Sinusoidal Spirals

Polar equation:
rp=apcos(pθ)r^{p} = a^{p} \cos(p \theta )


Sinusoidal spirals can have any rational number pp in the formula above. Many standard curves occur as sinusoidal spirals.

If p=1p = -1 we have a line.

If p=1p = 1 we have a circle.

If p=12p = \large\frac{1}{2}\normalsize we have a cardioid.

If p=12p = -\large\frac{1}{2}\normalsize we have a parabola.

If p=2p = -2 we have a hyperbola.

If p=2p = 2 we have a lemniscate of Bernoulli.

Sinusoidal spirals were first studied by Maclaurin.
They are not, of course, true spirals.

The pedal curve of sinusoidal spirals, when the pedal point is the pole, is another sinusoidal spiral.

The sinusoidal spiral rp=apcos(pθ)r^{p} = a^{p} \cos(p\theta) inverts to rp=ap/cos(pθ)r^{p} = a^{p}/\cos(p\theta) if the centre of inversion is taken at the pole.