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That is beautiful work.

It may even be possible to set it up so that the movement has a more constant velocity along the path.

Well if you want the laser to produce better images then one needs to use the dark art of cam design. As it turns out, simply determining cam geometry necessary to go through all those points is not enough, one must also take into account the velocity, acceleration, and even jerk profiles(the derivative of acceleration). If one does not take these into account the follower being driven can't spring back fast enough and the follower does not follow the intended profile. Most automobile cams must be produced to very high tolerances in order to achieve low vibration. The fact that CNC machine tools move in discretized distances can cause problems in some cases.

Yes- just like you are saying, the intention with the Mechancial Laser Show was not to maintain a constant laser speed, but accelerate and decelerate the laser gently. You can see the mechanism by which that is done by pausing the video at https://www.youtube.com/watch?v=_dtBUiaAqRE

Since the time taken to travel to each next point is constant, closer points together mean slower speed and spread out points mean high speed. You can see that the laser will slow down for taking the corners and speed up for the straights. There was no math behind this- I just eyeballed it but you could write software/math to help you determine the perfect spacing/acceleration.

Maybe a mechanical governor.

https://www.youtube.com/watch?v=u3lEnbrp9pI

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