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 2-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

Some modern commentators also call the Tusi couple a "rolling device" and describe it as a small circle rolling inside a large fixed circle. However, Tusi himself described it differently:

Relation to al-Tusi's astronomical theories

, born in the town of Tus, Iran in 1201, is acknowledged throughout the Islamic world as one of the 'Great Wisdoms'. Tusi was the first astronomer to attempt a solution which would provide for latitudinal motion without introducing a longitudinal component. To do so, he proposed in a work called Tahrir al-Majisti, which was completed in 1247, that the oscillatory motion be produced by the combined uniform circular motions of two identical circles, one riding on the circumference of the other. At that point, Tusi simply states that if one of these circles were to move at a uniform speed equal to twice the speed of the other, and in a direction to it, then any point on the circumference of the first circle would oscillate in a straight line along one of the diameters of the second 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.

Hypotrochoid

A property of the Tusi couple is that points on the inner circle that are not on the circumference trace ellipses. These ellipses, and the straight line traced by the classic Tusi couple, are special cases of hypotrochoids.