Supersphere
Supersphere
The supersphere is the algebraic surface that is the special case of the superellipse with a=b=c. It has equation
| |x/a|^n+|y/a|^n+|z/a|^n=1 |
(1)
|
or
| |x|^n+|y|^n+|z|^n=a^n |
(2)
|
for radius a and exponent n.
Special cases are summarized in the following table, together with their volumes.
n surface volume with
a=1
1 regular octahedron 4/3
2 sphere (4pi)/3
![画像:([Gamma(1/4)]^4)/(6sqrt(2)pi)](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/Supersphere/Inline8.svg){kind=link}
6 Hauser's "cube" [画像:(2[Gamma(1/6)]^3)/(27sqrt(pi))]
![画像:(2[Gamma(1/6)]^3)/(27sqrt(pi))](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/Supersphere/Inline9.svg){kind=link}
infty cube 8
The surface area is given by
| S_n=48int_0^(pi/4)int_0^(sec^(-1)(sqrt(2+tan^2phi)))sqrt((cos^mthetasin^2theta+sin^(2n)theta(cos^mphi+sin^mphi))/([cos^ntheta+sin^ntheta(cos^nphi+sin^nphi)^(n+1/2)]))dthetadphi |
(3)
|
(Trott 2006, p. 301), where m=2n-2.
The volume enclosed is given by
![画像:(8[Gamma(1+1/n)]^3)/(Gamma(1+3/n))a^3.](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/Supersphere/Inline17.svg){kind=link}
As n->infty, the solid becomes a cube, so
| lim_(n->infty)V_n=8a^3 |
(6)
|
as it must. This is a special case of the integral 3.2.2.2
| intintint_(x>=0,y>=0,z>=0; (x/a)^p+(y/b)^q+(z/c)^r<=1)x^(alpha-1)y^(beta-1)z^(gamma-1)dxdydz=(a^alphab^betac^gamma)/(pqr)(Gamma(alpha/p)Gamma(beta/q)Gamma(gamma/r))/(Gamma(alpha/p+beta/q+gamma/r+1)) |
(7)
|
in Prudnikov et al. (1986, p. 583). The cases n=2 and n=6 appear to be the only integers whose corresponding solids have simple moment of inertia tensors, given by
I_2 = [画像:[2/5Ma^2 0 0; 0 2/5Ma^2 0; 0 0 2/5Ma^2]]
![画像:[2/5Ma^2 0 0; 0 2/5Ma^2 0; 0 0 2/5Ma^2]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/Supersphere/Inline23.svg){kind=link}
(8)
I_6 = [画像:[3/5Ma^2 0 0; 0 3/5Ma^2 0; 0 0 3/5Ma^2].]
![画像:[3/5Ma^2 0 0; 0 3/5Ma^2 0; 0 0 3/5Ma^2].](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/Supersphere/Inline26.svg){kind=link}
(9)
See also
Sphere, Superegg, SuperellipsoidExplore with Wolfram|Alpha
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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. https://www.mathematicaguidebooks.org/.Referenced on Wolfram|Alpha
SupersphereCite this as:
Weisstein, Eric W. "Supersphere." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Supersphere.html