Tusi Couple is Diameter of Deferent

Theorem
A Cusi couple is a straight line that forms a diameter of the containing circle.

Thus the diameter of the containing circle is a degenerate case of the hypocycloid.

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

Let $C_2$ be embedded in a cartesian coordinate 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 $A = \left({a, 0}\right)$ 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:


 * $x = \left({a - b}\right) \cos \theta + b \cos \left({\left({\dfrac {a - b} b}\right) \theta}\right)$
 * $y = \left({a - b}\right) \sin \theta - b \sin \left({\left({\dfrac {a - b} b}\right) \theta}\right)$

Hence:

Thus the $x$ coordinate of the $2$ cusp hypocycloid has a range $\left[{-b \,.\,.\, b}\right]$.

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:
 * $x \in \left[{-b \,.\,.\, b}\right], y = 0$.

which is a diameter of the containing circle.