Trochoid

The trajectory of a point at fixed distance $a$ from the centre of a circle of radius $r$ rolling along a line. In coordinates aligned with that line it has a linear centre displacement plus sinusoidal components. The case $a=r$ is a cycloid; $a<r$ gives a shortened trajectory and $a>r$ gives loops. An inclined rolling line rotates this geometry relative to horizontal and vertical coordinates.

A point at distance $a$ from the axis of a radius-$r$ cylinder rolling downhill at constant axis speed $v$ has $\dot{x}=v[\cos \alpha +(a/r)\cos \theta ]$, $\dot{y}=-v[\sin \alpha +(a/r)\sin \theta ]$. For $0\le \alpha <\pi /2$, the existence of $\dot{x}=0$, $\dot{y}>0$ is equivalent to $a>r$. At $a=r$ the relevant zero-horizontal-velocity point is stationary, so a point within or on the rolling cylinder never has strictly vertical upward motion.