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Sinusoidal spiral

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Family of curves of the form r^n = a^n cos(nθ)
Sinusoidal spirals (r = –1 cos(nθ), θ = π/2) in polar coordinates and their equivalents in rectangular coordinates:   n = −2: Equilateral hyperbola   n = −1: Line   n = −1/2: Parabola   n = 1/2: Cardioid   n = 1: Circle   n = 2: Lemniscate of Bernoulli

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

r n = a n cos ⁡ ( n θ ) {\displaystyle r^{n}=a^{n}\cos(n\theta )\,}

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

r n = a n sin ⁡ ( n θ ) . {\displaystyle r^{n}=a^{n}\sin(n\theta ).\,}

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

r n = a n cos ⁡ ( n θ ) {\displaystyle r^{n}=a^{n}\cos(n\theta )\,}

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

d r d θ cos ⁡ n θ + r sin ⁡ n θ = 0. {\displaystyle {\frac {dr}{d\theta }}\cos n\theta +r\sin n\theta =0.}

Then

( d r d s ,   r d θ d s ) cos ⁡ n θ d s d θ = ( − r sin ⁡ n θ ,   r cos ⁡ n θ ) = r ( − sin ⁡ n θ ,   cos ⁡ n θ ) {\displaystyle \left({\frac {dr}{ds}},\ r{\frac {d\theta }{ds}}\right)\cos n\theta {\frac {ds}{d\theta }}=\left(-r\sin n\theta ,\ r\cos n\theta \right)=r\left(-\sin n\theta ,\ \cos n\theta \right)}

which implies that the polar tangential angle is

ψ = n θ ± π / 2 {\displaystyle \psi =n\theta \pm \pi /2}

and so the tangential angle is

φ = ( n + 1 ) θ ± π / 2. {\displaystyle \varphi =(n+1)\theta \pm \pi /2.}

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

The unit tangent vector,

( d r d s ,   r d θ d s ) , {\displaystyle \left({\frac {dr}{ds}},\ r{\frac {d\theta }{ds}}\right),}

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

d s d θ = r cos − 1 ⁡ n θ = a cos − 1 + 1 n ⁡ n θ . {\displaystyle {\frac {ds}{d\theta }}=r\cos ^{-1}n\theta =a\cos ^{-1+{\tfrac {1}{n}}}n\theta .}

In particular, the length of a single loop when n > 0 {\displaystyle n>0} is:

a ∫ − π 2 n π 2 n cos − 1 + 1 n ⁡ n θ   d θ {\displaystyle a\int _{-{\tfrac {\pi }{2n}}}^{\tfrac {\pi }{2n}}\cos ^{-1+{\tfrac {1}{n}}}n\theta \ d\theta }

The curvature is given by

d φ d s = ( n + 1 ) d θ d s = n + 1 a cos 1 − 1 n ⁡ n θ . {\displaystyle {\frac {d\varphi }{ds}}=(n+1){\frac {d\theta }{ds}}={\frac {n+1}{a}}\cos ^{1-{\tfrac {1}{n}}}n\theta .}

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

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