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sinusoidal spiral |
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:
a |
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.