Bolt , Ferdinands , Kavlie : The most general planar transformations that map parabolas into parabolas (original) (raw)

Involve: A Journal of Mathematics

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• Volume 2, Number 1 (2009), 79-88.

The most general planar transformations that map parabolas into parabolas

Michael Bolt, Timothy Ferdinands, and Landon Kavlie

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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.

Article information

Source
Involve, Volume 2, Number 1 (2009), 79-88.

Dates
Received: 5 September 2008
Accepted: 11 February 2009
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1513799118

Digital Object Identifier
doi:10.2140/involve.2009.2.79

Mathematical Reviews number (MathSciNet)
MR2501346

Zentralblatt MATH identifier
1171.51001

Subjects
Primary: 51B15: Laguerre geometries

Keywords
dual number Laguerre transformation parabola

Citation

Bolt, Michael; Ferdinands, Timothy; Kavlie, Landon. The most general planar transformations that map parabolas into parabolas. Involve 2 (2009), no. 1, 79--88. doi:10.2140/involve.2009.2.79. https://projecteuclid.org/euclid.involve/1513799118

References

• D. E. Blair, Inversion theory and conformal mapping, Student Mathematical Library 9, American Mathematical Society, Providence, RI, 2000.
• C. Carathéodory, “The most general transformations of plane regions which transform circles into circles”, Bull. Amer. Math. Soc. 43 (1937), 573–579.
• T. Ferdinands and L. Kavlie, “A Beckman–Quarles type theorem for Laguerre transformations in the dual plane”, preprint, 2009.
• V. V. Kisil, “Starting with the group rmSLsb2(bfR){\rm SL}\sb 2({\bf R})rmSLsb2(bfR)”, Notices Amer. Math. Soc. 54:11 (2007), 1458–1465.
• I. M. Yaglom, Complex numbers in geometry, Translated from the Russian by Eric J. F. Primrose, Academic Press, New York, 1968.
  Mathematical Reviews (MathSciNet): MR0220134