Superellipsoid

In mathematics, a super-ellipsoid or superellipsoid is 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 resemble certain superquadrics, but neither family is contained in the other.

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

Formulas

Basic shape

The basic super-ellipsoid is defined by the implicit equation:$left(\; left|x\; ight|^\{r\}\; +\; left|y\; ight|^\{r\}\; ight)^\{t/r\}\; +\; left|z\; ight|^\{t\}\; leq\; 1$The parameters "r" and "t" are positive real numbers that control the amount of flattening at the tips and at the equator.

Any horizontal section (at any constant "z" between -1 and +1) is a Lamé curve with exponent "r", scaled by $a\; =\; (1\; -\; left|z\; ight|^\{t\})^\{1/t\}$:: $left|frac\{x\}\{a\}\; ight|^\{r\}\; +\; left|frac\{y\}\{a\}\; ight|^\{r\}\; leq\; 1$

Any section of a super-ellipsoid by a 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 $x\; =\; ucos\; heta$ and $y\; =\; usin\; heta$, for a fixed $heta$, then: $left|frac\{u\}\{w\}\; ight|^\{t\}\; +\; left|z\; ight|^\{t\}\; leq\; 1$where :$w\; =\; (left|cos\; heta\; ight|^\{r\}\; +\; left|sin\; heta\; ight|^\{r\})^\{-1/r\}$

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:$left(\; left|frac\{x\}\{A\}\; ight|^\{r\}\; +\; left|frac\{y\}\{B\}\; ight|^\{r\}\; ight)^\{t/r\}\; +\; left|frac\{z\}\{C\}\; ight|^\{t\}\; leq\; 1$Setting "r"=2, "t"=2.5, "A"="B"=3, "C"=4 one obtains Piet Hein's superegg.

Following Harvtxt|Barr|1992, the general superellipsoid has a parametric representation in terms of surface parameters "u" and "v" (longitude and latitude)::where the auxiliary functions are:and the sign function sgn("x") is :The volume inside this surface can be expressed in terms of beta functions, β("m","n") = Γ("m")Γ("n")/Γ("m"+"n"), as:

References

*Barr, A.H., "Superquadrics and Angle-Preserving Transformations", IEEE_CGA(1), No. 1, January 1981, pp. 11-23.

*Jaklič, A., Leonardis, A., "Solina, F., Segmentation and Recovery of Superquadrics". Kluwer Academic Publishers, Dordrecht, 2000.

See also

* Super ellipse

External links

* [http://iris.usc.edu/Vision-Notes/bibliography/describe461.html Bibliography: SuperQuadric Representations]
* [http://www.cs.utah.edu/~gk/papers/vissym04/ Superquadric Tensor Glyphs]
* [http://www.gamedev.net/reference/articles/article1172.asp SuperQuadric Ellipsoids and Toroids, OpenGL Lighting, and Timing]
* [http://demonstrations.wolfram.com/Superquadrics/ Superquadratics] by Robert Kragler, The Wolfram Demonstrations Project.