  Nagata's famous counterexample to Hilbert's fourteenth problem shows that the
ring of invariants of an algebraic group action on an affine algebraic variety
is not always finitely generated. In some sense, however, invariant rings are
not far from affine. Indeed, invariant rings are always quasi-affine, and there
always exist finite separating sets. In this paper, we give a new method for
finding a quasi-affine variety on which the ring of regular functions is equal
to a given invariant ring, and we give a criterion to recognize separating
algebras. The method and criterion are used on some known examples and in a new
construction.
