Quasicircle

Remarks

Every quasicircle is a bounded Jordan curve, since $f$ is a homeomorphism. Example: an ellipse with semi-axes $a \ge b > 0$ is the image of $\mathbb{T}$ under the real-linear map $z \mapsto \frac{a+b}{2} z + \frac{a-b}{2}\bar z$, which is $K$-quasiconformal with $K = a/b$; so an ellipse is a quasicircle. Ahlfors' three-point condition gives an equivalent form. A bounded Jordan curve $\Gamma$ is a quasicircle if and only if there is a constant $C$ with the following property. For any two points $z_1, z_2 \in \Gamma$, the arc of $\Gamma$ of smaller diameter between them has diameter at most $C \abs{z_1 - z_2}$. Non-example: the cardioid $r = 1 - \cos\theta$ has a cusp at $0$. Take the two points of the cardioid at angles $\pm\theta$ near the cusp. Their distance is of order $\theta^3$, while the arc between them through the cusp has diameter of order $\theta^2$. So the three-point condition fails, and the cardioid is not a quasicircle. Every $C^1$ Jordan curve and the Koch snowflake are quasicircles.