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.

Author Bio

Timothy will graduate from Calvin College in May 2010 with a major in Mathematics and a minor in Computer Science. He currently lives in Grand Rapids, Michigan and attends college at Calvin College. He intends to attend graduate school and pursue a doctorate in Mathematics upon graduation from Calvin. He enjoys running, volunteering at church, and spending time with family and friends.

Landon Kavlie is a senior mathematics and computer science major at Calvin College in Grand Rapids, Michigan. Originally from Bismarck, North Dakota, Landon will be graduating in Spring 2010 after which he plans to pursue a doctorate in mathematics. In his free time, Landon enjoys lifting weights, playing softball, and spending time with friends.