In geometry, the **sinusoidal spirals** are a family of curves defined by the equation in polar coordinates

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

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

and eliminating *a* produces a differential equation for *r* and θ:

- .

Then

which implies that the polar tangential angle is

and so the tangential angle is

- .

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

The unit tangent vector,

- ,

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

- .

In particular, the length of a single loop when is:

The curvature is given by

- .

## 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.

Wikimedia Commons has media related to .Sinusoidal spiral |

## 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".
*MathWorld*.