Cassini Ovals

Cassini Oval is a quartic plane curve defined as the locus of points in the plane such that the product of the distances to two fixed points is constant. In the dynamic sketch below, this means AF1 x AF2 = k for some constant k. Contrast this to an ellipse, for which the sum of the distances is constant, rather than the product. Cassini ovals are named after the astronomer Giovanni Domenico Cassini who studied them in 1680, who investigated these curves when he was studying the relative motions of the Earth and the Sun. Cassini believed that the Sun traveled around the Earth on one of these ovals, with the Earth at one focus of the oval. In the dynamic sketch, drag points A and F1 to explore different Cassini Ovals.