The Yamamoto egg curve is an egg-shaped oval defined, for parameters 
, by the quartic
curve
 |
(1)
|
It is a reflected single Kepler's folium, expressed in a convention used for egg-shaped curves (Yamamoto 2007, Chan et al. 2018). Its polar equation is
 |
(2)
|
for 
. This follows from the usual
Kepler's folium with parameters 
and 
, followed by reflection
in the 
-axis.
Yamamoto (2007) gives the parametric equations
 |  |  |
(3)
|
 |  |  |
(4)
|
where 
. Away from the origin,
the parameter 
equals twice the polar angle 
modulo 
. For 
, the real oval is
smooth even though the corresponding algebraic curve
has a singular point at the origin.
The limit 
is a circle with center 
and radius 
. At 
, the origin
becomes a cusp.
The construction above uses 
and 
. It illustrates the polar
equation by comparison with the circle 
, whose center
is 
and radius is 
. For 
, let 
be the intersection of a
ray from the origin 
with this circle. Shortening 
by 
gives the point 
on the egg. The gray circle
shows the limiting case 
, and the gray radial line
segment shows the shortening for a selected angle 
.
Green's theorem gives the enclosed area as
 |
(5)
|
