Are Bezier curves invariant under conformal mapping?

Question

I've spent quite a bit of time on google trying to find information on whether or not Bezier curves are invariant under conformal mapping (i.e. a conformal mapping of all points on the curve is the same as a curve formed with just the conformal mapped control points for the curve). It feels like this should be true, but I can't seem to find anything that either proves or disproves this.

Does anyone know whether Bezier curves are invariant under conformal mapping?

Edit: based on Hagen's observation of a straight line becoming a circle, it no longer feels like this "should" be true! Although that does raise the question "which conformal transforms are also affine transforms", which I'm also having a bit of trouble googling (although I did find http://www.leptonica.com/affine.html)

Answer 1

No. For example a straight line (a Bezier curve) might be conformally mapped to a circle (a non-Bezier curve).

However, Bezier curves are invariant under affine linear transformations.

Answer 2

The straight line is usually give as a counter example for this question. However, I think that the straight line is not a proper Bezier curve but a degenerate one. To resolve this, simply break up the line into smaller segments, and you will have circles and all other shapes. Smaller the line segments, better the final result.