# Superellipsoid

In mathematics, a **superellipsoid** or **super-ellipsoid** is a solid whose horizontal sections are superellipses (Lamé curves) with the same exponent *r*, and whose vertical sections through the center are superellipses with the same exponent *t*.

Superellipsoids as computer graphics primitives were popularized by Alan H. Barr (who used the name "superquadrics" to refer to both superellipsoids and supertoroids).^{[2]}^{[3]} However, while some superellipsoids are superquadrics, neither family is contained in the other.

The parameters *r* and *t* are positive real numbers that control the amount of flattening at the tips and at the equator. Note that the formula becomes a special case of the superquadric's equation if (and only if) *t* = *r*.

Any "meridian of longitude" (a section by any vertical plane through the origin) is a Lamé curve with exponent *t*, stretched horizontally by a factor *w* that depends on the sectioning plane. Namely, if *x* = *u* cos *θ* and *y* = *u* sin *θ*, for a fixed *θ*, then

In particular, if *r* is 2, the horizontal cross-sections are circles, and the horizontal stretching *w* of the vertical sections is 1 for all planes. In that case, the superellipsoid is a solid of revolution, obtained by rotating the Lamé curve with exponent *t* around the vertical axis.

The basic shape above extends from −1 to +1 along each coordinate axis. The general superellipsoid is obtained by scaling the basic shape along each axis by factors *A*, *B*, *C*, the semi-diameters of the resulting solid. The implicit inequality is

Setting *r* = 2, *t* = 2.5, *A* = *B* = 3, *C* = 4 one obtains Piet Hein's superegg.

The general superellipsoid has a parametric representation in terms of surface parameters -π/2 < *v* < π/2, -π < *u* < π.^{[3]}

The volume inside this surface can be expressed in terms of beta functions (and Gamma functions, because β(*m*,*n*) = Γ(*m*)Γ(*n*) / Γ(*m* + *n*) ), as: