Sinusoidal spiral

Short description: Family of curves of the form r^n = a^n cos(nθ)

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 = −2)
• Line (n = −1)
• Parabola (n = −1/2)
• Tschirnhausen cubic (n = −1/3)
• Cayley's sextic (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 θ:

$\frac{dr}{d\theta }\cos n\theta +r\sin n\theta =0.$

Then

$(\frac{dr}{ds},r\frac{d\theta }{ds})\cos n\theta \frac{ds}{d\theta }=(-r\sin n\theta ,r\cos n\theta )=r(-\sin n\theta ,\cos n\theta )$

which implies that the polar tangential angle is

$\psi =n\theta \pm \pi /2$

and so the tangential angle is

$\phi =(n+1)\theta \pm \pi /2.$

(The sign here is positive if r and cos nθ have the same sign and negative otherwise.)

The unit tangent vector,

$(\frac{dr}{ds},r\frac{d\theta }{ds}),$

has length one, so comparing the magnitude of the vectors on each side of the above equation gives

$\frac{ds}{d\theta }=r{\cos }^{-1}n\theta =a{\cos }^{-1+\frac{1}{n}}n\theta .$

In particular, the length of a single loop when $n>0$ is:

$a\underset{-\frac{\pi }{2n}}{\overset{\frac{\pi }{2n}}{\int }}{\cos }^{-1+\frac{1}{n}}n\theta d\theta$

The substitution $r={\cos }^{\frac{1}{n}}(n\theta )$ induces the transformation

$\underset{0}{\overset{\frac{\pi }{2n}}{\int }}{\cos }^{-1+\frac{1}{n}}n\theta d\theta =\underset{0}{\overset{1}{\int }}\frac{dr}{\sqrt{1-r^{2n}}}$

Hence the length of a single loop of the spiral may be reexpressed as:

$2a\underset{0}{\overset{1}{\int }}\frac{dr}{\sqrt{1-r^{2n}}}$

The curvature is given by

$\frac{d\phi }{ds}=(n+1)\frac{d\theta }{ds}=\frac{n+1}{a}{\cos }^{1-\frac{1}{n}}n\theta .$

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 a positive integer, the sinusoidal spiral $r^n=\cos (n\theta )$ can be described as follows. Let $n$ points be arranged symmetrically on a circle of radius $\frac{a}{2^{\frac{1}{n}}}$ centered at the origin with one point on the positive $x$-axis. Then the set of points, so that the product of the distances from that point to these $n$ points is $\frac{a^n}{2}$, is precisely this sinusoidal spiral. Consequenly the sinusoidal spiral can be identified with the set of complex numbers $z$ satisfying the equation $|z^n-\frac{a^n}{2}|=\frac{a^n}{2}$. Thus such a spiral is a polynomial lemniscate.

References

• Loney, S. L.: An Elementary Treatise on the Dynamics of a Particle and of Rigid Bodies, Cambridge University Press (1913), p. 57–58
• 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
• Weisstein, Eric W.. "Sinusoidal Spiral". http://mathworld.wolfram.com/SinusoidalSpiral.html.