Cassinian Eight f(x,y) = 0
and other level lines of f(x,y) = ((x-1)^2+y^2)*((x+1)^2+y^2)-1

The level curves of the function
    f(x,y) = ((x-1)^2+y^2)*((x+1)^2+y^2)-1
can be computed from the ODE:
    c'(t) = Rot90(grad f)(c(t))
where grad f vanishes only at [x,y] = [0,0].
The Cassini Eight has a double point singularity
at the origin.

The product-form of the function f shows that on
each level line the product of the distances from
[1,0] and [-1,0] is constant.