# 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 coordinate 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 $\left({x, y}\right)$ 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 $2$