Skip to main content
Log in

Harthong-Reeb analysis and digital circles

  • Published:
The Visual Computer Aims and scope Submit manuscript

Abstract

We present a new approach to derive efficient and precise algorithms of interest to computer graphics. This is illustrated here by the case of the circle. It also yields algorithms for straight lines and more infrequent curves, such as segments of exponentials. The method is, in essence, the numerical integration of vector fields using only integers.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Andres E (1989) Le cercle discret: rapport de stage de DEA. Université Louis Pasteur, Strasbourg

    Google Scholar 

  • Badler N (1977) Disk generators for a raster display device. Comput Graphics Image Processing 6:589–593

    Google Scholar 

  • Belser K (1976) Comment on an improved algorithm for the generation of nonparametric curves. IEEE Trans Comput C-25(1):103

    Google Scholar 

  • Bresenham J (1977) A linear algorithm for increamental digital display of circular arcs. Commun ACM 20(2):100–106

    Google Scholar 

  • Crouzeix M, Mignot AL (1984) Analyse numérique des équations différentielles. (Collection mathématique appliquées pour la maîtrise) Masson, Paris

    Google Scholar 

  • Danielsson E (1970) Incremental curve generation. IEEE Trans Comput C-19(9):783–793

    Google Scholar 

  • Danielsson E (1978) Comments on circle generator for display devices. Comput Graphics Image Processing 7:300–301

    Google Scholar 

  • Denert E (1973) A method for computing points of a circle using only integers. Comput Graphics Image Processing 2:83–91

    Google Scholar 

  • Devaux M, Macé P (1985) Algorithmique en nombres entiers pour lignes, surfaces, et volumes splines d'interpolation. In: Actes de la 4em conférence européenne sur la C.F.A.O. et l'inforgraphie-Proceedings of the 4th European Conference on CAD/CAM and Computer Graphics. MICAD/Hermès, Hermann

    Google Scholar 

  • Diener, F, Reeb G (1989) Analyse non standard. (Collection enseignement des sciences)

  • Diener M (1984) Deux outils pour l'étude des équations différentielles perturbées. In: Actes de l'école d'été ≪Analyse non standard et représentation du réel≫. Oran-Les Andalouses, 8.−12.9.1984. CNRS (Paris), OPU (Alger)

    Google Scholar 

  • Diener M (1988) Application du calcul de Harthong-Reeb aux routines graphiques. Université Paris 7, seminaire non standard

  • Diener M, Deledicq A (1989) Leçons de calcul infinitesimal; ACL/Armand Collin, Paris

    Google Scholar 

  • Doros M (1979) Algorithms for generation of discrete circles, rings and disks. Comput Graphics Image Processing 10:366–371

    Google Scholar 

  • Foley JD, Van Dam A (1984) Fundamentals of interactive computer graphics: the systems programming series

  • Harthong J (1980) Le moiré. IRMA, Strasbourg

    Google Scholar 

  • Holin H (1989) Harthong-Reeb circles. Université Paris 7, séminaire non standard

  • Holin H (1990) Sun and filigrees. Université Paris 7, séminaire non standard

  • Holin H (1991) Moyennisation spaciale et supports pixélisés. Thesis, University of Paris 7

  • Horn BKP (1976) Circle generators for display devices. Comput Graphics Image Processing 5:280–289

    Google Scholar 

  • Horn BKP (1979) Comments on: “A note on the paper by B.K.P. Horn” by Zenon Kulpa, Comput Graphics Image Processing 9:104

    Google Scholar 

  • Jordan BW, Lennon WJ Jr, Holm BD (1973) An improved algorithm for the generation of nonparametric curves. IEEE Trans Comput C-22(12):1052–1060

    Google Scholar 

  • Kulpa Z (1978) A note on the paper by B.K.P. Horn: “Circle generators for display devices”. Comput Graphics Image Processing 9:102–103

    Google Scholar 

  • Kulpa Z (1979) On the properties of discrete circles, rings and disks. Comput Graphics Image Processing 10:348–365

    Google Scholar 

  • Nakamura A, Aizawa K (1984) Digital circles. Comput Vision, Graphics, and Image Processing 26:242–255

    Google Scholar 

  • Nelson E (1977) Internal set theory, a new approach to N.S.A. Bull Am Math Soc 83:1965–1998

    Google Scholar 

  • Newman WM, Sproull RF (1973) Principles of interactive computer graphics, 1st edn. McGraw-Hill, New York

    Google Scholar 

  • Pitteway MLV (1967) Algorithm for drawing ellipses or hyperbolae with a digital plotter. The Computer Journal 10–11:282–289

    Google Scholar 

  • Pitteway MLV (1974) Integer circles, etc. Some further thoughts. Comput Graphics Image Processing (1974) 3:262–263

    Google Scholar 

  • Pitteway MLV, Botting RJ (1974) “Integer Circles, Etc.” —Three moves extension of Bresenham's algorithm. Comput Graphics Image Processing

  • Pletinckx D (1989) Quaternion calculus as a basic tool in computer graphics. The Visual Computer 5(1/2):2–13

    Google Scholar 

  • Posch KC, Fellner WD (1989) The circle-brush algorithm. ACM Trans Graphics 8(1):1–23

    Google Scholar 

  • Ramot J (1976) Nonparametric curves. IEEE Trans Comput C-25(1):103–104

    Google Scholar 

  • Rao CVK (1975) Comment on a method for computing points of a circle using only integers. Comput Graphics Image Processing 4:79

    Google Scholar 

  • Reder C (1984) Observation macroscopique de phénomènes microscopiques. In: Actes de l'école d'été “Analyse non standard et représentation du réel. Oran-Les Andalouses, 8.–12.9.1984. CNRS (Paris) OPU (Alger)

    Google Scholar 

  • Reeb G, Reveilles JP, Troesch A, Urlacher E (1987) Equations différentielles et nombres entiers. IRMA, Strasbourg

    Google Scholar 

  • Reveilles JP (1987a) Liberez les chiffres — continuité et microprocesseurs. IRMA, Strasbourg

    Google Scholar 

  • Reveilles JP (1987b) Simulation arithmetique du continu. IRMA, Strasbourg

    Google Scholar 

  • Reveilles JP (1988a) Structure arithmétique des droites de Bresenham. Université Paris 7, seminaire non standard

  • Reveilles JP (1988b) Les paliers des droites de Bresenham. Pixim: 81–101

  • Reveilles JP (1990) Les droites discrètes — une définition générale, leur intersections. (Rapport de recherche) INRIA, Sophia-Antipolis

    Google Scholar 

  • Robinson A (1966) Non-standard analysis. North-Holland, Amsterdam

    Google Scholar 

  • Rubin F (1976) Generation of nonparametric curves; IEEE Trans Comput C-25(1):103–103

    Google Scholar 

  • Sari T (1983) Stroboscopie infinitésimale et moyenisation dans les systèmes d'équations différentielles à solutions rapidement oscillantes. In: Outils et modèles mathématiques pour l'automatique, le contrôle, et le traitement du signal, vol III. CNRS, Paris, pp 345–353

    Google Scholar 

  • Schweitzer P (1987) Infographie (I & II). Presses Polytechniques Romandes

  • Seroul R (1987) Équations différentielles et nombres entiers ou la méthode TRRU (L'ouvert no. 48)

  • Suenaga Y, Kamae T, Kobayashi T (1979) A high-speed algorithm for the generation of straight lines and circular arcs. IEEE Trans Comput C28(10):728–793

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Holin, H. Harthong-Reeb analysis and digital circles. The Visual Computer 8, 8–17 (1991). https://doi.org/10.1007/BF01900842

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01900842

Key words

Navigation