Cissoid of Diocles

The Cissoid of Diocles is a cubic curve first studied by the ancient Greek mathematician Diocles around 180 BC, who used it to solve the classical problem of doubling the cube (constructing a cube with twice the volume of a given cube) — one of the three famous construction problems of antiquity, alongside squaring the circle and trisecting an angle. It is generated from a circle of radius $a$ centered at $(a,0)$ and a vertical line tangent to the far side of that circle, at $x=2a$: for a line through the origin at angle $t$, the cissoid point is the difference between where that line crosses the circle and where it crosses the far vertical line. The curve has a cusp at the origin and creeps ever closer to the line $x=2a$ without ever reaching it — a vertical asymptote. Its parametric equations, in terms of $t$, are:

$x(t) = 2a\sin^2 t$

$y(t) = 2a\sin^3 t / \cos t$