The astronomer Giovanni Cassini (1625-1712) studied the family of curves with po

Question

The astronomer Giovanni Cassini (1625-1712) studied the family of curves with polar equations

where a and c 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 a and c related to each other when the curve splits into two parts?

Select the correct answer.

Explanation / Answer

solution :

answer a)   Conclusions: For c<a the curve has one component. For c less than about three=fourths of a, the curve is "oval" or convex. For c between about three-fourths of a and a, the curve has a dimple. At c=a, the curve splits. For c>a, there are two components.

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