The six normal trigonometric functions relate to the unit circle and the coordinates of the endpoints of the radii making some angle $\theta$ with the x-axis (sine represents the y-coordinate, cosine the x-coordinate, and tangent the ratio of the two). Does there exist a analogous set of functions for the unit superellipse with equation
$$x^4+y^4=1?$$
I've been struggling to define such functions that are periodic and work for all values of $\theta$, $0$ through $2\pi$.