Arc of Cycloid is Concave
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Theorem
Consider a circle of radius $a$ rolling without slipping along the x-axis of a cartesian plane.
Consider the point $P$ on the circumference of this circle which is at the origin when its center is on the y-axis.
Consider the cycloid traced out by the point $P$.
Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane.
Then the locus of $P$ is concave.
Proof
From Second Derivative of Locus of Cycloid:
- $y'' = -\dfrac a {y^2}$
As $y \ge 0$ throughout, then $y'' < 0$ wherever $y \ne 0$, which is at the cusps.
The result follows from Second Derivative of Concave Real Function is Non-Positive.
$\blacksquare$
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.21$: The Cycloid: Problem $2$