
Re: Mapping from straight line to straight line
Posted:
Nov 2, 2013 10:28 PM


Timothy Murphy wrote:
>> If L is a bijective continuous mapping from R^n to R^n that maps every >> straight line to straight line and L(0)=0, then L is linear. (P) >> >> If P is true, how to prove it? If P is false, what is a counter example? > > I believe it is true if n>=2. > (It is not true if n=1.) > Roughly speaking, choose coordinates on a line so each point P > is defined by x in R, say P = P(x). > Then I think you can find constructions in 2 dimensions, using just lines, > to define P(x+y) and P(xy). > It follows that your bijection defines an automorphism of R as a field. > It is easy to show that if this is continuous it is linear. > I think the result follows from this.
The argument can be put more simply as follows. It is easy to construct the midpoint C of two points A,B using only straight lines. (Eg take any point P not on the line AB, and take any line l parallel to AB not going through P or A. Suppose the line cuts AP,BP at E,F. Let EB,FA meet in X. Then PX cuts AB in C. This is the standard construction of 4 points with crossratio 1, the 4th point in this case being the point where AB meets the line at infinity.) By the same argument, given A,B we can construct the point D such that B is the midpoint of AD, etc.
It follows that a bijective map sending straight lines into straight lines must preserve midpoints. It is evident that this will give a dense set of points on the line AB which must be mapped into corresponding points on any line A'B'.
 Timothy Murphy email: gayleard /at/ eircom.net School of Mathematics, Trinity College, Dublin 2, Ireland

