How to get the edges of a duoprism?

Question

Let $P_1$ and $P_2$ be two polygons, and $V_1$, $V_2$, their respective sets of vertices. Then the set of vertices of the 4-dimensional duoprism $D$ formed by the Cartesian product of $P_1$ and $P_2$ is $V=\{(x,y,u,v) \mid (x,y) \in V_1, (u,v) \in V_2\}$. But how to get the edges of $D$? Currently, for convex polygons $P_1$ and $P_2$, then $D$ is convex and then I'm using a program to get the convex hull of $V$, and which provides the edges. Isn't there a straightforward mathematical way to get the edges?

Answer 1

Why, of course there is. Say, the vertices $(x_1,y_1)$ and $(x_2,y_2)\in V_1$ are joined by an edge, and so are $(u_1,v_1),\;(u_2,v_2)\in V_2$. Then the duoprism will have the following edges: $$\begin{array}{ccc} (x_1,y_1,u_1,v_1) & - & (x_1,y_1,u_2,v_2)\\ |& &| \\ (x_2,y_2,u_1,v_1) & - & (x_2,y_2,u_2,v_2)\\ \end{array} $$