Sinusoidal spiral

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:
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 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 formula_10 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.