Euler polygon division algorithm I am trying to enumerate and display all possible divisions of an n-polygon into (n-2) triangles (as in the following mathworld picture)

I know how to count how many divisions (Catalan numbers, 1, 2, 5, 14, 42, ... ), but I am struggling to find a way to generate all the C(n-2) triangulations. Can anyone point to existing software implementation, or at least to an algorithm that generates these triangles?
See: Hurtado and Noy, Graph of triangulations of a convex polygon and tree of triangulations, Computational Geometry 13 (1999) pp179–188.
Section 3 of that paper gives a relatively straightforward method for constructing all triangulations of $n+1$-gons given all triangulations of $n$-gons.