In this presentation, we consider the problem of characterizing maps that preserve pairs of right hyperbolas or lines in the extended double plane whose hyperbolic angle of intersection is zero. We consider two disjoint spaces of right hyperbolas and lines in the extended double plane $\mathscr{H}^+$ and $\mathscr{H}^-$ and prove that bijective mappings on the respective spaces that preserve tangency between pairs of hyperbolas or lines must be induced by a linear fractional transformation.

Author Bio

Joshua Keilman is a senior at Calvin College in Grand Rapids, Michigan. He will graduate in May 2012 with a major in Mathematics and a minor in Computer Science, at which point he plans to begin a career in a related field. During his free time, Josh enjoys playing sports and spending time with close friends and family.

Andrew Mis graduated from Calvin College in May 2011 with a major in Mathematics and a minor in Classical Greek. He now seeks entry into a graduate program in order to aspire to a doctorate in Mathematics. During his free time, Andrew enjoys hiking, traveling and gardening; he also enjoys reading classical literature, poetry and dramas.