Goursat's surface is a parametrized quartic surface defined by
 |
(1)
|
(Gray 1997, p. 314). Goursat surfaces are illustrated above for various values of 
.
Special cases are summarized in the table below.
 | surface |
 | "rounded cube" (superellipsoid
with  ) |
 | tooth surface |
The related surface
 |
(2)
|
for 
an even integer is also considered by Gray (1997, p. 292). These are superspheres
of unit radius, and hence superellipsoids.
The 
case is the "rounded cube" in the table, while 
is Hauser's cube surface.
The volume of the case 
is given by
 |  |  |
(3)
|
 |  | ![int_0^infty-(pi^2)/(16sqrt(2))Re{((-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/Inline15.svg) |
(4)
|
where
 |
(5)
|
![Re[z]](/images/equations/GoursatsSurface/Inline16.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.
