The astronomer Giovanni Cassini (1625–1712) studied the family of curves with polar equations r4 – 2c2r2 cos 2θ + c4 – a4 = 0 where and are positive real numbers. These curves are called the ovals of Cassini even though they are oval shaped only for certain values of a and c. (Cassini thought that these curves might represent planetary orbits better than Kepler’s ellipses.) Investigate the variety of shapes that these curves may have. In particular, how are and related to each other when the curve splits into two parts?