Abstract
This paper gives a perfect, ideal, discretization of continuous notions. This is a very convenient frame to treat continuous problems or theories with the help of a computer. This is illustrated by the conversion of algorithms using real numbers into algorithms using integers only and the founding of discrete geometry.
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References
J. Bresenham, Algorithm for computer control of a digital plotter, IBM Syst. J. 4(1965)25–30.
C.C. Chang and H.J. Keisler,Model Theory, 2nd ed. (North-Holland, Amsterdam, 1977).
J.M. Chassery and A. Montanvert,Géométrie Discrète (Editions Hermès, 1991).
M. Davis,Applied Non-Standard Analysis (Wiley-Interscience, 1977).
M. Diener, Application du calcul de Harthong-Reeb aux routines graphiques,Séminaire Non-Standard, Université Paris 7 (1988).
G.H. Hardy and E.M. Wright,An Introduction to the Theory of Numbers, 5th ed. (Oxford University Press, 1989).
J. Harthong, Eléments pour une théorie du continu, Astérisque 235(1983)109–110.
M.A. Jacob, Applications quasi-affines, Thèse, Université Louis Pasteur, Strasbourg (1993).
J. Keisler,Elementary Calculus (Prindle-Weber & Schmidt, Boston, MA).
P. Nehlig, Applications quasi-affines discrètes et antialiassage, Thèse, Université Louis Pasteur, Strasbourg (1992).
E. Nelson, Internal set theory, BAMS 83(1977)1165–1198.
G. Reeb, J.P. Reveillès, A. Troesch and E. Urlacher, Equations différentielles et nombres entiers, Preprint, IRMA Université Louis Pasteur, Strasbourg (1987).
J.P. Reveillès, Simulation arithmétique du continu, Preprint, IRMA, Université Louis Pasteur, Strabourg (1987).
J.P. Reveillès, Les paliers des droites de Bresenham, in:PIXIM 88 (Hermès Ed, Paris, 1988) pp. 81–101.
J.P. Reveillès, Géométrie Discrète, Calcul en nombres entiers et algorithmique, Thèse, Université Louis Pasteur, Strasbourg (1991).
J.P. Reveillès, Libérez les chiffres — Continuité et microprocesseur, Preprint, IRMA, Université Louis Pasteur, Stransbourg (1987).
J.P. Reveillès, Structure arithmétique du continu, Preprint, IRMA, Université Louis Pasteur, Stransbourg (1987).
J.P. Reveillès, Les droites discrètes — Une définition générale, leurs intersections, Preprint, INRIA, Sohpia-Antipolis (1990).
D. Richard, De la structure additive la saturation des modèles de Peano et à une classification des sous-langages de l'arithmétique,Lecture Notes in Mathematics 890, eds. C. Berline, K. McAloon and J.P. Ressayre (Springer, 1981) pp. 270–296.
D. Richard, On extremal properties of non standard models of arithmetics, Preprint, Dépt. Math., Université de Lyon (1977) t. 14-4, pp. 57–75.
P. Dehornoy, S. Grigorieff, D. Richard, R. Sami and J. Stern, CIMPA — L'Analyse non-standard, in Cours de logique du CIMPA (1984).
A. Robinson,Non-Standard Analysis (North-Holland, Amsterdam, 1966).
A. Rosenfeld and R.A. Melter, Digital geometry, Math. Intell. 11(3)(1989)69–72.
R. Seroul, Equations différentielles et nombres entiers ou la méthode TRRU,L'ouvert No. 48 (1987).
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Reveillès, JP., Richard, D. Back and forth between continuous and discrete for the working computer scientist. Ann Math Artif Intell 16, 89–152 (1996). https://doi.org/10.1007/BF02127796
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DOI: https://doi.org/10.1007/BF02127796