Stereographic Projection: Does Conformality Imply Circle Preservation?

Question

This may be a silly question. I have learned to prove that circles are preserved at the infinitesimal scale, however, does this ALONE imply that circles are mapped as circles for the stereographic projection? Why/Why not.

Answer 1

Consider the map $z \mapsto z^2$, which is conformal on the punctured plane $\mathbb{C} \setminus \{0\}$. A circle in this domain is $C : |z-2| = 1$. Suppose $z \in C$, so $z = 2+e^{i \theta}$ for $\theta \in [0, 2\pi)$. We have

$$z^2 = (2+e^{i \theta})^2= 4 + 4e^{i \theta} + e^{2i \theta},$$

which is clearly not a circle. So the answer is no, conformality alone does not imply that circles are preserved. Conformality means that the angle between contours are preserved. So for example, if you have two orthogonal circles, they will be mapped to two orthogonal contours under a conformal map, albeit not necessarily to two new circles.

However, circles are preserved under stereographic projection, or more generally circles and lines are mapped to circles under the steoreographic projection, which can be shown by simply applying the formula for the steoreographic projection to a generalized circle.