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Topic: Mapping from straight line to straight line
Replies: 11   Last Post: Nov 3, 2013 4:35 PM

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Timothy Murphy

Posts: 495
Registered: 12/18/07
Re: Mapping from straight line to straight line
Posted: Nov 2, 2013 10:28 PM
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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 mid-point 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 cross-ratio -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 mid-point of AD, etc.

It follows that a bijective map sending straight lines into straight lines
must preserve mid-points.
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
e-mail: gayleard /at/ eircom.net
School of Mathematics, Trinity College, Dublin 2, Ireland




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