Superquadrics

Instead of using polynomials to define $f_i$, many generalizations can be constructed. One popular primitive is a superquadric, which generalizes quadric surfaces. One example is a superellipsoid, which is given for ${\cal W}= {\mathbb{R}}^3$ by

$$\big( \vert x/a\vert^{n_1} + \vert y/b\vert^{n_2} \big)^{n_1/n_2} + \vert z/c\vert^{n_1} - 1 \leq 0 ,$$ (3.20)

in which $n_1 \geq 2$ and $n_2 \geq 2$. If $n_1 = n_2 = 2$, an ellipse is generated. As $n_1$ and $n_2$ increase, the superellipsoid becomes shaped like a box with rounded corners.