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The most general planar transformations that map parabolas into parabolas. (English) Zbl 1171.51001

For the extended dual plane the authors prove the following theorem:
Every injective map from a closed region bounded by a vertical parabola or a nonvertical line that maps vertical parabolas and nonvertical lines to vertical parabolas and nonvertical lines is the composition of a nonisotropic dilation \(d_{\lambda}:(x,y)\to(\lambda\,x,\lambda^2y)\), \(\lambda\in{\mathbb R}\setminus\{0\}\), with a direct or indirect Laguerre transformation.
The proof is modelled on Carathéodory’s proof of the analogous theorem for the extended complex plane (Möbius plane); [cf. C. Carathéodory, Bull. Am. Math. Soc. 43, 573–579 (1937; JFM 63.0294.03)].

MSC:

51B15 Laguerre geometries

Citations:

JFM 63.0294.03
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