Consider the space of vertical parabolas in the plane interpreted generally to include nonvertical lines. It is proved that an injective map from a closed region bounded by one such parabola into the plane that maps vertical parabolas to other vertical parabolas must be the composition of a Laguerre transformation with a nonisotropic dilation. Here, a Laguerre transformation refers to a linear fractional or antilinear fractional transformation of the underlying dual plane.
"The most general planar transformations that map parabolas into parabolas." Involve 2 (1) 79 - 88, 2009. https://doi.org/10.2140/involve.2009.2.79