5.1.3 Superellipsoid

A shape between sphere and box is the superellipsoid. The syntax of the superquadric ellipsoid object is:

SUPERELLIPSOID:
  superellipsoid { < K1, K2 > [OBJECT_MODIFIERS] }

See also:

• [pov:6.5.1.11]Superquadric Ellipsoid
• [pov:3.6.5]Superquadric Ellipsoid Object

This object is given via two parameters K1$=e>0$ and K2$=n>0$ by

$$\{\langle x,y,z\rangle: \Bigl\Vert(\Vert(x,y)\Vert _{2/e},z)\Bigr... ...(\vert x\vert^{2/e} + \vert y\vert^{2/e}\right)^{e/n}+\vert z\vert^{2/n}==1\},$$

where the $p$-norm of a 2d-vector $\langle x,y\rangle$ is given by

$$\Vert\langle x,y\rangle\Vert _p:=\left\{\begin{array}{ll} (\vert ... ...\\ \max\{\vert x\vert,\vert y\vert\}&\text{ for }p={\infty}\end{array}\right.$$

Hence we get a cube for $e=n=0$ and a sphere for $e=n=1$.

Figure: Superellipsoids with $e=n=1$ and with $e=n=1/2$

    

Figure: Superellipsoids with $e=n=2$ and with $e=n=4$

    

Figure: Superellipsoids with $e=2,n=1/2$ and with $e=1/2,n=2$