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 Question for JLo by Bractals)

Your first interpretation is the good one; I assumed that the image of a line is a full line, not only a subset of one. More formally, if l is a line then so is f(l):={f(P)|P in l}. Same for circles. This is how the phrase "f maps X's to Y's" is commonly used (I hope!).

Posted by JLo on 2006-11-17 12:46:52