## You are here

HomeCassini oval

## Primary tabs

# Cassini oval

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

$\displaystyle\sqrt{(x+c)^{2}+y^{2}}\sqrt{(x-c)^{2}+y^{2}}\;=\;a^{2},$ | (1) |

which after squaring may be written

$a^{4}\;=\;(x^{2}\!+\!y^{2}\!+\!c^{2}\!+\!2cx)(x^{2}\!+\!y^{2}\!+\!c^{2}\!-\!2% cx)\;\equiv\;(x^{2}\!+\!y^{2}\!+\!c^{2})^{2}-(2cx)^{2},$ |

i.e.

$\displaystyle(x^{2}\!+\!y^{2}\!+\!c^{2})^{2}-4c^{2}x^{2}\;=\;a^{4}.$ | (2) |

One sees that the curve is symmetric both in regard to $x$-axis and in regard to $y$-axis, whence it suffices to examine it in the first quadrant ($x\geqq 0$, $y\geqq 0$). If (2) is written as

$\displaystyle y^{2}\;=\;\sqrt{a^{4}\!+\!4c^{2}x^{2}}-(x^{2}\!+\!c^{2}),$ | (3) |

it appears that $y$ is real only for $\sqrt{a^{4}\!+\!4c^{2}x^{2}}\geqq x^{2}\!+\!c^{2}$, which condition can be simplified to

$\displaystyle|x^{2}\!-\!c^{2}|\;\leqq\;a^{2}.$ | (4) |

In order to $y$ being real, (4) gives the three cases:

$1^{{\underline{o}}}$. $a<c$. We have $\sqrt{c^{2}\!-\!a^{2}}\leqq x\leqq\sqrt{c^{2}\!+\!a^{2}}$; thus the curve consists of two separate loops.

$2^{{\underline{o}}}$. $a=c$. Now $0\leqq x\leqq c\sqrt{2}$; the two loops meet in the origin (the lemniscate of Bernoulli).

$3^{{\underline{o}}}$. $a>c$. Then $0\leqq x\leqq\sqrt{c^{2}\!+\!a^{2}}$; there is one loop surrounding the origin.

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 $x^{2}$,

$x^{2}\;=\;c^{2}\!-\!y^{2}\pm\sqrt{a^{4}\!-\!4c^{2}y^{2}},$ |

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”.

# References

- 1 F. Iversen: Analyyttisen geometrian oppikirja. Second edition. Kustannusosakeyhtiö Otava, Helsinki (1963).

## Mathematics Subject Classification

51N20*no label found*51-00

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections