Sinusoidal spiral

From Wikipedia, the free encyclopedia
Jump to: navigation, search
Sinusoidal spirals: equilateral hyperbola (n = -2), line (n = -1), parabola (n = -1/2), cardioid (n = 1/2), circle (n = 1) and lemniscate of Bernoulli (n = 2), where rn = 1n cos(nθ) in polar coordinates and their equivalents in rectangular coordinates.

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[edit]

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 nθ 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[edit]

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 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[edit]