In algebraic geometry, the sinusoidal spirals are a family of curves defined by the equation in polar coordinates
rn=an\cos(n\theta)
where is a nonzero constant and is a rational number other than 0. With a rotation about the origin, this can also be written
rn=an\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:
The curves were first studied by Colin Maclaurin.
Differentiating
rn=an\cos(n\theta)
| dr | |
| d\theta |
\cosn\theta+r\sinn\theta=0.
Then
| \left( | dr | , r |
| ds |
| d\theta | |
| ds |
\right)\cosn\theta
| ds | |
| d\theta |
=\left(-r\sinn\theta, r\cosn\theta\right) =r\left(-\sinn\theta, \cosn\theta\right)
\psi=n\theta\pm\pi/2
\varphi=(n+1)\theta\pm\pi/2.
The unit tangent vector,
| \left( | dr | , r |
| ds |
| d\theta | |
| ds |
\right),
| ds | |
| d\theta |
=r\cos-1n\theta=a\cos-1+\tfrac{1{n}}n\theta.
n>0
a\int-\tfrac{\pi{2n}}\tfrac{\pi{2n}}\cos-1+\tfrac{1{n}}n\theta d\theta
The curvature is given by
| d\varphi | |
| ds |
=(n+1)
| d\theta | |
| ds |
=
| n+1 | |
| a |
\cos1-\tfrac{1{n}}n\theta.
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.