Definition:Ovals of Cassini

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Definition

Let $P_1$ and $P_2$ be points in the plane such that $P_1 P_2 = 2 a$ for some constant $a$.

The ovals of Cassini are the loci of points $M$ in the plane such that:

$P_1 M \times P_2 M = b^2$

for a real constant $b$.


Shape

When $b > a$, $M$ is in one continuous piece, either oval or bone-shaped.

When $b < a$, $M$ is in two separate pieces, each surrounding one of the foci of $M$.

When $b = a$, $M$ degenerates into the lemniscate of Bernoulli.


Ovals-of-Cassini.png


Focus

Each of the two points $P_1$ and $P_2$ can be referred to as a focus of the ovals.


Also known as

The ovals of Cassini are also known as:

Cassini's ovals
the Cassini ovals
the Cassini ellipses (despite the fact that they are not actually ellipses).


Also see

  • Results about ovals of Cassini can be found here.


Source of Name

This entry was named for Giovanni Domenico Cassini.


Historical Note

The curves now known as the ovals of Cassini were first investigated by Giovanni Domenico Cassini in $1680$, during the course of his study of the relative motions of Earth and the Sun.

Cassini believed that the Sun orbited Earth on just such an oval, with Earth at one of its foci.


Sources

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