 Solution

For a fixed plane embedding, place one dual vertex in each face of a plane graph and draw a dual edge across each original edge, joining its incident faces. These dual edges can be drawn without mutual crossings, so this constructs a plane dual graph. Each original vertex corresponds to a dual face of a plane graph, while each edge gives exactly one dual edge. HenceBridges give dual loops and parallel dual edges are allowed. On the sphere the dual of the resulting embedding is the original embedding, so planarity is preserved in both directions for this dual pair. The geometric dual graph is defined using an embedding; the converse is understood in that setting, rather than as a definition for an arbitrary nonplanar drawing.