Superellipsoid

A superellipsoid is a surface defined as the set of points $$(x, y, z)$$ in 3D space such that $$(x^{2 / \epsilon_2} + y^{2 / \epsilon_2})^{\epsilon_2 / \epsilon_1} + z^{2 / \epsilon_1} = 1$$ where $$\epsilon_1, \epsilon_2 > 0$$, or any affine transformation of such a surface. "Superellipsoid" refers to both the surface and the solid that it encloses.

Superellipsoids come from the field of computer graphics. Special settings produce spheres, Steinmetz solids, bicones, and the regular octahedron. If the constants $$\epsilon_1, \epsilon_2$$ are permitted to tend to infinity, as they often are in computer graphics, cylinders and cuboids can also be produced.