Cotes's spiral
In physics and in the mathematics of plane curves, a Cotes's spiral (also written Cotes' spiral and Cotes spiral) is one of a family of spirals classified by Roger Cotes. Cotes introduces his analysis of these curves as follows: “It is proposed to list the different types of trajectories which bodies can move along when acted on by centripetal forces in the inverse ratio of the cubes of their distances, proceeding from a given place, with given speed, and direction.” (N. b. he does not describe them as spirals). The shape of spirals in the family depends on the parameters. The curves in polar coordinates, (r, θ), r > 0 are defined by one of the following five equations: A > 0, k > 0 and ε are arbitrary real number constants. A determines the size, k determines the shape, and ε determines the angular position of the spiral. Cotes referred to the different forms as "cases". The equations of the curves above correspond respectively to his 5 cases. The Diagram shows representative examples of the different curves. The centre is marked by ‘O’ and the radius from O to the curve is shown when θ is zero. The value of ε is zero unless shown. The first and third forms are Poinsot's spirals; the second is the equiangular spiral; the fourth is the hyperbolic spiral (more correctly called by its alternative name: the "Reciprocal Spiral" since it has no connection with the hyperbola, or the hyperbolic functions which feature in the Poinsot's spirals); the fifth is the epispiral. For more information about their properties, reference should be made to the individual curves. (Wikipedia).
