A prolate ellipsoid. Interaction Potential The general ellipsoid, also called a triaxial ellipsoid, is a quadratic surface which is given in Cartesian coordinates by
x
2
a
2
+
y
2
b
2
+
z
2
c
2
=
1
{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1}
where
a
{\displaystyle a}
b
{\displaystyle b}
c
{\displaystyle c}
Overlap algorithm The most widely used overlap algorithm is that of Perram and Wertheim:
Geometric properties The mean radius of curvature is given by (Refs. 2 and 3)
R
=
a
2
[
1
+
ϵ
b
1
+
ϵ
c
+
ϵ
c
{
1
ϵ
c
F
(
φ
,
k
1
)
+
E
(
φ
,
k
1
)
}
]
,
{\displaystyle R={\frac {a}{2}}\left[{\sqrt {\frac {1+\epsilon _{b}}{1+\epsilon _{c}}}}+{\sqrt {\epsilon }}_{c}\left\{{\frac {1}{\epsilon _{c}}}F(\varphi ,k_{1})+E(\varphi ,k_{1})\right\}\right],}
and the surface area is given by
S
=
2
π
a
2
[
1
+
ϵ
c
(
1
+
ϵ
b
)
{
1
ϵ
c
F
(
φ
,
k
2
)
+
E
(
φ
,
k
2
)
}
]
,
{\displaystyle S=2\pi a^{2}\left[1+{\sqrt {\epsilon _{c}(1+\epsilon _{b})}}\left\{{\frac {1}{\epsilon _{c}}}F(\varphi ,k_{2})+E(\varphi ,k_{2})\right\}\right],}
where
F
(
φ
,
k
)
{\displaystyle F(\varphi ,k)}
elliptic integral of the first kind and
E
(
φ
,
k
)
{\displaystyle E(\varphi ,k)}
φ
=
tan
−
1
(
ϵ
c
)
,
{\displaystyle \varphi =\tan ^{-1}({\sqrt {\epsilon }}_{c}),}
and the moduli
k
1
=
ϵ
c
−
ϵ
b
ϵ
c
,
{\displaystyle k_{1}={\sqrt {\frac {\epsilon _{c}-\epsilon _{b}}{\epsilon _{c}}}},}
and
k
2
=
ϵ
b
(
1
+
ϵ
c
)
ϵ
c
(
1
+
ϵ
b
)
,
{\displaystyle k_{2}={\sqrt {\frac {\epsilon _{b}(1+\epsilon _{c})}{\epsilon _{c}(1+\epsilon _{b})}}},}
where the anisotropy parameters,
ϵ
b
{\displaystyle \epsilon _{b}}
ϵ
c
{\displaystyle \epsilon _{c}}
ϵ
b
=
(
b
a
)
2
−
1
,
{\displaystyle \epsilon _{b}=\left({\frac {b}{a}}\right)^{2}-1,}
and
ϵ
c
=
(
c
a
)
2
−
1.
{\displaystyle \epsilon _{c}=\left({\frac {c}{a}}\right)^{2}-1.}
The volume of the ellipsoid is given by the well known
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V = \frac{4 \pi}{3}abc.}
Mathematica notebook file for calculating the surface area and mean radius of curvature of an ellipsoid
See also References Carl McBride and Enrique Lomba "Hard biaxial ellipsoids revisited: Numerical results", Fluid Phase Equilibria 255 pp. 37-45 (2007) G. S. Singh and B. Kumar "Geometry of hard ellipsoidal fluids and their virial coefficients", Journal of Chemical Physics 105 pp. 2429-2435 (1996) G. S. Singh and B. Kumar "Molecular Fluids and Liquid Crystals in Convex-Body Coordinate Systems", Annals of Physics 294 pp. 24-47 (2001) 
![{\displaystyle R={\frac {a}{2}}\left[{\sqrt {\frac {1+\epsilon _{b}}{1+\epsilon _{c}}}}+{\sqrt {\epsilon }}_{c}\left\{{\frac {1}{\epsilon _{c}}}F(\varphi ,k_{1})+E(\varphi ,k_{1})\right\}\right],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e881e50476a138c1b82d0eff3fa835b1212160d)
![{\displaystyle S=2\pi a^{2}\left[1+{\sqrt {\epsilon _{c}(1+\epsilon _{b})}}\left\{{\frac {1}{\epsilon _{c}}}F(\varphi ,k_{2})+E(\varphi ,k_{2})\right\}\right],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a0370210651aa0d817cf6bd9f11fc9898f4a23b)
