How can one, in a smooth way, produce a semicircle wave, as opposed to a sinusoidal wave ?

Imagining the machinery and linkages behind the movement. I mean the trace. For example a point fixed in wheel of a car may engender a cycloid. Others curves may be a Cyclogon, or a curve produced by a point fixed inside a sharkwheel, when rolling(not gyration).

The motivation behind the question, it is because in the usenet, there is a user called Archimedes Plutonium that has semicircle waves instead of sinusoidal. I have curiosity if it will be possible to produce such kind movement in at least one way. The mechanism that produce the curve does not need to be contained inside a plane.

Other way of thinking on it may be in the sense of producing such movement in a non-trivial way, similar to producing a straight line using the Peaucellier–Lipkin linkage.

Question re-phrased by @Transistor:

enter image description here

Figure 1. A cycloid can be generated by a point on the circumference of a circle. Image source: Britannica.

Can a similar waveform consisting of a series of semi-circles be generated by a mechanical linkage?

  • $\begingroup$ I have edited your question. Please check to see if I have interpreted correctly what you are asking. $\endgroup$
    – Transistor
    Commented May 15, 2021 at 9:44
  • $\begingroup$ Thanks @Transistor, I have researched a bit more, even there a couple of topics in MO and SE, but, I am not satisfied with the anwers provided in those topic. In the sense of not just producing one semicircle alone, rather, a bit more than one, in the "semicircle waveform" ;) $\endgroup$ Commented May 29, 2021 at 0:14
  • 1
    $\begingroup$ geneva mechanism? $\endgroup$
    – Pete W
    Commented Mar 7 at 2:07

3 Answers 3


If I remember correctly in the series Jayce & The Wheeled Warriors, there was one vehicle with crazy wheels, it looked like this.

enter image description here


This can be done very easily in air, as follows:

Make an audio sound file using a semicircular waveform instead of a sinusoid. You will probably have to do this by hand and approximate the waveform with lots of thin "slices", and then spline them together and smooth them using audio sound synthesis software.

Play the resulting audio file through an audio amplifier and into a loudspeaker. The sound waves in air will now be semicircular instead of sinusoidal.

  • $\begingroup$ make me remember the phonopaper, browsing the pixilang web. Rather a unconventional approach... $\endgroup$ Commented May 29, 2021 at 0:18
  • $\begingroup$ The sound waves aren't actually shaped like a waveform drawn on paper however. The y axis represents pressure, not vertical displacement I think. $\endgroup$
    – Drew
    Commented Oct 19, 2021 at 17:08

There are MANY ways to accomplish what you are asking for, as this is a fairly abstract mechanical problem we will need more parameters. You will need to constrain your requirements down more to make one solution more applicable than another.

I have two small things to comment on as well, 1) you mention a "smooth" motion, which is immediately problematic because a cusp waveform is by nature, discontinuous. 2) the Usenet user you mention (Ludwig Poehlmann) is curious as I cannot see how it pertains to a mechanical movement?

The two easiest mechanical ways I can see to produce a repeating wave like this would be a rack and pinion where the form is traced out by the pinion like you already showed above but more controllable with gears.

Another way is to think about what that curve is mathematically; one form is y=|cos(x)| which you can accomplish mechanically by first generating a cosine wave with a rotating-crank mechanism. Then place it on a rack which only moves forward on the portion of the stroke which is half of the crank stroke (this is the absolute value part). This could be done with a ratchet and pawl which only engages for half or the stroke.

Like I said there are many ways this could be done for a repeating wave, and many more for just one stationary half circle shape. If these are unsatisfactory answers, I think on it more I can probably come up with more ideas.

  • $\begingroup$ 1. For example, in other SE forum I saw one post that shows how to generate a semicircle. But extending the mechanism to "repeat" is the interesting part. I found several, but none is "smooth movement" (in the generation of the curve). $\endgroup$ Commented Jun 28, 2022 at 23:06
  • $\begingroup$ 2. mmm, the center of the question comes from his material. $\endgroup$ Commented Jun 28, 2022 at 23:08
  • $\begingroup$ Still I m curious for differents answers :) $\endgroup$ Commented Jun 28, 2022 at 23:09
  • $\begingroup$ Please refine your requirements more so that my answers can be more helpful to you. $\endgroup$
    – Austin Fox
    Commented Jun 30, 2022 at 10:53
  • $\begingroup$ Having been on the receiving end of some of Archie's rants I can assure you that this is worth no effort. However, your immediate problem is that using the word smooth is hard to justify given the discontinuity in gradient at y=0. $\endgroup$ Commented Mar 12, 2023 at 18:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.