My conclusions, right or wrong: Motion is relative to the linear gear at the "pitch line."

The roller pinion is NOT rolling along on the outer radius.
It is rolling along at the "pitch line" (?), formed by the point of contact of the rollers with the teeth. (?)
X axis Velocity is determined by the speed of rotation at this radius.

So this point on the roller determines the a radius, and traces out a full radius of a cycloid.
So the radius at the innermost point of the roller would trace out a slightly curtate cycloid.
The radius at the outermost point of the roller would trace out a slightly prolate cycloid
Other points on the perimeter of the roller cylinder would plot slightly curtate or prolate cycloids, out of phase a bit.
The whole envelope determines the tooth profile at the top.

The very outer radius of the roller pinion is just support material. It actually would trace a prolate cycloid.

It is confusing.

- Brian