  Whitney's theorem states that every 3-connected planar graph is uniquely
embeddable on the sphere. On the other hand, it has many inequivalent
embeddings on another surface. We shall characterize structures of a
$3$-connected $3$-regular planar graph $G$ embedded on the projective-plane,
the torus and the Klein bottle, and give a one-to-one correspondence between
inequivalent embeddings of $G$ on each surface and some subgraphs of the dual
of $G$ embedded on the sphere. These results enable us to give explicit bounds
for the number of inequivalent embeddings of $G$ on each surface, and propose
effective algorithms for enumerating and counting these embeddings.
