  We propose efficient algorithms for enumerating the notorious combinatorial
structures of maximal planar graphs, called canonical orderings and Schnyder
woods, and the related classical graph drawings by de Fraysseix, Pach, and
Pollack [Combinatorica, 1990] and by Schnyder [SODA, 1990], called canonical
drawings and Schnyder drawings, respectively. To this aim (i) we devise an
algorithm for enumerating special $e$-bipolar orientations of maximal planar
graphs, called canonical orientations; (ii) we establish bijections between
canonical orientations and canonical drawings, and between canonical
orientations and Schnyder drawings; and (iii) we exploit the known
correspondence between canonical orientations and canonical orderings, and the
known bijection between canonical orientations and Schnyder woods. All our
enumeration algorithms have $O(n)$ setup time, space usage, and delay between
any two consecutively listed outputs, for an $n$-vertex maximal planar graph.
