I have a superellipse which I will rotate it and then translate it. My question is that how to determine the origin will still be inside the superellipse after all the action?
Take the equation of the Lamé curve to be
Taking your specified order of operations, the result after rotating by an anticlockwise angle $\varphi$ and translating the center of the superellipse to $(h,k)$ is
To test if some point $(x,y)$ you have is within that superellipse, all you need to do is to change the "$=$" to a "$\lt$"...