Problem: Let $\{A_r\},\{B_r\},\{C_r\}$ each be families of (determinant 1) 2x2 matrices in $SL(2,\mathbb{R})$ such that each family is continuously indexed by a parameter $0
Progress thus far: The indexing is sufficiently complicated that simply multiplying and looking for a pattern has not been successful. Each entry of each family is a ratio of quadratics in $r$, and I do not remember the right computational linear algebra to attack this problem.
Motivation: the families of matrices represent isometries of the hyperbolic plane, and I am interested in knowing when such isometries change from elliptic to parabolic to hyperbolic