At a time when satellites and computers have accustomed us to the most detailed and accurate maps, transportation networks, particularly subways, seem to be one of the last refuges for an almost medieval mapmaking sensibility. Yes, not all subway maps have a little drawing of a gothic cathedral smack in the middle like the diagram of the Vienna U-Bahn, but many subway networks, from Tokyo to Buenos Aires, relish in the reductive geometries that remind you more of the "mappa mundi" of the 1300s than of the excruciatingly precise maps of the 21st century.
And like the medieval maps drawn within the capricious limits of animal-skin parchments, the Vienna U-Bahn Linienplan has to fit within the extremely elongated rectangles above the subway-car doors. How do you do that? You have to really understand the city to pull it off.
With the Stephansdom at the center, the U2 (violet) loop from Karlsplatz to Schottenring establishes the path of the Ringstrasse with surprising ease. The central portion of the U4 (green) line completes the Ring, before paralleling the Donaukanal from Landestrasse all the way north to Heilingenstadt. The right portion of the U3 (orange) line parallels the Danube to the southeast. Further west (to the left in the map) the larger loop of the U6 (brown) line describes the Gürtel--the second concentric ring, or belt-road of Vienna--before continuing south (to the left) and north (to the right) across the Danube. Yes, once the core of the city is established, the lines extend left and right to fit the constrains of the long rectangle.
(By the way, why isn't there a U5 line?)