Superellipsoid - Alchetron, The Free Social Encyclopedia

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.

Basic shape

The basic super-ellipsoid is defined by the implicit equation

( | x | r + | y | r ) t / r + | z | t ≤ 1

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 :

| x a | r + | y a | r ≤ 1

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

| u w | t + | z | t ≤ 1

where

w = ( | cos ⁡ θ | r + | sin ⁡ θ | r ) − 1 / r .

In particular, if r is 2, the horizontal cross-sections are circles, and the horizontal stretching w of the vertical sections is 1 for all planes. In that case, the super-ellipsoid is a solid of revolution, obtained by rotating the Lamé curve with exponent t around the vertical axis.

The basic shape above extends from −1 to +1 along each coordinate axis. The general super-ellipsoid is obtained by scaling the basic shape along each axis by factors A, B, C, the semi-diameters of the resulting solid. The implicit equation is

( | x A | r + | y B | r ) t / r + | z C | t ≤ 1

Setting r = 2, t = 2.5, A = B = 3, C = 4 one obtains Piet Hein's superegg.

The general superellipsoid has a parametric representation in terms of surface parameters -π/2 < v < π/2, -π < u < π

x ( u , v ) = A c ( v , 2 t ) c ( u , 2 r ) y ( u , v ) = B c ( v , 2 t ) s ( u , 2 r ) z ( u , v ) = C s ( v , 2 t )

where the auxiliary functions are

c ( ω , m ) = sgn ⁡ ( cos ⁡ ω ) | cos ⁡ ω | m s ( ω , m ) = sgn ⁡ ( sin ⁡ ω ) | sin ⁡ ω | m

and the sign function sgn(x) is

sgn ⁡ ( x ) = { − 1 , x < 0 0 , x = 0 + 1 , x > 0.

The volume inside this surface can be expressed in terms of beta functions (and Gamma functions, because β(m,n) = Γ(m)Γ(n) / Γ(m + n) ), as:

V = 2 3 A B C 4 r t β ( 1 r , 1 r ) β ( 2 t , 1 t ) .

References

Superellipsoid Wikipedia

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