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:

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

$\displaystyle \{\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

$\displaystyle \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$
Image /home/andreas/tex/Books/computer-graphics/img//obj-superellipsoid-1-1.png     Image /home/andreas/tex/Books/computer-graphics/img//obj-superellipsoid-h-h.png

Figure: Superellipsoids with $ e=n=2$ and with $ e=n=4$
Image /home/andreas/tex/Books/computer-graphics/img//obj-superellipsoid-2-2.png     Image /home/andreas/tex/Books/computer-graphics/img//obj-superellipsoid-4-4.png

Figure: Superellipsoids with $ e=2,n=1/2$ and with $ e=1/2,n=2$
Image /home/andreas/tex/Books/computer-graphics/img//obj-superellipsoid-2-h.png     Image /home/andreas/tex/Books/computer-graphics/img//obj-superellipsoid-h-2.png

Andreas Kriegl 2003-07-23