If I'm not mistaken, the square wave case should in essence be the limiting case of the general case you get when looking at "trigonometric" functions which don't arise as the parametrisation of xˆ2 + yˆ2 = 1, but xˆn + yˆn = 1 for n > 2. It turns out that people have looked at that, and the study of those functions have been given the funny name "squigonometry". There's a Springer textbook out there.