How to determine a point is inside a superellipse?

Question

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?

Thanks

Answer 1

Take the equation of the Lamé curve to be

$$\left|\frac{x}{a}\right|^p+\left|\frac{y}{b}\right|^p=1$$

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

$$\left|\frac{(x-h)\cos\,\varphi-(y-k)\sin\,\varphi}{a}\right|^p+\left|\frac{(x-h)\sin\,\varphi+(y-k)\cos\,\varphi}{b}\right|^p=1$$

To test if some point $(x,y)$ you have is within that superellipse, all you need to do is to change the "$=$" to a "$\lt$"...