Taking inspiration from the question What kind of curve is made of half circles? I'm trying to build up a function $f$ whose graph is similar to a sine function made of semicircles. While finding an expression for the absolute value of this function is not so difficult and the formula looks like
$$ |f(x)|=\sqrt{1-\left(\frac{\arcsin\left(\sin\left(\pi(x-1)+\frac{\pi}{2}\right)\right)+\frac{\pi}{2}}{\pi}\right)^2} $$
an explicit formula for the original problem seems way more involved. I'm searching for a formula that have the following properties:
- No functions defined by cases.
- Domain of $f$ has to be $\mathbb{R}$ so if possible no tricks with the sign function.
- The expression should involve only elementary functions and their inverses (No special functions, series or something more involved) since it has to be understandable with a minimum analysis background.
Any suggestion will be very appreciated.