Tusi Couple is Diameter of Deferent

Theorem
A Tusi couple is a degenerate case of the hypocycloid whose form is a straight line that forms a diameter of the deferent.

Proof
Let $C_1$ be a circle of radius $b$ rolling without slipping around the inside of a circle $C_2$ of radius $a$.

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

Let $P$ be a point on the circumference of $C_1$.

Let $C_1$ be initially positioned so that $P$ is its point of tangency to $C_2$, located at point $A = \tuple {a, 0}$ on the $x$-axis.

Let $H$ be the hypocycloid formed by the locus of $P$.

From Number of Cusps of Hypocycloid from Integral Ratio of Circle Radii we have that $H$ will have $2$ cusps :


 * $a = 2 b$

By Equation of Hypocycloid a hypocycloid can be expressed in parametric form as:


 * $x = \paren {a - b} \cos \theta + b \map \cos {\paren {\dfrac {a - b} b} \theta}$
 * $y = \paren {a - b} \sin \theta - b \map \sin {\paren {\dfrac {a - b} b} \theta}$

Hence:

Thus the $x$ coordinate of the $2$ cusp hypocycloid has a range $\closedint {-b} b$.

Similarly:

Thus the $y$ coordinate of the $2$ cusp hypocycloid is fixed at $y = 0$.

Thus the $2$ cusp hypocycloid consists of the line segment:
 * $x \in \closedint {-b} b, y = 0$.

which is a diameter of the containing circle.