Nasir al-Din al-Tusi

Nasir al-Din al-Tusi developed geometric constructions for mathematical astronomy, including the Tusi couple. In its planar form, two equal-radius uniform circular motions with opposite angular rates can produce a straight oscillating resultant.

For example, adding the planar vectors $R(\cos t,\sin t)$ and $R(\cos t,-\sin t)$ gives $(2R\cos t,0)$. The transverse components cancel while the remaining coordinate oscillates. This supplies a useful building block for representing astronomical motion through circular components. The planar identity should be distinguished from spherical variants, which require their own geometric argument.^[ref]