Based on my undergraduate research, we will review basics of the real hyperbolic plane, *RH*^2 , as it is contained in the complex hyperbolic plane, *CH*^2 . We see that special matrix groups from algebra will help us find the metric on a cylinder/tube which surrounds *RH*^2 . The talk will be an overview of the different relationships between the hyperbolic planes and the rotations/translations which lead to the algebra.