Tusi couple ($R(e^{i\theta }+e^{-i\theta })=2R\cos \theta$)

Two uniform circular motions of radius $R$ and opposite angular rates sum to rectilinear oscillation: $R(\cos \theta ,\sin \theta )+R(\cos \theta ,-\sin \theta )=(2R\cos \theta ,0)$. Geometrically, a circle of radius $R$ rolls inside one of radius $2R$; a marked point traces a diameter. This plane construction must be distinguished from spherical variants, which need their own geometric argument.^[ref]