sinusoidal spiral


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.

For many values of a the spiral coincides with another curve:


- 2 hyperbola
- 1 line
- 1/2 parabola
- 1/3 Tschirnhausen's cubic
   0 logarithmic spiral
   1/3 Cayley's sextic
   1/2 cardioid
   1 circle
   2 lemniscate of Bernoulli
  3 Kiepert's curve

The smaller parameter a the more the curve has a spiral form.


There are several operations that keep the sinusoidal spiral intact:

  • the isoptic of a sinusoidal spiral is again a sinusoidal spiral
  • it is easy to see that the polar inverse of a sinusoidal spiral with parameter a is the sinusoidal spiral with parameter -a
  • the pedal of a sinusoidal spiral (with the center as pedal point) is again a sinusoidal spiral

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.