance, and Christ et al. give a complete CDS in ℤ2 with optimal Hausdorff distance. Christ et al. also give a characterization of CDRs in ℤ2, and they leave open the question on how to characterize CDSes in ℤ2. In this paper, we answer one of the most important open question regarding CDSes in ℤ2 by giving the characterization asked for by Christ et al. We obtain the characterization by giving a set of necessary and sufficient conditions that a CDS must satisfy.
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References
Andres, E.: Discrete linear objects in dimension n: the standard model. Graphical Models 65(1-3), 92–111 (2003)
Berthé, V., Labbé, S.: An arithmetic and combinatorial approach to three-dimensional discrete lines. In: Discrete Geometry for Computer Imagery - 16th IAPR International Conference, Nancy, France, April 6-8, pp. 47–58 (2011)
Christ, T.: Discrete Descriptions of Geometric Objects. PhD thesis, Zürich (2011)
Christ, T., Pálvölgyi, D., Stojakovic, M.: Consistent digital line segments. Discrete & Computational Geometry 47(4), 691–710 (2012)
Chun, J., Korman, M., Nöllenburg, M., Tokuyama, T.: Consistent digital rays. Discrete & Computational Geometry 42(3), 359–378 (2009)
Cohen-Or, D., Kaufman, A.E.: 3d line voxelization and connectivity control. IEEE Computer Graphics and Applications 17(6), 80–87 (1997)
Eckhardt, U.: Digital lines and digital convexity. In: Bertrand, G., Imiya, A., Klette, R. (eds.) Digital and Image Geometry. LNCS, vol. 2243, pp. 209–228. Springer, Heidelberg (2002)
Franklin, W.R.: Problems with raster graphics algorithm. In: Peters, F.J., Kessener, L.R.A., van Lierop, M.L.P. (eds.) Data Structures for Raster Graphics, Steensel, Netherlands (1985)
Goodrich, M.T., Guibas, L.J., Hershberger, J., Tanenbaum, P.J.: Snap rounding line segments efficiently in two and three dimensions. In: Symposium on Computational Geometry, pp. 284–293 (1997)
Greene, D.H., Yao, F.F.: Finite-resolution computational geometry. In: 27th Annual Symposium on Foundations of Computer Science, Toronto, Canada, October 27-29, pp. 143–152. IEEE Computer Society (1986)
Klette, R., Rosenfeld, A.: Digital straightness - a review. Discrete Applied Mathematics 139(1-3), 197–230 (2004)
Lachaud, J.-O., Said, M.: Two efficient algorithms for computing the characteristics of a subsegment of a digital straight line. Discrete Applied Mathematics 161(15), 2293–2315 (2013)
Luby, M.G.: Grid geometries which preserve properties of euclidean geometry: A study of graphics line drawing algorithms. In: Earnshaw, R.A. (ed.) Theoretical Foundations of Computer Graphics and CAD, vol. 40, pp. 397–432 (1988)
Sivignon, I., Dupont, F., Chassery, J.-M.: Digital intersections: minimal carrier, connectivity, and periodicity properties. Graphical Models 66(4), 226–244 (2004)
Sugihara, K.: Robust geometric computation based on topological consistency. In: International Conference on Computational Science (1), pp. 12–26 (2001)
van Lierop, M.L.P., van Overveld, C.W.A.M., van de Wetering, H.M.M.: Line rasterization algorithms that satisfy the subset line property. Computer Vision, Graphics, and Image Processing 41(2), 210–228 (1988)
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Chowdhury, I., Gibson, M. (2015). A Characterization of Consistent Digital Line Segments in ℤ2 . In: Bansal, N., Finocchi, I. (eds) Algorithms - ESA 2015. Lecture Notes in Computer Science(), vol 9294. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48350-3_29
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DOI: https://doi.org/10.1007/978-3-662-48350-3_29
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