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On a model of plane geometry. (English) Zbl 0896.51011

The authors deal with following model of plane geometry. The points are the points of the Euclidean plane \(\mathbb{R}^2\). There are four types of lines: a vertical Euclidean line; a horizontal Euclidean line; a translate of the hyperbola \(L= \{(x,y): x>0, y={1\over x}\}\); and a translate of the hyperbola \(L^*= \{(x,y): x<0, y=- {1\over x}\}\). They label it \(G2\) [see B. Grünbaum and L. Mycielski, Am. Math. Monthly 97, No. 9, 839-846 (1990; Zbl 0799.51007)], define two functions \(f:I\to \mathbb{R}\) and \(g:J \to\mathbb{R}\), where \(I\) and \(J\) are subintervals of \(\mathbb{R}\), substitute them instead of \(y={1\over x}\) and \(y=-{1\over x}\) in \(G2\) and label this interpretation \(M_{(f,g)}\).
The main result of this article: If there exists a continuous bijection \(\emptyset\) of \(\mathbb{R}^2\) onto itself which induces a map from \(G2\) onto \(M_{(f,g)}\), then, up to translations, \(f(x)= {a\over x}\) for \(x>0\) and \(g(x)= -{a\over x}\) for \(x<0\) for some positive constant \(a\).
Reviewer: P.Burda (Ostrava)

MSC:

51G05 Ordered geometries (ordered incidence structures, etc.)
51A30 Desarguesian and Pappian geometries

Citations:

Zbl 0799.51007
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