Superellipsoid

A superellipsoid is a surface defined as the set of points ${\displaystyle (x,y,z)}$ in 3D space such that ${\displaystyle (x^{2/\epsilon _{2}}+y^{2/\epsilon _{2}})^{\epsilon _{2}/\epsilon _{1}}+z^{2/\epsilon _{1}}=1}$ where ${\displaystyle \epsilon _{1},\epsilon _{2}>0}$, or any scaling (possibly nonuniform) of such a surface. "Superellipsoid" refers to both the surface and the solid that it encloses. All cross sections of a superellipsoid in planes parallel to the xy-plane are superellipses of exponent ${\displaystyle \epsilon _{1}}$, and the cross sections in the xz- and yz-planes are superellipses of exponent ${\displaystyle \epsilon _{1}}$.
Superellipsoids come from the field of computer graphics. Special settings produce spheres, Steinmetz solids, bicones, supereggs, and the regular octahedron. If the constants ${\displaystyle \epsilon _{1},\epsilon _{2}}$ are permitted to tend to infinity, as they often are in computer graphics, cylinders and cuboids can also be produced.