Length of Tangent to Astroid between Axes equals Radius of Stator
Theorem
Let $C_1$ be a circle of radius $b$ roll without slipping around the inside of a circle $C_2$ of radius $a = 4 b$.
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 astroid formed by the locus of $P$.
The segment of the tangent to $H$ between the $x$-axis and the $y$-axis is constant and equals the radius of the stator $C_2$.
Proof
From Equation of Astroid, $H$ can be expressed as:
- $\begin{cases}
x & = a \cos^3 \theta \\ y & = a \sin^3 \theta \end{cases}$
Thus the slope of the tangent to $H$ at $\tuple {x, y}$ is:
\(\ds \frac {\d y} {\d x}\) | \(=\) | \(\ds \frac {3 a \sin^2 \theta \cos \theta \rd \theta} {-3 a \cos^2 \theta \sin \theta \rd \theta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\tan \theta\) |
Thus the equation of the tangent to $H$ is given by:
- $y - a \sin^3 \theta = -\tan \theta \paren {x - a \cos^3 \theta}$
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The $x$-intercept is found by setting $y = 0$ and solving for $x$:
\(\ds x\) | \(=\) | \(\ds a \cos^3 \theta + a \sin^2 \theta \cos \theta\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a \cos \theta \paren {\cos^2 \theta + \sin^2 \theta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a \cos \theta\) | Sum of Squares of Sine and Cosine |
Similarly, the $y$-intercept is found by setting $x = 0$ and solving for $y$, which gives:
- $y = a \sin \theta$
The length of the part of the tangent to $H$ between the $x$-axis and the $y$-axis is given by:
\(\ds \sqrt {a^2 \cos^2 \theta + a^2 \sin^2 \theta}\) | \(=\) | \(\ds a \sqrt {\cos^2 \theta + \sin^2 \theta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a\) | Sum of Squares of Sine and Cosine |
which is constant.
$\blacksquare$
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.21$: The Cycloid
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): astroid