# Function mapping a Hilbert plant to itself (Geometry)

1. Apr 7, 2009

### dismo

Let F: H ->H be a map of a Hilbert plane into itself. For any point A, denote F(A) by A`. Assume that AB is congruent to A'B' for any two points A,B.

How can I prove that this map is in fact a bijection?

In a arbitrary Hilbert plane, one can not be certain that square roots exist, so congruence of line segments is defined using the square of the normal distance function (i.e. dist^2=(x2-x1)^2+(y2-y1)^2.

To check if it is 1-1, I could use F(A)=F(P) if and only if A=P, but based on the small amount of info given about the map F. It seems like it would be possible to design a function that maps all the points in the plane onto a single line...

Also, I'm not clear on how I could prove onto...