  In this paper we will use b-groups to construct coordinates for the
Teichm\"uller spaces of 2-orbifolds. The main technical tool is the
parametrization of triangle groups, which allows us to compute explicitly
formul\ae\ for generators of b-groups uniformizing orbifolds. In this way, we
obtain a technique to pass from the abstract objects of deformation spaces to
concrete calculations with M\"obius transformations. We explore this
computational character of our work by finding the expressions of certain
classical isomorphisms between Teichm\"uller spaces.
