Tusi Couple is Diameter of Stator

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Theorem

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


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 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 if and only if:

$a = 2 b$


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:

\(\displaystyle x\) \(=\) \(\displaystyle \left({2 b - b}\right) \cos \theta + b \cos \left({\left({\dfrac {2 b - b} b}\right) \theta}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle b \cos \theta + b \cos \theta\)
\(\displaystyle \) \(=\) \(\displaystyle 2 b \cos \theta\)

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


Similarly:

\(\displaystyle y\) \(=\) \(\displaystyle \left({2 b - b}\right) \sin \theta - b \sin \left({\left({\dfrac {2 b - b} b}\right) \theta}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle b \sin \theta - b \sin \theta\)
\(\displaystyle \) \(=\) \(\displaystyle 0\)


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.

$\blacksquare$


Sources