Sinusoidal spiral

In algebraic 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.

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 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.

One path of a particle moving according to a central force proportional to a power of r is a sinusoidal spiral.