Basic (normalized) superellipsoid

The basic superellipsoid is defined by the implicit function

${\displaystyle f(x,y,z)=\left(x^{\frac {2}{\epsilon _{2}}}+y^{\frac {2}{\epsilon _{2}}}\right)^{\epsilon _{2}/\epsilon _{1}}+z^{\frac {2}{\epsilon _{1}}}}$

The parameters ${\displaystyle \epsilon _{1}}$ and ${\displaystyle \epsilon _{2}}$ are positive real numbers that control the squareness of the shape.

The surface of the superellipsoid is defined by the equation:

${\displaystyle f(x,y,z)=1}$

For any given point ${\displaystyle (x,y,z)\in \mathbb {R} ^{3}}$, the point lies inside the superellipsoid if ${\displaystyle f(x,y,z)<1}$, and outside if ${\displaystyle f(x,y,z)>1}$.

Any "parallel of latitude" of the superellipsoid (a horizontal section at any constant z between -1 and +1) is a Lamé curve with exponent ${\displaystyle 2/\epsilon _{2}}$, scaled by ${\displaystyle a=(1-z^{\frac {2}{\epsilon _{1}}})^{\frac {\epsilon _{1}}{2}}}$, which is

${\displaystyle \left({\frac {x}{a}}\right)^{\frac {2}{\epsilon _{2}}}+\left({\frac {y}{a}}\right)^{\frac {2}{\epsilon _{2}}}=1.}$

Any "meridian of longitude" (a section by any vertical plane through the origin) is a Lamé curve with exponent ${\displaystyle 2/\epsilon _{1}}$, stretched horizontally by a factor w that depends on the sectioning plane. Namely, if ${\displaystyle x=u\cos \theta }$ and ${\displaystyle y=u\sin \theta }$, for a given ${\displaystyle \theta }$, then the section is

${\displaystyle \left({\frac {u}{w}}\right)^{\frac {2}{\epsilon _{1}}}+z^{\frac {2}{\epsilon _{1}}}=1,}$

where

${\displaystyle w=(\cos ^{\frac {2}{\epsilon _{2}}}\theta +\sin ^{\frac {2}{\epsilon _{2}}}\theta )^{-{\frac {\epsilon _{2}}{2}}}.}$

In particular, if ${\displaystyle \epsilon _{2}}$ is 1, the horizontal cross-sections are circles, and the horizontal stretching ${\displaystyle w}$ 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 ${\displaystyle 2/\epsilon _{1}}$ around the vertical axis.

Superellipsoid

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 ${\displaystyle a_{x}}$, ${\displaystyle a_{y}}$, ${\displaystyle a_{z}}$, the semi-diameters of the resulting solid. The implicit function is ^[2]

${\displaystyle F(x,y,z)=\left(\left({\frac {x}{a_{x}}}\right)^{\frac {2}{\epsilon _{2}}}+\left({\frac {y}{a_{y}}}\right)^{\frac {2}{\epsilon _{2}}}\right)^{\frac {\epsilon _{2}}{\epsilon _{1}}}+\left({\frac {z}{a_{z}}}\right)^{\frac {2}{\epsilon _{1}}}}$.

Similarly, the surface of the superellipsoid is defined by the equation

${\displaystyle F(x,y,z)=1}$

For any given point ${\displaystyle (x,y,z)\in \mathbb {R} ^{3}}$, the point lies inside the superellipsoid if ${\displaystyle f(x,y,z)<1}$, and outside if ${\displaystyle f(x,y,z)>1}$.

Therefore, the implicit function is also called the inside-outside function of the superellipsoid.^[2]

The superellipsoid has a parametric representation in terms of surface parameters ${\displaystyle \eta \in [-\pi /2,\pi /2)}$, ${\displaystyle \omega \in [-\pi ,\pi )}$.^[3]

${\displaystyle x(\eta ,\omega )=a_{x}\cos ^{\epsilon _{1}}\eta \cos ^{\epsilon _{2}}\omega }$

${\displaystyle y(\eta ,\omega )=a_{y}\cos ^{\epsilon _{1}}\eta \sin ^{\epsilon _{2}}\omega }$

${\displaystyle z(\eta ,\omega )=a_{z}\sin ^{\epsilon _{1}}\eta }$

General posed superellipsoid

In computer vision and robotic applications, a superellipsoid with a general pose in the 3D Euclidean space is usually of more interest.^[6][5]

For a given Euclidean transformation of the superellipsoid frame ${\displaystyle g=[\mathbf {R} \in SO(3),\mathbf {t} \in \mathbb {R} ^{3}]\in SE(3)}$ relative to the world frame, the implicit function of a general posed superellipsoid surface defined the world frame is^[6]

${\displaystyle F\left(g^{-1}\circ (x,y,z)\right)=1}$

where ${\displaystyle \circ }$ is the transformation operation that maps the point ${\displaystyle (x,y,z)\in \mathbb {R} ^{3}}$ in the world frame into the canonical superellipsoid frame.

Volume of superellipsoid

The volume encompassed by the superelllipsoid surface can be expressed in terms of the beta functions ${\displaystyle \beta (\cdot ,\cdot )}$,^[10]

${\displaystyle V(\epsilon _{1},\epsilon _{2},a_{x},a_{y},a_{z})=2a_{x}a_{y}a_{z}\epsilon _{1}\epsilon _{2}\beta ({\frac {\epsilon _{1}}{2}},\epsilon _{1}+1)\beta ({\frac {\epsilon _{2}}{2}},{\frac {\epsilon _{2}+2}{2}})}$

or equivalently with the Gamma function ${\displaystyle \Gamma (\cdot )}$, since

${\displaystyle \beta (m,n)={\frac {\Gamma (m)\Gamma (n)}{\Gamma (m+n)}}}$