Sinusoidal spirals

Colin Maclaurin first studied the family rⁿ = aⁿ cos nθ. The name misleads: they are not spirals, and many look like flowers. As n runs through the rationals the family takes in the rectangular hyperbola at n = -2, the parabola at -1/2, the cardioid at 1/2, the circle at 1 and the lemniscate of Bernoulli at 2. For a positive whole number n, the curve holds the points whose distances to n poles, evenly spaced on a circle, multiply to the same number.

rⁿ = aⁿ cos(nθ)

On mathcurve.com as Spirale sinusoïdale