Superellipsoid

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 1The 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 1where :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 1Setting "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)::egin{align} x(u,v) &{}= A c(v,frac{2}{t}) c(u,frac{2}{r}) \ y(u,v) &{}= B c(v,frac{2}{t}) s(u,frac{2}{r}) \ z(u,v) &{}= C s(v,frac{2}{t}) \ & -pi/2 le v le pi/2, quad -pi le u < pi ,end{align}where the auxiliary functions are:egin{align} c(omega,m) &{}= sgn(cos omega) |cos omega|^m \ s(omega,m) &{}= sgn(sin omega) |sin omega|^mend{align}and the sign function sgn("x") is : sgn(x) = egin{cases} -1, & x < 0 \ 0, & x = 0 \ +1, & x > 0 .end{cases}The volume inside this surface can be expressed in terms of beta functions, β("m","n") = Γ("m")Γ("n")/Γ("m"+"n"), as: V = frac23 A B C frac{4}{r t} eta left( frac{1}{r},frac{1}{r} ight) eta left(frac{2}{t},frac{1}{t} ight).

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.


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