Rose Curve

A rose curve (or rhodonea curve, from the Greek word for rose) is a polar curve that traces a pattern of petals radiating from the origin. For the equation $r = a\cos(k\theta)$, the number of petals depends on $k$: when $k$ is an odd integer, the curve has exactly $k$ petals; when $k$ is an even integer, it has $2k$ petals, because the curve has to sweep all the way around twice before it closes on itself. When $k$ is a non-integer rational number, the curve still eventually closes but produces overlapping petals of uneven size — a striking example of how much a single parameter can change a polar curve's appearance. The equation is:

$r = a\cos(k\theta)$