I'm sure we've all seen images such as the following, from wikipedia: link. They give us some nice intuition on what the sine and cosine functions are. Some people may also have seen images such as this one, or even this one, where instead of a circle, we take the 'sine of an arbitrary shape,' hence the title of this question. I'll say that the shape generates a periodic function in this way.
I now have a couple questions:
- Given a periodic function, is there always a shape that generates that function as in the second image (with the period adjusted as necessary, of course)?
- Are such shapes necessarily unique for a given periodic function (provided they exist)?
- In particular, what are the shapes (if any) that generate triangle, square, and sawtooth waves?
Also, if this is something that has been studied in detail, please enlighten me as to the terminology that people might use in this context.