The supersphere is the algebraic surface that is the special case of the superellipse with . It has equation

(1)

or

(2)

for radius
and exponent .

Special cases are summarized in the following table, together with their volumes.

The surface area is given by

(3)

(Trott 2006, p. 301), where .

The volume enclosed is given by

As , the solid becomes a cube ,
so

(6)

as it must. This is a special case of the integral 3.2.2.2

(7)

in Prudnikov et al. (1986, p. 583). The cases and appear to be the only integers whose corresponding solids
have simple moment of inertia tensors, given by

See also Sphere ,

Superegg ,

Superellipsoid
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References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca
Raton, FL: CRC Press, p. 292, 1997. Hauser, H. "Gallery of
Singular Algebraic Surfaces: Cube." https://homepage.univie.ac.at/herwig.hauser/gallery.html . POV-Ray
Team. "Superquadratic Ellipsoid." §4.5.1.10 in Persistence of Vision
Ray-Tracer Version 3.1g User's Documentation, p. 199, May 1999. Prudnikov,
A. P.; Brychkov, Yu. A.; and Marichev, O. I. Integrals
and Series, Vol. 1: Elementary Functions. New York: Gordon and Breach,
1986. Trott, M. The
Mathematica GuideBook for Numerics. New York: Springer-Verlag, pp. 301-303,
2006. http://www.mathematicaguidebooks.org/ .
Cite this as:
Weisstein, Eric W. "Supersphere." From
MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/Supersphere.html

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