Superquadrics

In mathematics, the superquadrics or super-quadrics are a family of geometric shapes defined by formulas that resemble those of elipsoids and other quadrics, except that the squaring operations are replaced by arbitrary powers. They can be seen as the three-dimensional relatives of the Lamé curves ("superellipses").

The superquadrics include many smooth shapes that resemble cubes, octahedra, cylinders, lozenges and spindles, with rounded corners. Because of their flexibility and relative simplicity, they are popular geometric modeling tools, especially in computer graphics.

Superquadrics include some shapes that resemble superellipsoids, but neither family is contained in the other.

Formulas

The basic superquadric has the formula:$left|x\; ight|^r\; +\; left|y\; ight|^s\; +\; left|z\; ight|^t\; leq\; 1$where "r", "s", and "t" are positive real numbers that determine the main features of the superquadric. Namely:

* less than 1: a pointy octahedron with concave faces and sharp edges.
* exactly 1: a regular octahedron.
* between 1 and 2: an octahedron with convex faces, blunt edges and blunt corners.
* exactly 2: a sphere
* greater than 2: a cube with rounded edges and corners.
* infinite (in the limit): a cube

If the three exponents are negative, the shape extends to infinity.

Each exponent can be varied independently to obtain combined shapes. For example, if "r"="s"=2, and "t"=4, one obtains a solid of revolution which resembles an ellipsoid with round cross-section but flattened ends.

The basic shape above spans from -1 to +1 along each coordinate axis. The general superquadric is the result of scaling this basic shape by different amounts "A", "B", "C" along each axis. Its general equation is:$left|frac\{x\}\{A\}\; ight|^r\; +\; left|frac\{y\}\{B\}\; ight|^s\; +\; left|frac\{z\}\{C\}\; ight|^t\; leq\; 1$

Parametric equations in terms of surface parameters "u" and "v" (longitude and latitude) are:where the auxiliary functions are:and the sign function sgn("x") is :

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

* Superellipse
* Superegg

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.