The map from the movie shows the continent of Antartica. While that continent had been discovered decades earlier, it had not been fully explored by 1862 and it was not known to be a continent at that time. It was still thought that it might be a collection of islands somewhat like the North Polar region.

The map from the Broadway production is an American map where the American continents are in the middle of the map and the the Europe/Asian continents are split at the sides. Anna almost certainly would have had the King purchase a map where the Eurasian continents were not split, as in the movie map.