The relation between the power of the radius and the angle φ has the
form of a sine. Some authors confine the parameter a to a rational number.
The smaller parameter a the more the curve has a spiral form.
There are several operations that keep the sinusoidal spiral intact:
It was the Scotsman Colin MacLaurin (1718) who was the first to study these group of curves.
The curve can be generalized to the Cassinian curve.