Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 39, No. 1, pp. 231-247 (1998)

Non-Standard Curves Embedded in the Non-Standard Euclidian Plane

Petru Dragos

Faculty of Sciences, University of Oradea
str. Armatei Romana, No. 5, Oradea, Romania

Abstract: Starting from the field $\bf R$ of the real numbers as a model for the formal system of the theory of the real ordered closed fields, complete in the topological sense and Archimedian. A. Robinson [Ro] has found the extension $^*\!\hbox R$ which is ordered, nonarchimedian, not complete and, moreover, the models of $^*\!\hbox R$ are not isomorphic. We work on a model [D2] based on a $\delta$-incomplete ultrafilter, fixed and built as a consequence of the Lowenheim-Skolem theorem [D1] which states that any infinite structure has an elementary extension. We use the three principles: of transfer, of idealization and of standardization, and the rules stated in [Ne].

Using relevant examples, we point out the efficiency of the infinitesimal methods resulting from the non-standard analysis in the study of geometry. The fundamental object is the point, which is supposed to be non-standard and finite [Go]. The point defines a standard frame and we consider that through the point is passing a differentiable manifold, a in this case standard differentiable curve. The point defines also the euclidian invariant as the curvature (see [Go] and [St]).

\item{[D1]} Dragos, P.: Elemente de geometrie diferentiala si calculate pe un model de analiza non-standard, I. An. Univ. din Oradea, fasc. mat., (1991), 68-71. \item{[D2]} Dragos, P.: Elemente de geometrie diferentiala si calculate pe un model de analiza non-standard, II. An. Univ. din Oradea, fasc. mat.,Vol.II (1992), 35-39. \item{[Go]} Goze, M.: Etudes d'un point infiniment petit, dans Le Labyrinthe de continuu. Colloq. Cerisy, Ed. Dinaceur, Springer V. (1993). \item{[Ne]} Nelson, E.: Internal Set Theory: A new approach to Nonstandard Analysis. Bull. Amer. Math. Soc. 83 (1977), 1165-1198. \item{[Ro]} Robinson,A.: Non-Standard Analysis. North Holland Publ. Comp. (1966). \item{[St]} Stroyan, K. D.: Infinitesimal Analysis of Curves and Surfaces. In: Handbook of Mathematical Logic, J. Barwise Editor, North Holland Publ. Comp. (1977), 197-231.

Keywords: non-standard curve, non-standard plane, circle of curvature, osculating circle, monad, infinitesimal element.

Classification (MSC91): 03H05, 53A99

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