Let f be a one-to-one correspondence of the points in a plane. Prove or disprove the following statement:

"If f maps circles to circles, then it maps straight lines to straight lines."

(In reply to Not a proof, but ... by Steve Herman)

Well you could imagine that the function f is based on all the circles that go through (0,1) and (0,-1).  f might map each of these circles to the circle that perpendicularly intersects the original at (0,1) and (0,-1).  The problem with this function is that the unit circle is mapped to a line, so it doesn't count.

Actually, even then, I imagine some discontinuities are required, since to map the unit circle to a line, you have to map the half-closed interval [0,2pi) to the open interval (-∞,∞).

Posted by Tristan on 2006-11-18 14:49:20