Sinusoidal spiral

Equations

Differentiating :r^n = a^n \cos(n \theta), and eliminating a produces a differential equation for r and θ: :\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 nθ 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 n0 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 constant is a sinusoidal spiral. In this case the sinusoidal spiral is a polynomial lemniscate.

References

• Yates, R. C.: A Handbook on Curves and Their Properties, J. W. Edwards (1952), "Spiral" p. 213–214
• "Sinusoidal spiral" at www.2dcurves.com
• "Sinusoidal Spirals" at The MacTutor History of Mathematics