In mathematics, a superellipsoid (or super-ellipsoid) is a solid whose horizontal sections are superellipses (Lamé curves) with the same squareness parameter ${\displaystyle \epsilon _{2}}$, and whose vertical sections through the center are superellipses with the squareness parameter ${\displaystyle \epsilon _{1}}$. It is a generalization of an ellipsoid, which is a special case when ${\displaystyle \epsilon _{1}=\epsilon _{2}=1}$.

Superellipsoids as computer graphics primitives were popularized by Alan H. Barr (who used the name "superquadrics" to refer to both superellipsoids and supertoroids). In modern computer vision and robotics literatures, superquadrics and superellipsoids are used interchangeably, since superellipsoids are the most representative and widely utilized shape among all the superquadrics.

Superellipsoids have a rich shape vocabulary, including cuboids, cylinders, ellipsoids, octahedra and their intermediates. It becomes an important geometric primitive widely used in computer vision, robotics, and physical simulation. The main advantage of describing objects and envirionment with superellipsoids is its conciseness and expressiveness in shape. Furthermore, a closed-form expression of the Minkowski sum between two superellipsoids is available. This makes it a desirable geometric primitive for robot grasping, collision detection, and motion planning.

A handful of notable mathematical figures can arise as special cases of superellipsoids given the correct set of values, which are depicted in the above graphic:

Piet Hein's supereggs are also special cases of superellipsoids.

The basic superellipsoid is defined by the implicit function

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: