It's not hard to find multiple trigonometric functions of period $2\pi$ that added to self shifted by some constant offset result in a constant.
In classic pythagorean identity, you have
$F(x)+F\left(x+\frac{\pi}{2} \right) = 1 $
where $F(x)=\sin^2 x$
Or you can use symmetry of the sine wave and create
$F(x)+F(x+\pi ) = 0$
where $F(x) = \sin x$
Now what I'm looking for is a transformation of the sine function that while retaining the $2\pi$ period, gives you identity if repeated three times, with $\frac 2 3 \pi$ shift in each appearance:
$F(x) + F\left(x + \frac 2 3 \pi \right) + F\left(x+\frac 4 3 \pi\right) = \mathrm{const}$
Can you find such a function?
Reason and purpose:
I've been trying to develop a better RGB$<=>$HSV color space conversion - all the common ones use sawtooth style variant functions with variant equations, and I think using trigonometric functions could result in more smooth color passages, never mind much simpler algorithm.