Simple Harmonic Motion of Point on Tusi Couple

Theorem
Let $C_1$ and $C_2$ be the epicycle and deferent respectively of a Tusi couple $H$.

Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin.

Let the center of $C_1$ move at a constant angular velocity $\omega$ around the center of $C_2$.

Let $P$ be the point on the circumference of $C_1$ whose locus is $H$.

Let $C_1$ be initially positioned so that $P$ its point of tangency to $C_2$, located on the $x$-axis.

Then $P$ moves back and forward on the $x$-axis with simple harmonic motion with period $\dfrac {2 \pi} \omega$ and maximum speed $a \omega$.