Computing coordinates from distances is known in cartography as trilateration. If one takes road distances from a Rand McNally road atlas and uses these distances to compute the location of the places one can then interpolate the latitude/longitude graticule, and from this draw a map with state boundaries and coastlines. The resulting map projection illustrates the distortion introduced by the road system.
Furthermore, Tissot’s indicatrix can be used to calculate the angular and areal distortion, as well as the distance distortion, in every direction at each map location.