Sinusoidal spiral

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

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 in polar coordinates and their equivalents in rectangular 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 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

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. 213214
  • "Sinusoidal spiral" at www.2dcurves.com
  • "Sinusoidal Spirals" at The MacTutor History of Mathematics
  • Weisstein, Eric W. "Sinusoidal Spiral". MathWorld.
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