Superellipsoid

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

Super-ellipsoids as computer graphics primitives were popularized by Alan H. Barr (who used the name "superquadrics" to refer to both superellipsoids and supertoroids). However, while some super-ellipsoids are superquadrics, neither family is contained in the other.

Piet Hein's supereggs are special cases of super-ellipsoids.

The basic super-ellipsoid is defined by the implicit equation

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 ${\displaystyle a=(1-\left|z\right|^{t})^{1/t}}$:

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