Sinusoidal spiral

{{short description|1=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 {{mvar|a}} is a nonzero constant and {{mvar|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 ({{math|1=n = −2}})
Line ({{math|1=n = −1}})
Parabola ({{math|1=n = −1/2}})
Tschirnhausen cubic ({{math|1=n = −1/3}})
Cayley's sextet ({{math|1=n = 1/3}})
Cardioid ({{math|1=n = 1/2}})
Circle ({{math|1=n = 1}})
Lemniscate of Bernoulli ({{math|1=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

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

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 $n>0$ 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.

References

Yates, R. C.: A Handbook on Curves and Their Properties, J. W. Edwards (1952), "Spiral" p. 213–214
[http://www.2dcurves.com/spiral/spirals.html "Sinusoidal spiral" at www.2dcurves.com]
[http://www-groups.dcs.st-and.ac.uk/~history/Curves/Sinusoidal.html "Sinusoidal Spirals" at The MacTutor History of Mathematics]
{{MathWorld |title=Sinusoidal Spiral |urlname=SinusoidalSpiral}}

Category:Plane curves

Category:Algebraic curves