A formal specification of line representations on graphics devices

  • Lynn S. Marshall
Colloquium On Software Engineering Concepts And Methods In Software Development
Part of the Lecture Notes in Computer Science book series (LNCS, volume 186)


To show that a computer graphics system functions properly it is necessary to prove that the images it produces are correct. Most graphical devices are unable to exactly represent an image, or even just a straight line. Thus each device must display an approximation to the ideal. This paper presents a formal specification of the properties any reasonable approximation to a straight line should have. Bresenham's algorithm is shown to satisfy this specification and extensions to the specification are discussed.


Chain Code Graphic Device Foreground Colour Vienna Development Method Draft International Standard 
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.


  1. 1.
    C. Arcelli and A. Massarotti, "Regular Arcs in Digital Contours," Computer Graphics and Image Processing Vol. 4 pp. 339–360 (1975).Google Scholar
  2. 2.
    G. Bongiovanni, F. Luccio, and A. Zorat, "The Discrete Equation of a Straight Line," IEEE Transactions on Computers Vol. C-24(3) pp. 310–313 (March 1975).Google Scholar
  3. 3.
    J. Boothroyd and P. A. Hamilton, "Exactly Reversible Plotter Paths," Australian Computer Journal Vol. 2(1) pp. 20–21 (1970).Google Scholar
  4. 4.
    J. E. Bresenham, "Algorithm for Computer Control of a Digital Plotter," IBM Systems Journal Vol. 4(1) pp. 25–30 (1965).Google Scholar
  5. 5.
    K. W. Brodlie, M. C. Maguire, and G. E. Pfaff, "A Practical Strategy for Certifying GKS Implementations," in EUROGRAPHICS 82 International Conference and Exhibition UMIST 8–10 Sept 1982, eds. D. S. Greenaway and E. A. Warman, North-Holland (1982).Google Scholar
  6. 6.
    R. Brons, "Linguistic Methods for the Description of a Straight Line on a Grid," Computer Graphics and Image Processing Vol. 3 pp. 48–62 (1974).Google Scholar
  7. 7.
    George S. Carson, "The Specification of Computer Graphics Systems," IEEE Computer Graphics and Applications Vol. 3(6) pp. 27–41 (1974).Google Scholar
  8. 8.
    D. A. Duce, E. V. C. Fielding, and L. S. Marshall, Formal Specification and Graphics Software, Rutherford Appleton Laboratory Report RAL-84-068, August 1984.Google Scholar
  9. 9.
    R. A. Earnshaw, Display Algorithms — History, Developments and Applications, University Computing Service, University of Leeds.Google Scholar
  10. 10.
    Eugene Fiume and Alain Fournier, A Programme for the Development of a Mathematical Theory of Computer Graphics, Computer Systems Research Group, Department of Computer Science, University of Toronto, Toronto, Ontario (1984).Google Scholar
  11. 11.
    J. D. Foley and A. van Dam, Fundamentals of Interactive Computer Graphics, Addison-Wesley Publishing Company (1982).Google Scholar
  12. 12.
    Herbert Freeman, "Boundary Encoding and Processing," pp. 241–266 in Picture Processing and Psychopictorics, eds. Bernice Sacks Lipkin and Azriel Rosenfeld, Academic Press, New York-London (1970).Google Scholar
  13. 13.
    Graphical Kernel System (GKS) 7.2 Functional Description, Draft International Standard ISO/DIS 7942 (November 14th, 1982).Google Scholar
  14. 14.
    R. Gnatz, Approaching a Formal Framework for Graphics Software Standards, Technical University of Munich.Google Scholar
  15. 15.
    Leo J. Guibas and Jorge Stolfi, "A Language for Bitmap Manipulation," ACM Transactions on Graphics Vol. 1(3) pp. 191–214 (July 1982).Google Scholar
  16. 16.
    Satish Gupta and Robert F. Sproull, "Filtering Edges for Grey-Scale Displays," ACM Computer Graphics Vol. 15(3) pp. 1–5 (August 1981).Google Scholar
  17. 17.
    John Guttag and James J. Horning, Formal Specification as a Design Tool, XEROX PARC Technical Report CSL-80-1, Palo Alto, CA (June 1982).Google Scholar
  18. 18.
    C. B. Jones, Software Development: A Rigorous Approach, Prentice-Hall, Englewood Cliffs, NJ (1980).Google Scholar
  19. 19.
    William R. Mallgren, Formal Specification of Interactive Graphics Programming Languages, ACM Distinguished Dissertation 1982, MIT Press (1983).Google Scholar
  20. 20.
    Lynn S. Marshall, A Formal Specification of Line Representations on Graphics Devices, University of Manchester Transfer Report, September 1984.Google Scholar
  21. 21.
    Lynn S. Marshall, GKS Workstations: Formal Specification and Proofs of Correctness for Specific Devices, University of Manchester Transfer Report, September 1984.Google Scholar
  22. 22.
    William M. Newman and Robert F. Sproull, Principles of Interactive Computer Graphics, McGraw Hill Kogakuska Limited (1973).Google Scholar
  23. 23.
    William M. Newman and Robert F. Sproull, Principles of Interactive Computer Graphics, Second Edition, McGraw Hill International Book Company (1981).Google Scholar
  24. 24.
    M. L. V. Pitteway and D. J. Watkinson, "Bresenham's Algorithm with Grey Scale," Communications of the ACM Vol. 23(11) pp. 625–626 (1980).Google Scholar
  25. 25.
    Azriel Rosenfeld, "Digital Straight Line Segments," IEEE Transactions on Computers Vol. C-23(12) pp. 1264–1269 (December 1974).Google Scholar
  26. 26.
    Jerome Rothstein and Carl Weiman, "Parallel and Sequential Specification of a Context Sensitive Language for Straight Lines on Grids," Computer Graphics and Image Processing Vol. 5 pp. 106–124 (1976).Google Scholar
  27. 27.
    Robert F. Sproull, "Using Program Transformations to Derive Line Drawing Algorithms," ACM Transactions on Graphics Vol. 1(4) pp. 259–273 (October 1982).Google Scholar
  28. 28.
    University of Manchester Computer Graphics Unit, Interactive Computer Graphics Course Notes (March 1984).Google Scholar
  29. 29.
    John Warnock and Douglas K. Wyatt, "A Device Independent Graphics Imaging Model for Use with Raster Devices," ACM Computer Graphics Vol. 16(3) pp. 313–319 (July 1982).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • Lynn S. Marshall
    • 1
  1. 1.Department of Computer ScienceUniversity of ManchesterManchesterUK

Personalised recommendations