# 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

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

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.,

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

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

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

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

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

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

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

## References

1. ^ Nathaniel Grossman (1996). The sheer joy of celestial mechanics. Springer. p. 34. ISBN 978-0-8176-3832-0.