Geometry of Analytic Functions

chaoseverlasting

There's an example mapping [tex]w=z^2[/tex] in sec.12.5 p674, Kreyzig. In the example, two planes (w-plane and the z-plane) are used. As w=f(z), z is mapped onto w. Now, at first, polar coordinates are used to map this region (concentric circles with [tex]R=r^2[/tex] and [tex]\phi =2\theta[/tex]). Next, cartesian coordinates are used where the axes u and v are defined.

Now, from the function, [tex]u=Re(z^2)=x^2-y^2[/tex] and [tex]v=Im(z^2)=2xy[/tex]. So far, everything is clear. The book now says that 'vertical lines x=c are mapped onto u and hence y is eliminated'. What does this mean? I understand that using c as a parameter, you can eliminate y and write v as a function of u, but why?