a critical point where conformality fails

Conformal maps preserve angles — but only on the condition that the derivative never vanishes. So a natural question is: what happens at a spot where f'(z) actually does hit zero? Such a spot is called a critical point of the map. There, the angle-preserving spell breaks, and curves crossing through the point get their angle distorted in a very specific, predictable way: it gets multiplied.

Here is the mechanism. If f'(z_0) = 0 but f is not constant, then near z_0 the lowest surviving term in the Taylor expansion is of some order k at least 2: f(z) = f(z_0) + a_k (z - z_0)^k + (higher terms), with a_k not zero. Near such a point f behaves like the model map w = z^k, and z^k multiplies angles at the origin by k. Concretely, two curves that cross at z_0 with an angle theta between them have images crossing at angle k*theta. The integer k is the order of the critical point (it equals 1 plus the order of the zero of f' there). At an ordinary point f' is nonzero, k = 1, and angles are unchanged — the conformal case.

This is exactly why z -> z^2 opens a right angle at the origin into a straight angle: there k = 2, so the angle pi/2 becomes pi. The lesson is a warning attached to every conformal-mapping claim: to map one region conformally ONTO another you must avoid critical points, or the corners of your region get folded open (or, going the other way through a local inverse, folded shut). Critical points are also where a holomorphic map fails to be locally one-to-one, so they are precisely the obstacles to building a clean inverse map.