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  "body": "\n## Definition\n\nLet $K \\ge 1$ and let $\\mathbb{T} = \\set{z \\in \\C : \\abs{z} = 1}$ be the unit [[circle|060W5CBZP6]].\nA subset $\\Gamma$ of $\\C$ is a *$K$-quasicircle* if $\\Gamma = f(\\mathbb{T})$ for some $K$-[[quasiconformal mapping|A9J2KK4CEJ]] $f \\colon \\C \\to \\C$.\nIt is a *quasicircle* if it is a $K$-quasicircle for some $K \\ge 1$.\n\n## Remarks\n\nEvery quasicircle is a bounded Jordan curve, since $f$ is a homeomorphism.\nExample: an ellipse with semi-axes $a \\ge b \u003e 0$ 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.\nAhlfors' three-point condition gives an equivalent form.\nA bounded Jordan curve $\\Gamma$ is a quasicircle if and only if there is a constant $C$ with the following property.\nFor any two points $z_1, z_2 \\in \\Gamma$, the arc of $\\Gamma$ of smaller diameter between them has diameter at most $C \\abs{z_1 - z_2}$.\nNon-example: the cardioid $r = 1 - \\cos\\theta$ has a cusp at $0$.\nTake the two points of the cardioid at angles $\\pm\\theta$ near the cusp.\nTheir distance is of order $\\theta^3$, while the arc between them through the cusp has diameter of order $\\theta^2$.\nSo the three-point condition fails, and the cardioid is not a quasicircle.\nEvery $C^1$ Jordan curve and the Koch snowflake are quasicircles.\n",
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