Trisectrix of Maclaurin

The Trisectrix of Maclaurin, described by Colin Maclaurin in 1742, is built from two lines through two fixed points a distance $2a$ apart on the x-axis: one line rotates at angle $\theta$ around the left-hand point, while the other rotates three times as fast, at angle $3\theta$, around the right-hand point. As $\theta$ varies, the intersection of the two lines traces out this curve — and that exact 1:3 relationship is what makes it useful for angle trisection: given an angle to trisect, the curve (together with a short auxiliary construction) lets you recover one-third of it directly with a straightedge. Working through the geometry of the two rotating lines gives a clean pair of parametric equations:

$x(\theta) = 2a\cos(2\theta)$

$y(\theta) = 2a\tan(\theta)\cos(2\theta)$