Sinusoidal spiral

In algebraic geometry, the sinusoidal spirals are a family of curves defined by the equation in polar coordinates


 * $$r^n = a^n \cos(n \theta)\,$$

where $a$ is a nonzero constant and $n$ is a rational number other than 0. With a rotation about the origin, this can also be written


 * $$r^n = a^n \sin(n \theta).\,$$

The term "spiral" is a misnomer, because they are not actually spirals, and often have a flower-like shape. Many well known curves are sinusoidal spirals including:
 * Rectangular hyperbola ($n = &minus;2$)
 * Line ($n = &minus;1$)
 * Parabola ($n = &minus;1/2$)
 * Tschirnhausen cubic ($n = &minus;1/3$)
 * Cayley's sextet ($n = 1/3$)
 * Cardioid ($n = 1/2$)
 * Circle ($n = 1$)
 * Lemniscate of Bernoulli ($n = 2$)

The curves were first studied by Colin Maclaurin.

Equations
Differentiating
 * $$r^n = a^n \cos(n \theta)\,$$

and eliminating a produces a differential equation for r and &theta;:
 * $$\frac{dr}{d\theta}\cos n\theta + r\sin n\theta =0. $$

Then
 * $$\left(\frac{dr}{ds},\ r\frac{d\theta}{ds}\right)\cos n\theta \frac{ds}{d\theta}

= \left(-r\sin n\theta ,\ r \cos n\theta \right) = r\left(-\sin n\theta ,\ \cos n\theta \right)$$ which implies that the polar tangential angle is
 * $$\psi = n\theta \pm \pi/2$$

and so the tangential angle is
 * $$\varphi = (n+1)\theta \pm \pi/2. $$

(The sign here is positive if r and cos n&theta; have the same sign and negative otherwise.)

The unit tangent vector,
 * $$\left(\frac{dr}{ds},\ r\frac{d\theta}{ds}\right),$$

has length one, so comparing the magnitude of the vectors on each side of the above equation gives
 * $$\frac{ds}{d\theta} = r \cos^{-1} n\theta = a \cos^{-1+\tfrac{1}{n}} n\theta. $$

In particular, the length of a single loop when $$n>0$$ is:
 * $$a\int_{-\tfrac{\pi}{2n}}^{\tfrac{\pi}{2n}} \cos^{-1+\tfrac{1}{n}} n\theta\ d\theta$$

The curvature is given by
 * $$\frac{d\varphi}{ds} = (n+1)\frac{d\theta}{ds} = \frac{n+1}{a} \cos^{1-\tfrac{1}{n}} n\theta. $$

Properties
The inverse of a sinusoidal spiral with respect to a circle with center at the origin is another sinusoidal spiral whose value of n is the negative of the original curve's value of n. For example, the inverse of the lemniscate of Bernoulli is a rectangular hyperbola.

The isoptic, pedal and negative pedal of a sinusoidal spiral are different sinusoidal spirals.

One path of a particle moving according to a central force proportional to a power of r is a sinusoidal spiral.

When n is an integer, and n points are arranged regularly on a circle of radius a, then the set of points so that the geometric mean of the distances from the point to the n points is a sinusoidal spiral. In this case the sinusoidal spiral is a polynomial lemniscate.