  We will establish several arithmetic and geometric properties regarding the
bi-sequences of approximation coefficients (BAC) associated with the two
one-parameter families of piecewise-continuous Mobius transformations
introduced by Haas and Molnar. The Gauss and Renyi maps, which lead to the
expansions of irrational numbers on the interval as regular and backwards
continued fractions, are realized as special cases. The results are natural
generalizations of theorems from Diophantine approximation.
