Previously I made the assertion to you that if f is a mapping from a Euclidean space E^n into a Euclidean space E^m which maps straight lines into straight lines and whose range has three points in general position, then it follows that f is an affine transformation.
As Guowu Yao of Tsinghau University has pointed out to me this is false. For consider the case n=m=2 and f acts as the identity on a straight line and collapses the rest of the plane to a point not on the line.
However I believe that I can fix this. For example, I think that I can prove that if f is a mapping from a Euclidean space E^n into E^n, which maps straight lines into straight lines, and whose range contains n+2 distinct points any n+1 of which are in general position, then f is an affine transformation.
I hope to publish this sometime soon. I will keep you posted on the details.