Bolt, Michael; Ferdinands, Timothy; Kavlie, Landon The most general planar transformations that map parabolas into parabolas. (English) Zbl 1171.51001 Involve 2, No. 1, 79-88 (2009). 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)]. Reviewer: Rolf Riesinger (Wien) Cited in 1 Document MSC: 51B15 Laguerre geometries Keywords:extended dual plane; direct/indirect Laguerre transformation; Blaschke cylinder; parabola; dual number; linear/antilinear fractional transformation Citations:JFM 63.0294.03 × Cite Format Result Cite Review PDF Full Text: DOI