> 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.