Witch of Agnesi

The Witch of Agnesi is a bell-shaped curve named after Maria Gaetana Agnesi, who described it in her 1748 calculus textbook Instituzioni Analitiche — the name itself comes from a translation mix-up: Agnesi called the curve versiera (from the Latin for "to turn"), which an early English translator confused with avversiera ("witch" or "she-devil"), and the mistranslation stuck. It is generated from a circle of radius $a$, centered at $(0,a)$ and tangent to the x-axis at the origin: draw a line from the origin at angle $\theta$; that line crosses the circle at one point and the horizontal line $y=2a$ at another, and the Witch of Agnesi curve traces the point built from the second crossing's x-coordinate and the first crossing's y-coordinate. The result is a smooth, symmetric bump that flattens toward zero on both sides — and, scaled and centered, it is exactly the shape of a Cauchy distribution's probability density function. Its Cartesian equation is:

$y = \dfrac{8a^3}{x^2+4a^2}$