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Sinusoidal spiral

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Sinusoidal spiral

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

Contents

r n = a n cos ⁡ ( n θ )

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 ) .

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:

  • Equilateral hyperbola (n = −2)
  • Line (n = −1)
  • Parabola (n = −1/2)
  • Tschirnhausen cubic (n = −1/3)
  • Cayley's sextet (n = 1/3)
  • Cardioid (n = 1/2)
  • Circle (n = 1)
  • Lemniscate of Bernoulli (n = 2)

    • The curves were first studied by Colin Maclaurin.

    Equations

    Differentiating

    r n = a n cos ⁡ ( n θ )

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

    d r d θ cos ⁡ n θ + r sin ⁡ n θ = 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 )

    which implies that the polar tangential angle is

    ψ = n θ ± π / 2

    and so the tangential angle is

    φ = ( n + 1 ) θ ± π / 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 ) ,

    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 .

    In particular, the length of a single loop when n > 0 is:

    a ∫ − π 2 n π 2 n cos − 1 + 1 n ⁡ n θ   d θ

    The curvature is given by

    d φ d s = ( n + 1 ) d θ d s = n + 1 a cos 1 − 1 n ⁡ n θ .

    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 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

    Sinusoidal spiral Wikipedia
    (Text) CC BY-SA