Cassini oval is the locus of the point $P$ in the plane having a constant product of the distances $PF_1$ and $PF_2$ measured from two fixed points $F_1$ and $F_2$ of the plane.

One obtains the simplest equation for the Cassini oval by choosing $F_1$ and $F_2$ on the other coordinate axis and equidistant ($= c > 0$ ) from the origin. Let $F_1 = (-c,\,0)$ , $F_2 = (c,\,0)$ and the locus condition $$PF_1 \cdot PF_2 = a^2 \quad (a > 0).$$ This reads in the Cartesian coordinates

(1)

(2)

(3)

(4)

When $a$ gets different values (the parametre $c$ being unchanged), (2) represents a family of curves. For any point $P$ of the plane (except $(\pm c,\,0)$ ), there is one representant of the family passing through $P$ , corresponding the value $a = \sqrt{PF_1 \cdot PF_2}$ .

As a matter of fact, the common name for all members of the family is Cassini curve, and only the special case, where $a = c\sqrt{2}$ , is the Cassini oval proper; it and the other members with $a \geqq c\sqrt{2}$ have the property of having only one highest point $(0,\,\sqrt{a^2-c^2})$ . All other members (with $a < c\sqrt{2}$ ) have two distinct highest points.

The locus of the highest and lowest points of any member with $a \leqq c\sqrt{2}$ is obtained by solving (2) with respect to $x^2$ , $$x^2 = c^2-y^2\pm\sqrt{a^4-4c^2y^2},$$ whence $y^2 \leqq \frac{a^4}{4c^2}$ . When the radicand vanishes, $|y|$ gets its maximum value and then we have $x^2 = c^2-y^2$ , which means a circle centered in the origin (green in the picture).

Note 1. Each Cassini oval is the intersection curve of a torus of revolution by a plane parallel to the axis of revolution.

Note 2. The astronomer Domenico Cassini found in 1680 the curve named after him; he thought that the orbit of Earth relative to the Sun was a cassinoid with the Sun in the other ``focus''.

Bibliography

1

F. IVERSEN: Analyyttisen geometrian oppikirja. Second edition. Kustannusosakeyhtiö Otava, Helsinki (1963).