A Superellipsoid or super-egg. You can control the shape using these values: r t A B C

The default values give Piet Hein's superegg. Other interesting cases to try are:

• r=2, t=2, ellipsoids
• r=1, t=1, octohedra
• r=0.7, t=0.7, pointy stars.

Superellipsoids are three dimensional versions of superellipses which in turn are a cross between a square and a circle. A circle has equation $x^2+y^2=1$ if we generalise that we get the superellipses $x^p+y^p=1$. You can play with the 2D version parametric version (faster) implicit verson (slower).

The equation for the superellipsoid is ${({\left|\frac{x}{A}\right|}^r+{\left|\frac{y}{B}\right|}^r)}^{t/r}+{\left|\frac{z}{C}\right|}^t=1$. To generate the mesh I have started with a mesh on a cube, project them onto a sphere, and find the spherical coordinates, $\rho =\sqrt{x^2+y^2+z^2}$, $\theta =\mathrm{atan}2(y,x)$, $\varphi =\mathrm{asin}(z/\rho )$. Using the polar representation for the superellipse $$\begin{array}{rl}X(\theta ,\varphi )=& Ac(\varphi ,2/t)c(\theta ,2/r)\\ Y(\theta ,\varphi )=& Bc(\varphi ,2/t)s(\theta ,2/r)\\ Z(\theta ,\varphi )=& Cs(\varphi ,2/t).\end{array}$$ Where $c(v,t)=\mathrm{sgn}(\cos v)|\cos v{|}^t$, $s(v,t)=\mathrm{sgn}(\sin v)|\sin v{|}^t$. This simplifies to $$\begin{array}{rl}X(x,y,z)=& Ap(m/\rho ,2/t)p(x/m,2/r)\\ X(x,y,z)=& Bp(m/\rho ,2/t)p(y/m,2/r)\\ X(x,y,z)=& Cp(z/\rho ,2/t)\end{array}$$ with $p(v,t)=\mathrm{sgn}(v)|v{|}^t$ and $m=\sqrt{x^2+y^2}$. See this answer at stackoverflow.

Readers comments

Definition of the theta and phi

Yibin Jiang Mon Nov 19 2018

The definition of theta and phi can be varied according to t and r. Theta is not always defined by atan2(x,y), neither as phi.

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