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

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



Steven M LaValle 2012-04-20