# Cotes's spiral

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In physics and in the mathematics of plane curves, Cotes's spiral (also written Cotes' spiral and Cotes spiral) is a spiral that is typically written in one of three forms

$\frac{1}{r} = A \cos\left( k\theta + \varepsilon \right)$
$\frac{1}{r} = A \cosh\left( k\theta + \varepsilon \right)$
$\frac{1}{r} = A \theta + \varepsilon$

where r and θ are the radius and azimuthal angle in a polar coordinate system, respectively, and A, k and ε are arbitrary real number constants. These spirals are named after Roger Cotes. The first form corresponds to an epispiral, and the second to one of Poinsot's spirals; the third form corresponds to a hyperbolic spiral, also known as a reciprocal spiral, which is sometimes not counted as a Cotes's spiral.[1]

The significance of Cotes's spirals for physics is in the field of classical mechanics. These spirals are the solutions for the motion of a particle moving under an inverse-cube central force, e.g.,

$F(r) = \frac{\mu}{r^3}$

where μ is any real number constant. A central force is one that depends only on the distance r between the moving particle and a point fixed in space, the center. In this case, the constant k of the spiral can be determined from μ and the areal velocity of the particle h by the formula

$k^{2} = 1 - \frac{\mu}{h^2}$

when μ < h 2 (cosine form of the spiral) and

$k^{2} = \frac{\mu}{h^2} - 1$

when μ > h 2 (hyperbolic cosine form of the spiral). When μ = h 2 exactly, the particle follows the third form of the spiral

$\frac{1}{r} = A \theta + \varepsilon.$