Hügelschäffer egg curve:
\begin{equation}
y
\!\,=\,\!
\pm\,
H
\mbox{$\displaystyle \sqrt{\frac{(\alpha\!-\!x)(x\!-\!\beta)}{x-\gamma}}$}\!
\end{equation}
where\begin{equation} \label{ABC} \alpha = a,\; \beta = -a,\; \gamma = -\frac{a^2+w^2}{2w},\; H = \displaystyle\frac{b}{\sqrt{2w}} \end{equation}
Hügelschäffer egg surface:
\begin{equation}
\label{egg33}
z=z(x,y)=\pm \sqrt{\frac{b^2(a^2 - x^2)}{2wx+w^2+a^2}-y^2}
\end{equation}
Hügelschäffer egg curve area:
\begin{equation}
\mathcal{A}_{egg}
=
\mathcal{A}_1
+
\mathcal{A}_2
\end{equation}
\begin{equation}
\begin{array}{rcl}
\mathcal{A}_1
\!\!&\!\!=\!\!&\!\!
\displaystyle\frac{4}{3} H \sqrt{\alpha-\gamma} \, {\Big (} (\alpha+\beta-2\gamma)\cdot E(\kappa ,p) -
2(\beta-\gamma)\cdot F(\kappa ,p) {\Big )} + \\[2.5 ex]
\!\!&\!\! \!\!&\!\!
\displaystyle\frac{4}{3} H
(\mbox{$u$}+\gamma-\alpha-\beta)\sqrt{\frac{(\alpha-\mbox{$u$})(\mbox{$u$}-\beta)}{\mbox{$u$}-\gamma}}
\end{array}
\end{equation}
\(...\)
\begin{equation}
\begin{array}{rcl}
\mathcal{A}_2
\!\!&\!\!=\!\!&\!\!
\displaystyle\frac{4}{3} H \sqrt{\alpha-\gamma} \, {\Big (} (\alpha+\beta-2\gamma)\cdot E(\lambda ,p) -
2(\beta-\gamma)\cdot F(\lambda ,p) {\Big )} - \\[2.5 ex]
\!\!&\!\! \!\!&\!\!
\displaystyle\frac{4}{3} H \sqrt{(\alpha-\mbox{$u$})(\mbox{$u$}-\beta)(\mbox{$u$}-\gamma)}
\end{array}
\end{equation}
\(...\)\(...\)
Hügelschäffer egg surface area:
$$
{\cal S}_{egg}
=
\frac{b \pi}{4 w^2} \!\!\!\displaystyle\int\limits_{-a-\gamma}^{a-\gamma}
\!\!\!\frac{\sqrt{Q_5(t)}}{t^2}\,\, dt \,,
$$
where
$$
Q_5(t)
=
a_5 t^5 + a_4 t^4 + a_3 t^3 + a_2 t^2 + a_1 t + a_0
$$
$$
\begin{array}{rcl}
a_5 \!\!&\!\!=\!\!&\!\! -32 w^3 \\[1.5 ex]
a_4 \!\!&\!\!=\!\!&\!\! 4 w^2(8a^2+b^2+8w^2) \\[1.5 ex]
a_3 \!\!&\!\!=\!\!&\!\! -8 w (a^2-w^2)^2 \\[1.5 ex]
a_2 \!\!&\!\!=\!\!&\!\! -2 b^2 (a^2-w^2)^2 \\[1.5 ex]
a_1 \!\!&\!\!=\!\!&\!\! 0 \\[1.5 ex]
a_0 \!\!&\!\!=\!\!&\!\! \frac{b^2}{4w^2} (a^2-w^2)^4.
\end{array}
$$
\(...\)
Hügelschäffer egg surface volume:
\begin{equation}
V_{egg}
=
\frac {\pi b^2}{4w^3} \left( (a^2-w^2)^2 \ln \left |\frac{a-w}{a+w}\right |+ 2aw (a^2+w^2) \right)
\end{equation}
\(...\)