A general quartic surface defined by
(1)
(Gray 1997, p. 314). The above two images correspond to
Additional cases are illustrated above.
The "rounded cube" case corresponding to superellipsoid
with volume
(2)
where gamma function .
The volume of the case
where
(5)
real
part of Bessel function of the first kind
(E. W. Weisstein and M. Trott, pers. comm., Nov. 9, 2008), which
can probably be expressed in closed form in terms of bivariate hypergeometric functions.
The related surface
(6)
for superellipsoid .
See also Chmutov Surface ,
Cube ,
Superellipsoid ,
Tooth
Surface
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References Banchoff, T. F. "Computer Graphics Tools for Rendering Algebraic Surfaces and for Geometry of Order." In Geometric
Analysis and Computer Graphics: Proceedings of a Workshop Held May 23-25, 1988 Goursat, E. "Étude des surfaces qui
admettent tous les plans de symétrie d'un polyèdre régulier."
Ann. Sci. École Norm. Sup. 4 , 159-2000, 1897. Gray,
A. Modern
Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed.
Cite this as:
Weisstein, Eric W. "Goursat's Surface."
From MathWorld https://mathworld.wolfram.com/GoursatsSurface.html
Subject classifications
, and 
, respectively.
is a superellipsoid
with volume
is the gamma function.
is given by
![int_0^infty-(pi^2)/(16sqrt(2))R{((-1)^(5/8))/(t^(7/4))[(-1)^(1/4)e^(it/8)t^(3/4)×[(-1)^(3/4)J_(1/4)(t/8)-J_(-1/4)(t/8)]^3+(12sqrt(t)Gamma(1/4))/(pi^2)+(2(1+i)Gamma^3(1/4))/(pi^3)]}dt](/images/equations/GoursatsSurface/Inline11.svg)
![R[z]](/images/equations/GoursatsSurface/Inline12.svg)
is the real
part of 
and 
is a Bessel function of the first kind
(E. W. Weisstein and M. Trott, pers. comm., Nov. 9, 2008), which
can probably be expressed in closed form in terms of bivariate hypergeometric functions.
an even integer is also considered by Gray (1997, p. 292) and is a special case
of the superellipsoid.
