I've added a rate of change to my cycloid construction so it gets 'faster'. The radius of each circle is only linked to the next line segment it will 'roll' into, so there's a jump from rotation to rotation. The splines that plot these points seem to smooth this out. Or at least that's what I think I'm up to. To keep working in this way though, it makes way more sense (and is cut and paste easier) to make each radius/line segment a repeating unit linked by coincidence with a further level of controlling geometry, that gets closer to 'bending' a cycloid along a curve.
Edit: never mind, there has to be a simpler way. I just end up with a mess.