Spheroellipsoids
Let $a,c$ denote the short and long ($z$-) semiaxes of an ellipse.
Truncate the ellipse at some distance $z_{0}$ from its center and
complete the figure by circle segments having the same tangent as
the ellipse at the point of fusion. Rotate the figure around the
long axis to produce a spheroellipsoid.
The pertinent equations are:
$z_{c}=z_{0}(1-a^{2}/c^{2})$ (center of circle segment);
$r_{c}^{2}=a^{2}[1-(z_{0}^{2}/c^{2})(1-a^{2}/c^{2})]$
(radius of circle segment).
The volume of a spheroellipsoid is $v_{1}=v_{body}+2v_{cap}$,
where $v_{body}=2 \pi c^{2}z_{0}(1-z_{0}^{2}/3c^{2})$ and
$v_{cap}=\pi\left[2r_{c}^{2}(z_{0}-z_{c})/3 + (z_{0}-z_{c})^{3}/3\right]$
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