Tusi couple

The Tusi couple is a geometric mechanism in which two circular motions combine to produce straight-line oscillation. Nasir al-Din al-Tusi devised it in the 13th century as a device for modeling planetary motion.

Epicycle: A small circle whose center moves along the circumference of a larger circle in a geometric model of motion. The Tusi couple uses nested circular motion, though its paired circles generate a line rather than a looping path.
Nasir al-Din al-Tusi: A 13th-century Persian astronomer, mathematician, and philosopher who directed the Maragheh observatory. He devised the mechanism and incorporated it into his account of celestial motions.
Planetary model: A mathematical representation of the positions and motions of planets over time. The Tusi couple supplied a way to represent nonuniform-looking motion through uniform rotations.
Eccentric circle: A circle whose center is displaced from the center of the system used to describe its motion. Eccentric circles model apparent variation through displaced centers, unlike the couple's paired rotations.
Ibn al-Shatir: A 14th-century Syrian astronomer whose planetary models used combinations of circular motions. His models include mathematical constructions related to techniques associated with al-Tusi.
Deferent: The larger circle in an ancient geocentric planetary model, along which an epicycle's center moves. Together with the epicycle, it supplies the circular-motion framework the couple reconfigures.
Maragheh observatory: A major 13th-century astronomical observatory established near Maragheh in northwestern Iran. Al-Tusi worked there amid mathematical efforts that produced new planetary models.
Lunar theory: Mathematical accounts of the Moon's changing position and motion relative to Earth and the stars. Al-Tusi's use of the couple helped model lunar motions while preserving circular components.
Equant: A point in Ptolemaic astronomy from which a planet's angular motion along its deferent appears uniform. The equant handles irregular planetary speed by a measuring point, not by converting rotations into a line.
Copernican heliocentrism: The astronomical system in which Earth and the other planets orbit the Sun. Similar mathematical devices appear in Copernicus's planetary models, though the route of transmission remains debated.

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