Sinusoidal spiral
In algebraic geometry, the sinusoidal spirals are a family of curves defined by the equation in polar coordinates
- [math]\displaystyle{ r^n = a^n \cos(n \theta)\, }[/math]
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
- [math]\displaystyle{ r^n = a^n \sin(n \theta).\, }[/math]
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 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
- [math]\displaystyle{ r^n = a^n \cos(n \theta)\, }[/math]
and eliminating a produces a differential equation for r and θ:
- [math]\displaystyle{ \frac{dr}{d\theta}\cos n\theta + r\sin n\theta =0 }[/math].
Then
- [math]\displaystyle{ \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) }[/math]
which implies that the polar tangential angle is
- [math]\displaystyle{ \psi = n\theta \pm \pi/2 }[/math]
and so the tangential angle is
- [math]\displaystyle{ \varphi = (n+1)\theta \pm \pi/2 }[/math].
(The sign here is positive if r and cos nθ have the same sign and negative otherwise.)
The unit tangent vector,
- [math]\displaystyle{ \left(\frac{dr}{ds},\ r\frac{d\theta}{ds}\right) }[/math],
has length one, so comparing the magnitude of the vectors on each side of the above equation gives
- [math]\displaystyle{ \frac{ds}{d\theta} = r \cos^{-1} n\theta = a \cos^{-1+\tfrac{1}{n}} n\theta }[/math].
In particular, the length of a single loop when [math]\displaystyle{ n\gt 0 }[/math] is:
- [math]\displaystyle{ a\int_{-\tfrac{\pi}{2n}}^{\tfrac{\pi}{2n}} \cos^{-1+\tfrac{1}{n}} n\theta\ d\theta }[/math]
The curvature is given by
- [math]\displaystyle{ \frac{d\varphi}{ds} = (n+1)\frac{d\theta}{ds} = \frac{n+1}{a} \cos^{1-\tfrac{1}{n}} n\theta }[/math].
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
- "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.
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