A Booth lemniscate is the completed pedal curve of a hyperbola with respect to its center, including the origin as a limiting double point. It is the figure eight branch of the hippopede family (Booth 1877, pp. 163-165; Ferréol). Its polar equation is
 |
where 
. The parent hyperbola
is
 |
The real loops occur where 
. The limiting
tangents at the origin have slopes 
. The double
point is approached as the point of tangency on the parent hyperbola
tends to infinity.
When 
, the parent is a rectangular
hyperbola and the curve is Bernoulli's lemniscate,
with 
. The figures use 
and 
, 
, and 
. Booth (1877, pp. 164-165)
called this family the hyperbolic lemniscate.
