Lemniscate of Bernoulli

The Lemniscate of Bernoulli is a figure-eight-shaped curve defined as the set of points where the product of the distances to two fixed points (its foci), a distance $2c$ apart, stays constant and equal to $c^2$ — a close cousin of the ellipse, which is instead defined by a constant sum of distances to its two foci. Jakob Bernoulli described it in 1694 while studying a variation on the ellipse, apparently without realizing it was a special case of the more general Cassini ovals his colleague Giovanni Domenico Cassini had already studied a few years earlier. The lemniscate's arc length also gives rise to the lemniscate constant, a close analogue of $\pi$ for this curve. In polar coordinates, with $a^2=2c^2$, it follows the equation:

$r^2 = a^2\cos(2\theta)$