Burleigh's oval is the ellipse negative pedal curve with a focus as the pedal point. Lockwood (1957) named it for his student M. J. Burleigh, who first drew his attention to the curve.
Place the focus 
at the origin and the center of the ellipse
to its left. The point 
at polar angle 
has polar coordinates 
, where
 |
(1)
|
is the semimajor
axis, and 
is the eccentricity. Let 
and 
. A point 
lies on the line
through 
perpendicular to 
exactly when 
. Differentiating
with 
fixed gives 
. Since 
and 
form an orthonormal basis,
the point 
of the envelope is
 |
(2)
|
The dashed curves in the figure are the generating ellipses. For 
, Burleigh's oval is a smooth ovoid,
as in the egg shown at 
in the left panel. The right panel shows the fish
curve at 
. The shape depends on 
alone, since 
sets the scale.
For 
, Green's
theorem and integration by parts give
the enclosed area
 |
(3)
|
In particular, 
for 
and 
. For the self-crossing fish
curve, the same integral gives the signed
area, which subtracts the tail's area from the head's.
