Vol. 2, No. 1, 2009

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The most general planar transformations that map parabolas into parabolas

Michael Bolt, Timothy Ferdinands and Landon Kavlie

Vol. 2 (2009), No. 1, 79–88
Abstract

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.

Keywords
dual number, Laguerre transformation, parabola
Mathematical Subject Classification 2000
Primary: 51B15
Milestones
Received: 5 September 2008
Accepted: 11 February 2009
Published: 18 March 2009

Communicated by Michael Dorff
Authors
Michael Bolt
Department of Mathematics and Statistics
Calvin College
1740 Knollcrest Circle SE
Grand Rapids, Michigan 49546-4403
United States
http://www.calvin.edu/~mdb7/
Timothy Ferdinands
Department of Mathematics and Statistics
Calvin College
1740 Knollcrest Circle SE
Grand Rapids, Michigan 49546-4403
United States
Landon Kavlie
Department of Mathematics and Statistics
1740 Knollcrest Circle SE
Calvin College
Grand Rapids, Michigan 49546-4403
United States