Evolution of Geometric Figures from the Euclidean to the Digital Era

  • Partha Bhowmick
Conference paper


Recent trends from the Euclidean to the digital geometry in solving various problems on the digital plane are presented in this paper. The notional difference of digital geometry with the Euclidean and the allied geometries has also been pointed out to show how the problems are conceivable in the digital paradigm. Significant contributions in solving these problems using number theory, theory of words, and theory of fractions in general, and digital geometry in particular, have been briefed. The paper is mainly focused on digital straightness and digital circularity, with their related problems, theories, and different perspectives in solving various geometric problems in the digital domain, such as analysis, characterization, segmentation, and approximation.


Euclidean Geometry Chain Code Discrete Apply Mathematic Digital Point Digital Plane 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    P. K. Agarwal and J. Erickson. Geometric range searching and its relatives. In B. Chazelle, J. Goodman, and R. Pollack, editors, Advances in Discrete and Computational Geometry, pages 1–56. American Mathematical Society, Providence, 1998.Google Scholar
  2. 2.
    H. Alt and L. J. Guibas. Discrete geometric shapes: Matching, interpolation, and approximation — A survey. Report B 96-11, 1996. Freie Universität, Berlin.Google Scholar
  3. 3.
    T. Asano, R. Klette, and C. Ronse, editors. Geometry, Morphology, and Computational Imaging, volume 2616 of LNCS. Springer, Berlin, 2003.Google Scholar
  4. 4.
    G. Bertrand, A. Imiya, and R. Klette, editors. Digital and Image Geometry: Advanced Lectures, volume 2243 of LNCS. Springer, Berlin, 2001.Google Scholar
  5. 5.
    P. Bhowmick and B. B. Bhattacharya. Approximate fingerprint matching using Kd-tree. In Proc. 17th Intl. Conf. Pattern Recognition (ICPR), IEEE CS Press, volume 1, pages 544–547, 2004.CrossRefGoogle Scholar
  6. 6.
    P. Bhowmick and B. B. Bhattacharya. Approximation of digital circles by regular polygons. In Proc. Intl. Conf. Advances in Pattern Recognition (ICAPR), volume 3686 of LNCS, pages 257–267. Springer, Berlin, 2005.Google Scholar
  7. 7.
    P. Bhowmick and B. B. Bhattacharya. Approximate matching of digital point sets using a novel angular tree. IEEE Trans. PAMI (, 2007.Google Scholar
  8. 8.
    P. Bhowmick and B. B. Bhattacharya. Fast polygonal approximation of digital curves using relaxed straightness properties. IEEE Trans. PAMI, 29(9):1590–1602, 2007.Google Scholar
  9. 9.
    P. Bhowmick and B. B. Bhattacharya. Number theoretic interpretation and construction of a digital circle. Discrete Applied Mathematics, 156(12):2381–2399, 2008.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    P. Bhowmick and B. B. Bhattacharya. Real polygonal covers of digital discs — Some theories and experiments. Fundamenta Informaticae, page (accepted), 2008.Google Scholar
  11. 11.
    P. Bhowmick, A. Biswas, and B. B. Bhattacharya. Ranking of optical character prototypes using cellular lengths. In Proc. Intl. Conf. Computing: Theory and Applications (ICCTA), IEEE CS Press, pages 422–426, 2007.Google Scholar
  12. 12.
    A. Biswas, P. Bhowmick, and B. B. Bhattacharya. CONFERM: Connectivity Features with Randomized Masks and their applications to image indexing. In Proc. Indian Conf. Computer Vision, Graphics and Image Processing (ICVGIP), pages 556–562, New Delhi, 2004. Allied Publishers Pvt. Ltd.Google Scholar
  13. 13.
    A. Biswas, P. Bhowmick, and B. B. Bhattacharya. Characterization of isothetic polygons for image indexing and retrieval. In Proc. Intl. Conf. Computing: Theory and Applications (ICCTA), IEEE CS Press, pages 590–594, 2007.Google Scholar
  14. 14.
    A. Biswas, P. Bhowmick, and B. B. Bhattacharya. SCOPE: Shape Complexity of Objects using isothetic Polygonal Envelope. In Proc. 6th Intl. Conf. Advances in Pattern Recognition (ICAPR), pages 356–360. World Scientific, Singapore, 2007.Google Scholar
  15. 15.
    J. F. Blinn. How many ways can you draw a circle? IEEE Computer Graphics and Applications, 7(8):39–44, 1987.CrossRefGoogle Scholar
  16. 16.
    C. B. Boyer. A History of Mathematics (2nd Ed.). John Wiley & Sons, Inc., 1991.Google Scholar
  17. 17.
    J. E. Bresenham. An incremental algorithm for digital plotting. In Proc. ACM Natl. Conf., 1963.Google Scholar
  18. 18.
    J. E. Bresenham. Algorithm for for computer control of a digital plotter. IBM Systems Journal, 4(1):25–30, 1965.CrossRefGoogle Scholar
  19. 19.
    J. E. Bresenham. A linear algorithm for incremental digital display of circular ares. Communications of the ACM, 20(2):100–106, 1977.MATHCrossRefGoogle Scholar
  20. 20.
    J. E. Bresenham. Run length slice algorithm for incremental lines. In R. A. Earnshaw, editor, Fundamental Algorithms for Computer Graphics, volume F17 of NATO ASI Series, pages 59–104. Springer-Verlag, New York, 1985.Google Scholar
  21. 21.
    V. Brimkov, D. Coeurjolly, and R. Klette. Digital planarity — A review. Discrete Applied Mathematics, 155(4):468–495, 2007.MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    V. E. Brimkov and R. P. Barneva. Plane digitization and related combinatorial problems. Discrete Appl. Math., 147(2–3):169–186, 2005.MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    R. Brons. Linguistic methods for description of a straight line on a grid. Comput. Graphics Image Process., 2:48–62, 1974.CrossRefMathSciNetGoogle Scholar
  24. 24.
    W. L. Chung. On circle generation algorithms. Computer Graphics and Image Processing, 6:196–198, 1977.CrossRefGoogle Scholar
  25. 25.
    D. Coeurjolly, Y. Gérard, J.-P. Reveillès, and L. Tougne. An elementary algorithm for digital are segmentation. Discrete Applied Mathematics, 139:31–50, 2004.MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    D. Coeurjolly, I. Sivignon, F. Dupont, F. Feschet, and J.-M. Chassery. On digital plane preimage structure. Discrete Appl. Math., 151(1–3):78–92, 2005.MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    B. Coifman, D. Beymer, P. Mclauchlan, and J. Malik. A real-time computer vision system for vehicle tracking and traffic surveillance. Transportation Research: Part C, 6(4):271–288, 1998.CrossRefGoogle Scholar
  28. 28.
    E. Creutzburg, A. Hübler, and V. Wedler. On-line recognition of digital straight line segments. In Proc. 2nd Intl. Conf. AI and Inf. Control Systems of Robots, pages 42–46, 1982.Google Scholar
  29. 29.
    P. E. Danielsson. Comments on circle generator for display devices. Computer Graphics and Image Processing, 7(2):300–301, 1978.CrossRefGoogle Scholar
  30. 30.
    A. K. Das and B. Chanda. A fast algorithm for skew detection of document images using morphology. IJDAR, 4(2):109–114, 2001.CrossRefGoogle Scholar
  31. 31.
    L. S. Davis, A. Rosenfeld, and A. K. Agrawala. On models for line detection. IEEE Trans. Sys., Man & Cybern., 6:127–133, 1976.MATHGoogle Scholar
  32. 32.
    I. Debled-Rennesson and J. P. Reveilles. A linear algorithm for segmentation of digital curves. Intl. J. Patt. Rec. Artif. Intell., 9:635–662, 1995.CrossRefGoogle Scholar
  33. 33.
    M. Doros. Algorithms for generation of discrete circles, rings, and disks. Computer Graphics and Image Processing, 10:366–371, 1979.CrossRefGoogle Scholar
  34. 34.
    J. D. Foley, A. v. Dam, S. K. Feiner, and J. F. Hughes. Computer Graphics — Principles and Practice. Addison-Wesley, Reading (Mass.), 1993.Google Scholar
  35. 35.
    H. Freeman. Techniques for the digital computer analysis of chain-encoded arbitrary plane curves. In Proc. National Electronics Conf., volume 17, pages 421–432, 1961.Google Scholar
  36. 36.
    Y. Gerard, I. Debled-Rennesson, and P. Zimmermann. An elementary digital plane recognition algorithm. Discrete Appl. Math., 151(1–3):169–183, 2005.MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    M. Gleicher. Animation from observation: Motion capture and motion editing. Computer Graphics, 33(4):51–55, 1999.CrossRefGoogle Scholar
  38. 38.
    R. C. Gonzalez and R. E. Woods. Digital Image Processing. Addison-Wesley, California, 1993.Google Scholar
  39. 39.
    M. J. Greenberg. Euclidean and non-Euclidean geometries: Development and history. W.H. Freeman, 1993.Google Scholar
  40. 40.
    Y.-H. Gu and T. Tjahjadi. Corner-based feature extraction for object retrieval. In Proc. Intl. Conf. Image Processing (ICIP), IEEE CS Press, pages 119–123, 1999.Google Scholar
  41. 41.
    S. Har-Peled. An output sensitive algorithm for discrete convex hulls. CGTA, 10:125–138, 1998.MATHMathSciNetGoogle Scholar
  42. 42.
    F. Hoffmann, K. Kriegel, and C. Wenk. An applied point pattern matching problem: Comparing 2D patterns of protein spots. Discrete Applied Mathematics, 93:75–88, 1999.MATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    L. Holm and C. Sander. Mapping the protein universe. Science, 273(5275):595–602, August 1996.CrossRefGoogle Scholar
  44. 44.
    B. K. P. Horn. Circle generators for display devices. Computer Graphics and Image Processing, 5(2):280–288, 1976.CrossRefMathSciNetGoogle Scholar
  45. 45.
    S. Y. Hsu, L. R. Chow, and C. H. Liu. A new approach for the generation of circles. Computer Graphics Forum 12, 2:105–109, 1993.CrossRefGoogle Scholar
  46. 46.
    A. K. Jain, L. Hong, and R. Bolle. On-line fingerprint verification. IEEE Trans. PAMI, 19:302–313, 1997.Google Scholar
  47. 47.
    A. Jobbágy, E. Furnée, B. Romhányi, L. Gyöngy, and G. Soós. Resolution and accuracy of passive marker-based motion analysis. Automatika, 40:25–29, 1999.Google Scholar
  48. 48.
    H. S. Kim and J. H. Kim. A two-step circle detection algorithm from the intersecting chords. Pattern Recognition Letters, 22(6–7):787–798, 2001.MATHCrossRefGoogle Scholar
  49. 49.
    R. Klette. Digital geometry — The birth of a new discipline. In L. S. Davis, editor, Foundations of Image Understanding, pages 33–71. Kluwer, Boston, Massachusetts, 2001.Google Scholar
  50. 50.
    R. Klette and A. Rosenfeld. Digital Geometry: Geometric Methods for Digital Picture Analysis. Morgan Kaufmann Series in Computer Graphics and Geometric Modeling. Morgan Kaufmann, San Francisco, 2004.MATHGoogle Scholar
  51. 51.
    R. Klette and A. Rosenfeld. Digital straightness: A review. Discrete Applied Mathematics, 139(1–3):197–230, 2004.MATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    R. Klette, A. Rosenfeld, and F. Sloboda, editors. Advances in Digital and Computational Geometry. Springer, Singapore, 1998.MATHGoogle Scholar
  53. 53.
    J. Koplowitz, M. Lindenbaum, and A. Bruckstein. The number of digital straight lines on an n×n grid. IEEE Trans. Information Theory, 36:192–197, 1990.MATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    V. A. Kovalevsky. New definition and fast recognition of digital straight segments and ares. In Proc. 10th Intl. Conf. Pattern Recognition (ICPR), IEEE CS Press, pages 31–34, 1990.Google Scholar
  55. 55.
    Z. Kulpa. A note on “circle generator for display devices”. Computer Graphics and Image Processing, 9:102–103, January 1979.CrossRefGoogle Scholar
  56. 56.
    B. Li and H. Holstein. Using k-d trees for robust 3D point pattern matching. In Proc. 4th Intl. Conf. 3-D Digital Imaging and Modeling, pages 95–102, 2003.Google Scholar
  57. 57.
    R. Liang, C. Chen, and J. Bu. Real-time facial features tracker with motion estimation and feedback. In Proc. IEEE Intl. Conf. Systems, Man and Cybernetics, pages 3744–3749, 2003.Google Scholar
  58. 58.
    B. Likar and F. Pernus. Automatic extraction of corresponding points for the registration of medical images. Med. Phys., 26(8):1678–1686, 1999.CrossRefGoogle Scholar
  59. 59.
    M. Lothaire. Algebraic Combinatorics on Words. Cambridge Mathematical Library, 2002.Google Scholar
  60. 60.
    J. Maintz and M. Viergever. A survey of medical image registration. IEEE Engineering in Medicine and Biology Magazine, 2(1):1–36, 1998.Google Scholar
  61. 61.
    D. Maltoni, D. Maio, A. K. Jain, and S. Prabhakar. Handbook of Fingerprint Recognition. Springer-Verlag, New York, 2003.MATHGoogle Scholar
  62. 62.
    M. D. Mcllroy. Best approximate circles on integer grids. ACM Trans. Graphics, 2(4):237–263 1983.CrossRefGoogle Scholar
  63. 63.
    V. Märgner, M. Pechwitz, and H. ElAbed. ICDAR 2005 Arabic handwriting recognition competition. In Proc. Intl. Conf. Document Analysis and Recognition (ICDAR), pages 70–74, 2005.Google Scholar
  64. 64.
    J. A. V. Mieghem, H. I. Avi-Itzhak, and R. D. Melen. Straight line extraction using iterative total least squares methods. J. Visual Commun. and Image Representation, 6:59–68, 1995.CrossRefGoogle Scholar
  65. 65.
    F. Mignosi. On the number of factors of Sturmian words. Theoretical Computer Science, 82(1):71–84, 1991.MATHCrossRefMathSciNetGoogle Scholar
  66. 66.
    E. Mohanna and E. Mokhtarian. Robust corner tracking for unconstrained motions. In IEEE Intl. Conf. Acoustics, Speech, and Signal Processing, pages 804–807, 2003.Google Scholar
  67. 67.
    F. Mokhtarian and F. Mohanna. Content-based video database retrieval through robust corner tracking. In Proc. IEEE Workshop on Multimedia Signal Processing, pages 224–228, 2002.Google Scholar
  68. 68.
    D. Mount, N. Netanyahu, and J. Lemoigne. Improved algorithms for robust point pattern matching and applications to image registration. In Proc. 14th Annual ACM Symposium on Computational Geometry, pages 155–164, 1998.Google Scholar
  69. 69.
    B. Nagy. Characterization of digital circles in triangular grid. Pattern Recognition Letters, 25(11):1231–1242, 2004.CrossRefGoogle Scholar
  70. 70.
    J. Panek and J. Vohradsky. Point pattern matching in the analysis of two-dimensional gel electropherograms. Electrophoresis, 20:3483–3491, 1999.CrossRefGoogle Scholar
  71. 71.
    I. Pavlidis, R. Singh, and N. P. Papanikolopoulos. An on-line handwritten note recognition method using shape metamorphosis. In 4th Intl. Conf. Document Analysis and Recognition (ICDAR), pages 914–918, 1997.Google Scholar
  72. 72.
    T. Pavlidis. Structural Pattern Recognition, Springer, New York, 1977.MATHGoogle Scholar
  73. 73.
    M. L. V. Pitteway. Integer circles, etc. — Some further thoughts. Computer Graphics and Image Processing, 3:262–265, 1974.CrossRefGoogle Scholar
  74. 74.
    I. Povazan and L. Uher. The structure of digital straight line segments and Euclid’s algorithm. In Proc. Spring Conf. Computer Graphics, pages 205–209, 1998.Google Scholar
  75. 75.
    J. Richards. The measurement of human motion: A comparison of commercially available systems. Human Movement Science, 18(5):589–602, 1999.CrossRefGoogle Scholar
  76. 76.
    A. Rosenfeld. Digital straight line segments. IEEE Trans. Computers, 23(12):1264–1268, 1974.MATHCrossRefMathSciNetGoogle Scholar
  77. 77.
    A. Rosenfeld and R. Klette. Digital straightness. Electronic Notes in Theoretical Computer Sc., 46, 2001. Scholar
  78. 78.
    P. L. Rosin. Shape partitioning by convexity. IEEE Trans. Sys., Man & Cybern., 30(2):202–210, 2000.CrossRefMathSciNetGoogle Scholar
  79. 79.
    P. L. Rosin. Measuring shape: Ellipticity, rectangularity, and triangularity. Machine Vision and Applications, 14:172–184, 2003.Google Scholar
  80. 80.
    P. L. Rosin and G. A. W. West. Detection of circular arcs in images. In Proc. 4th. Alvey Vision Conf., Manchester, pages 259–263, 1988.Google Scholar
  81. 81.
    A. W. M. Smeulders and L. Dorst. Decomposition of discrete curves into piecewise segments in linear time. Contemporary Math., 119:169–195, 1991.MathSciNetGoogle Scholar
  82. 82.
    Y. Suenaga, T. Kamae, and T. Kobayashi. A high speed algorithm for the generation of straight lines and circular arcs. IEEE Trans. Comput., 28:728–736, 1979.MATHCrossRefMathSciNetGoogle Scholar
  83. 83.
    G. Tian, D. Gledhill, and D. Taylor. Comprehensive interest points based imaging mosaic. Pattern Recognition Letters, 24:1171–1179, 2003.MATHCrossRefGoogle Scholar
  84. 84.
    K. Voss. Coding of digital straight lines by continued fractions. Comput. Artif. Intelligence, 10:75–80, 1991.MathSciNetGoogle Scholar
  85. 85.
    J. H. Wegstein. An Automated Fingerprint Identification System. US Government Publication, Washington, 1982.Google Scholar
  86. 86.
    J. Williams and M. Bennamoun. Simultaneous registration of multiple corresponding point sets. Computer Vision and Image Understanding, 81:117–142, 2001.MATHCrossRefGoogle Scholar
  87. 87.
    C. Wolf, J.-M. Jolion, W. Kropatsch, and H. Bischof. Content based image retrieval using interest points and texture features. In Proc. 15th Intl. Conf. Pattern Recognition (ICPR), IEEE CS Press, volume 4, pages 234–237, 2000.CrossRefGoogle Scholar
  88. 88.
    M. Worring and A. W. M. Smeulders. Digitized circular arcs: Characterization and parameter estimation. IEEE Trans. PAMI, 17(6):587–598, 1995.Google Scholar
  89. 89.
    W. E. Wright. Parallelization of Bresenham’s line and circle algorithms. IEEE Computer Graphics and Applications, 10(5):60–67, 1990.CrossRefGoogle Scholar
  90. 90.
    X. Wu and J. G. Rokne. Double-step incremental generation of lines and circles. Computer Vision, Graphics, and Image Processing, 37(3):331–344, 1987.CrossRefGoogle Scholar
  91. 91.
    C. Yao and J. G. Rokne. Hybrid scan-conversion of circles. IEEE Trans. Visualization and Computer Graphics, 1(4):311–318, 1995.CrossRefGoogle Scholar
  92. 92.
    Q. Zang and R. Klette. Evaluation of an adaptive composite Gaussian model in video surveillance. In Proc. 10th Intl Conf. Computer Analysis of Images and Patterns (CAIP), pages 165–172, 2003.Google Scholar
  93. 93.
    D. Zhang, G. Lu, W. K. Kong, and M. Wong Online palmprint authentication system for civil applications. J. Computers Science and Technology, 20(1):70–76, 2005.MATHCrossRefGoogle Scholar
  94. 94.
    H. Zhu, X. L. Tang, and P. Liu. An MLP-orthogonal Gaussian mixture model hybrid model for Chinese bank check printed numeral recognition. IJDAR, 8(1):27–34, 2006.CrossRefGoogle Scholar
  95. 95.
    J. Zunic and P. L. Rosin. A new convexity measure for polygons. IEEE Trans. PAMI, 26(7):923–934, 2004.Google Scholar

Copyright information

© Indian Institute of Information Technology, India 2009

Authors and Affiliations

  • Partha Bhowmick
    • 1
  1. 1.Computer Science and Engineering DepartmentIndian Institute of TechnologyKharagpurIndia

Personalised recommendations