Main Concept

A superellipsoid is a 3-dimensional solid whose horizontal cross-sections are superellipses with the same exponent r, and whose vertical cross-sections through the centre are superellipses with the same exponent t. The general implicit equation for a superellipsoid is

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

The parameters t and r are positive real numbers which control the amount of flattening at the tips and equator of the solid. Factors A, B, and C scale the basic shape along each axis and are called the semi-diameters of the solid.

If $t \= r$, the equation for a superellipsoid becomes a special case of the superquadric equation.

Interesting Solids

When $r \= 2$, the horizontal cross sections of the solid are circles, so the superellipsoid is a solid of revolution: it can be obtained by rotating a superellipse of exponent t around the vertical axis. When $t \= r \=2$, the solid is an ordinary ellipsoid. In particular, if $A \= B \= C$, the solid is a sphere of radius A. When $A \= B \= 3\, C \= 4\, t \= 2.5$ and $r \= 2$, the superellipsoid is a special solid known as Piet Hein's "superegg".

Parameter, t

Parameter, r

Semi-Diameter, A

Semi-Diameter, B

Semi-Diameter, C

More MathApps

MathApps/Calculus