Consider a trapezoidal map used in point location algorithms in computational geometry, as follows:
Consider $n$ 2D line segments. Two line segments are said to be non crossing if their intersection is either empty or one of the end points. All the $n$ line segments are non crossing pairwise. Also no two distinct end points in the set of endpoints of those $n$ segments have same $x$ coordinate. Now enclose all the line segments in an axis parallel rectangle. Draw two vertical lines from each end point. Those verticals stop either at the outer rectangle or when they meet another segment. Create a vertex at each of such intersections. Each face in this planar graph is either a triangle or a trapezoid.
Suppose there were $n$ segments. Also, assume that the entire graph is inside an axis parallel to a rectangle. Now, according to the text Computational geometry, Algorithms and applications by Mark de Berg et al. the upper bound on the number of vertices is $6n+4$ . This I get. It's also stated there that the upper bound on the number of faces is $3n+1$ . The proof given there is not based on Euler's formula for planar graphs, but the authors hint that the bound can also be proved using the said formula by using the bound on the vertices.
Now Euler's formula for a planar graph is f >= 2+e-v. If e and v are not independent, it is not necessary that f reaches maximum when e reaches maximum and v reaches minimum. Even if they are independent, I don't get how upper bound on faces is attained when number of vertices reaches it's maximum. What am I missing here?