238 CHAPTER 7. PLANAR ORTHOGONAL AND POLYLINE DRAWINGS
for the external face. Again, since each uni t corresponds to π/2 radians, we know the sum
of these angles is equal to π(d(f ) − 2) for an internal face and π(d(f) + 2) for the external
face. Thus, each face is pr operly closed, and we can se e that any valid flow φ on the network
corresponds to a proper orthogonal shape for G.
We now interpret the cost associated with a specific flow. For arcs of type (s, v) the cost
is 1 and the flow is fixed. So, for this case, the total cost is exactly 4n. Similarly, all the
arcs of type (f, t) have cost that sum to exactly 4n. Since all the arc s of type (v, f) have to
release the commodity sent from the source s, we know that the sum of these arcs is also
4n