The lemniscate is the locus of the points for which the product of the distances from the two points (-1, 0) and (1, 0) is constant a^{2}.

Its equation is
(x^{2} + y^{2})^{2} - 2 (x^{2} - y^{2}) + 1 - a^{4} = 0. The curve will change as the value of parameter "a" changes.