A Booth oval is the pedal curve of an ellipse with respect to its center (Booth 1877, p. 163). It is the oval branch of the hippopede family. For a noncircular ellipse, the polar equation can be written
 |
where 
. The corresponding parent ellipse
is
 |
The oval is indented when 
and convex when 
. At 
, its curvature
vanishes at the two minor-axis endpoints. When 
,
the curvature is positive everywhere. The figures use
and 
,
2, and 3.
The center pedal curve of a circle is the same circle. This circular case is a limiting shape
of the positive-parameter hippopede family, rather
than a member with finite 
. As 
decreases to 1 with 
fixed, the oval approaches two externally
tangent circles. Booth (1877, p. 163) used
the name elliptic lemniscate for the ellipse-pedal family.
