The Hügelschäffer egg curve is the cubic curve
 |
(1)
|
where 
, 
, and 
are positive real numbers,
with 
. Its real locus has a
closed oval component with 
, and an unbounded component with 
. The closed component gives the
egg shape (Petrović and Malešević 2023).
The circle construction starts with 
and 
. For a point 
on the first circle,
take 
on the second circle where the ray
from 
through 
meets it. The point 
traces the closed component. Equivalently, that component
has the parametric equations
 |  |  |
(2)
|
 |  |  |
(3)
|
for 
.
Writing
 |
(4)
|
makes both real components explicit. At 
the equation reduces to an ellipse,
while at 
it factors as ![(x+a)[2ay^2+b^2(x-a)]=0](/images/equations/HugelschafferEggCurve/Inline23.svg)
, a line
together with a parabola.
The area enclosed by the oval is
![A=(2b(a+w))/(3w^2)[(a^2+w^2)E(k)-(a-w)^2K(k)],](/images/equations/HugelschafferEggCurve/NumberedEquation3.svg) |
(5)
|
where 
, and 
and 
are the complete
elliptic integral of the first kind and complete
elliptic integral of the second kind, respectively, in the modulus convention.
This follows by integrating the two branches over ![[-a,a]](/images/equations/HugelschafferEggCurve/Inline27.svg)
(Petrović and Malešević 2023).
