Superellipsoid

The basic superellipsoid is defined by the implicit inequality

${\displaystyle \left(\left|x\right|^{r}+\left|y\right|^{r}\right)^{t/r}+\left|z\right|^{t}\leq 1.}$

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 "parallel of latitude" of the superellipsoid (a horizontal section at any constant z between -1 and +1) is a Lamé curve with exponent r, scaled by ${\displaystyle a=(1-\left|z\right|^{t})^{1/t}}$:

${\displaystyle \left|{\frac {x}{a}}\right|^{r}+\left|{\frac {y}{a}}\right|^{r}\leq 1.}$

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

${\displaystyle \left|{\frac {u}{w}}\right|^{t}+\left|z\right|^{t}\leq 1,}$

where

${\displaystyle w=(\left|\cos \theta \right|^{r}+\left|\sin \theta \right|^{r})^{-1/r}.}$

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

${\displaystyle \left(\left|{\frac {x}{A}}\right|^{r}+\left|{\frac {y}{B}}\right|^{r}\right)^{t/r}+\left|{\frac {z}{C}}\right|^{t}\leq 1.}$

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]

${\displaystyle x(u,v)=Ac\left(v,{\frac {2}{t}}\right)c\left(u,{\frac {2}{r}}\right)}$

${\displaystyle y(u,v)=Bc\left(v,{\frac {2}{t}}\right)s\left(u,{\frac {2}{r}}\right)}$

${\displaystyle z(u,v)=Cs\left(v,{\frac {2}{t}}\right)}$

where the auxiliary functions are

${\displaystyle c(\omega ,m)=\operatorname {sgn}(\cos \omega )|\cos \omega |^{m}}$

${\displaystyle s(\omega ,m)=\operatorname {sgn}(\sin \omega )|\sin \omega |^{m}}$

and the sign function sgn(x) is

${\displaystyle \operatorname {sgn}(x)={\begin{cases}-1,&x<0\\0,&x=0\\+1,&x>0.\end{cases}}}$

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:

${\displaystyle V={\frac {2}{3}}ABC{\frac {4}{rt}}\beta \left({\frac {1}{r}},{\frac {1}{r}}\right)\beta \left({\frac {2}{t}},{\frac {1}{t}}\right).}$