  We compare two families of continued fractions algorithms, the symmetrized
Rosen algorithm and the Veech algorithm. Each of these algorithms expands real
numbers in terms of certain algebraic integers. We give explicit models of the
natural extension of the maps associated with these algorithms; prove that
these natural extensions are in fact conjugate to the first return map of the
geodesic flow on a related surface; and, deduce that, up to a conjugacy, almost
every real number has an infinite number of common approximants for both
algorithms.
