The NYC subway map - like most subway maps - isn't shown to scale. Is there a way to visually represent the distortion of the map in terms of area? More specifically, suppose you have the set of coordinates representing stations on the distorted map, $(x_1,y_1), ..., (x_n,y_n)$. Suppose you also have the set of coordinates for the original map, $(u_1,v_1), ..., (u_n, v_n)$.
Is there a good way to plot the distorted map and overlay a color-coded image which represents the localized area compression at each point? My only lead so far is to use something akin to Tissot's indicatrix to denote local compression. But this approach has some technical issues given a finite set of points. Any pointers would be greatly appreciated.
Edit:
I appreciate the suggestions so far but I really would like to create a completely colored map with the following simple properties: Darker colors indicate compressed distances and lighted colors indicate stretched distances. A really simple example is shown in the plot below. The points originally lie in a square but get transformed into a rhombus.
Take an image of an evenly spaced grid, and distort it such that each initial point moves to the corresponding final point. Then place the resulting image over the map and the density and shape of the lines will visualize the distortion.