In 1953, Beckman and Quarles proved a well-known result in Euclidean Geometry that any transformation preserving a distance r must be a rigid motion. In 1991, June Lester published an analogous result for circle-preserving transformations in the complex plane. In our paper, we introduce the notion of dual numbers and the geometry of the dual plan. We forcus on the set of vertical parabolas and non-vertical linear P with a distance between pairs of parabolas defined to be the difference of slopes at their point(s) of intersection. We then prove that any bijective transformation from P to itself which preserves our distance 1 induces a fractional linear or Laguerre transformation of the dual plane.
Michael Bolt, Department of Mathematics and Statistics, Calvin College firstname.lastname@example.org
Ferdinands, Timothy and Kavlie, Landon
"A Beckman-Quarles Type Theorem for Laguerre Transformations in the Dual Plane,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 10
, Article 7.
Available at: http://scholar.rose-hulman.edu/rhumj/vol10/iss1/7