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 if and only if:
- $a = 2 b$
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:
\(\ds x\) | \(=\) | \(\ds \paren {2 b - b} \cos \theta + b \map \cos {\paren {\dfrac {2 b - b} b} \theta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds b \cos \theta + b \cos \theta\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 b \cos \theta\) |
Thus the $x$ coordinate of the $2$ cusp hypocycloid has a range $\closedint {-b} b$.
Similarly:
\(\ds y\) | \(=\) | \(\ds \paren {2 b - b} \sin \theta - b \map \sin {\paren {\dfrac {2 b - b} b} \theta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds b \sin \theta - b \sin \theta\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
Thus the $y$ coordinate of the $2$ cusp hypocycloid is fixed at $y = 0$.
Thus the $2$ cusp hypocycloid consists of the line segment:
- $x \in \closedint {-b} b, y = 0$.
which is a diameter of the containing circle.
$\blacksquare$
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.21$: The Cycloid: Problem $9$