# Are Bezier curves invariant under conformal mapping?

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)

• Could you give some examples, I find the claim suspicious, but haven't experimented with it. In particular, if one thinks about the control polygon, most conformal maps wouldn't preserve the straight lines of the control polygon. If you can understand the behavior of the control polygon, perhaps you will have some success. Commented Nov 29, 2015 at 15:49
• A Bezier curve cannot lie on a sphere unless it degnerates to a point. Commented Dec 3, 2015 at 13:57

• (+1) Affine linear transformation are the most general such transformations: If $T$ is not affine, pick $q_{1}$, $q_{2}$ so that the segment from $q_{1}$ to $q_{2}$ is not mapped into a line, and consider the control points $p_{1} = q_{1}$, $p_{2} = p_{3} = p_{4} = q_{2}$. Commented Nov 29, 2015 at 16:23