  We consider the problem of generating all ideals of a poset. It is a long
standing open problem, whether or not the ideals of any poset can be generated
in constant amortized time, CAT for short. We refine the tree traversal, a
method introduced by Pruesse and Ruskey in 1993, to obtain a CAT-generator for
two large classes of posets: posets of interval dimension at most two and so
called locally planar posets. This includes all posets for which a
CAT-generator was known before. Posets of interval dimension at most two
generalize both, interval orders and 2-dimensional posets. Locally planar
posets generalize for example posets with a planar cover graph.
  We apply our results to CAT-generate all c-orientations of a planar graph. As
a special case this is a CAT-generator for many combinatorial objects like
domino and lozenge tilings, planar spanning trees, planar bipartite perfect
matchings, Schnyder woods, and others.
