P.S.: “I have to dig a little bit deeper …”, was what I mentioned in the post above.
The teeth of a cogwheel (pinion) while rolling describe a larger space than they have while standing still. Which leads to a none working set up when the teeth become too long, and/or, too tight.
Having said that, please have a look at the file below:
https://www.amazon.com/clouddrive/share/YeQyuhCIfQoHBO0uhZmW2NqWU1lMyse0d7dnDPYoOnM
Press play and change carefully the parameters of the cogwheel in the scene. For each point of the spline, one frame has to be played. Each frame, the cogwheel is explored 0.1% and then the next.
Animation>Play Mode> Simple, if played from frame zero, it stops then on the end, the “Spline0-1000” object can then be copied/saved and used. It should be a close fit to the cogwheel, Scaling a little bit perhaps is needed.
A short and quick overview is screen captured here: (one minute)
https://www.amazon.com/clouddrive/share/N98F7RLZUnqKkx8OS8gp43HDlzKf5DgiqA63alLKQhC
When done, uncheck the Xpresso “enabled” checkbox.
As a side note, I had first a continuously rotation as exploration of the cogwheel, which distorted the shape. If you observe the Platonic object closely, you will see that it rotates slow, then fast, then slow, that finally allowed to explore the round wheel close enough to create the straight one.
Enjoy
P.S.: it is easier with an editable cogwheel. Scene file
https://www.amazon.com/clouddrive/share/MEyIyydwTdfEduDspmYd8lfC9IUQtMoRSzka4HTTTIP