There's a theorem that A fractional linear transformation maps circles in the extended complex plane to circles. However, can Fractional linear transformation map collinear points to 'elliptically distributed points'? For instance, can I map the points $\{−1,0,1/2,1\}$ in ω plane to the points $\{−1, ir, −ir, 1\}$ in the z-plane with a real $r < 1$?
Thanks!
For the purposes of a linear fractional transformation a line is a type of circle (like a circle with infinite radius). So, lines are mapped to circles (or lines). So, one way to put this is that linear fractional transformations map generalized circles (lines or circles) to generalized circles.
Edit: To answer the comment, no, I don't believe so. In that example, the real axis (a generalized circle) would be mapped to an ellipse (not a generalized circle).