As I understand it, a transformation mapping is rather like looking at an existing mapping from a different point of view, or frame of reference; whereas a general mapping uses a function to map one set on to another. An example of a transformation mapping would be a photograph. The objects in 3-dimensional space can each be given a location with three parameters in relation to a fixed point of reference, while the photograph shows the same scene in two dimensions via an optical mapping on to a screen. Any 3-D point will have a corresponding mapping in 2-D in relation to the aperture of the camera photographing the scene. The mapping is merely the rules of optics applied mathematically. By contrast, a function like f(x)=x^2 or f:X→X^2 maps one set of points on to another using a rule defined by the function.