NetNews Usenet Archive 1992 #18/NN_1992_18.iso/spool/sci/math/10245

Internet Message Format | 1992-08-13 | 2.3 KB

Path: sparky!uunet!olivea!mintaka.lcs.mit.edu!zurich.ai.mit.edu!ara
From: [e-mail] (Allan Adler)
Newsgroups: sci.math
Subject: Re: Collineations
Message-ID: <[e-mail]>
Date: 13 Aug 92 23:07:15 GMT
References: <[e-mail]>
Sender: [e-mail]
Organization: M.I.T. Artificial Intelligence Lab.
Lines: 34
In-Reply-To: [e-mail]'s message of Wed, 12 Aug 1992 19:42:04 GMT

In article <[e-mail]> [e-mail] (David G Radcliffe) writes:

   Suppose f is a one-to-one function from the plane to itself
   which maps lines into lines, and suppose the image of f is
   not contained in a line.  Must f be affine?

   I have found some partial results.  The restriction of f to
   Q^2 is equal to the restriction of a projective transformation.
   If one also assumes that f is surjective, or continuous, or that
   it preserves betweenness, then f must be affine.

By the plane, I assume you mean R^2. In that case, yes, it must be an
affine transformation. By composing your function with a suitable
affine transformation, you can assume that it fixes the points (0,0),
(1,0) and (0,1). Since it must preserve parallel lines, there
are functions g(x) and h(y) such that f(x,y)=(g(x),h(y)). And then
the fact that the line y=x is mapped to itself, we have g=h.
Furthermore, g(0)=0 and g(1)=1. Therefore, f maps the line with
slope y through the origin (i.e. the line joining (0,0) and (1,y))
to the line with slope g(y) through the origin (i.e. the line joining
(0,0) and (1,g(y)). In particular, it maps (x,xy) to (g(x),g(x)g(y)),
which proves that g(xy)=g(x)g(y). Also, since f preserves parallograms,
we have g(x+y)=g(x)+g(y). Therefore, g is an automorphism of R.
It is easy to see that g fixes every rational. To prove that g fixes
every real, it is enough to show that it g is order preserving. But that
follows from the fact that a number is nonnegative if and only if it has
a square root. So g is the identity and therefore so is f.

If one relaces R by another field, the proof shows that f is the composition
of an affine transformation with an automorphism of the field applied to all
the coordinates. This works in n dimensional space for all n>1.

Allan Adler
[e-mail]