Superellipsoid

In mathematics, a superellipsoid or super-ellipsoid is a solid whose horizontal sections are superellipses (Lamé curves) with the same exponent r, and whose vertical sections through the center are superellipses with the same exponent t. In mathematics, a superellipsoid or super-ellipsoid is a solid whose horizontal sections are superellipses (Lamé curves) with the same exponent r, and whose vertical sections through the center are superellipses with the same exponent t. Superellipsoids 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 superellipsoids are superquadrics, neither family is contained in the other. Piet Hein's supereggs are special cases of superellipsoids. The basic superellipsoid is defined by the implicit inequality 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 a = ( 1 − | z | t ) 1 / t {displaystyle a=(1-left|z ight|^{t})^{1/t}} : Any 'meridian of longitude' (a section by any vertical plane through the origin) is a Lamé curve with exponent t, stretched horizontally by a factor w that depends on the sectioning plane. Namely, if x = u cos θ and y = u sin θ, for a fixed θ, then