Definition:Ovals of Cassini
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:
- $P_1 M \times P_2 M = b^2$
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.
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
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 11$: Special Plane Curves: Ovals of Cassini: $11.31$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Cassini's ovals
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Cassini's ovals
- Weisstein, Eric W. "Cassini Ovals." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CassiniOvals.html