Tusi couple
The Tusi couple is a mathematical device in which a small circle rotates inside a larger circle twice the diameter of the smaller circle. Rotations of the circles cause a point on the circumference of the smaller circle to oscillate back and forth in linear motion along a diameter of the larger circle. The Tusi couple is a two-cusped hypocycloid.
The couple was first proposed by the 13th-century Persian astronomer and mathematician Nasir al-Din al-Tusi in his 1247 Tahrir al-Majisti as a solution for the latitudinal motion of the inferior planets and later used extensively as a substitute for the equant introduced over a thousand years earlier in Ptolemy's Almagest.
Original description
The translation of the copy of Tusi's original description of his geometrical model alludes to at least one inversion of the model to be seen in the diagrams:Algebraically, the model can be expressed with complex numbers as
Other commentators have observed that the Tusi couple can be interpreted as a rolling curve where the rotation of the inner circle satisfies a no-slip condition as its tangent point moves along the fixed outer circle.
Later examples
Although the Tusi couple was developed within an astronomical context, later mathematicians and engineers developed similar versions of what came to be called hypocycloid straight-line mechanisms. The mathematician Gerolamo Cardano designed a system known as Cardan's movement. Nineteenth-century engineers James White, Matthew Murray, as well as later designers, developed practical applications of the hypocycloid straight-line mechanism.A practical and mechanically simple version of the Tusi couple, which avoids the use of an external rim gear, was developed in 2021 by John Goodman in order to provide linear motion. It uses 3 standard spur gears. A rotating arm is mounted on a central shaft, to which a fixed gear is mounted. A idler gear on the arm meshes with the fixed gear. A third gear meshes with the idler. The third gear has half the number of teeth of the fixed gear. An arm is fixed to the third gear. If the length of the arm equals the distance between the fixed and outer gears = d, the arm will describe a straight line of throw = 2d. An advantage of this design is that, if standard modulus gears that do not provide the required throw, the idler gear does not have to be colinear with the other two gears