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."

Now that we know that our circular map is in fact a "line-ear" map, i.e. maps lines to lines, it is natural to ask which maps are line-ear. In fact these are exactly the affine maps (linear map plus translation), which is relatively easy to proof. The only circle-preserving affine maps are those that are angle-preserving ones, i.e. combinations of translations, rotations, magnifications and reflections. This confirms Steve's suspicion that there are not so many interesting circular maps.

Posted by JLo on 2006-12-21 13:36:50