# Equation of Ovals of Cassini/Cartesian Form

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## Contents

## Theorem

Let $P_1$ and $P_2$ be points in the cartesian coordinate plane located at $\tuple {a, 0}$ and $\tuple {-a, 0}$ for some constant $a \ne 0$.

The Cartesian equation:

- $\paren {x^2 + y^2 + a^2}^2 - 4 a^2 x^2 = b^4$

describes the **ovals of Cassini**.

## Proof

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

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

Let $b$ be chosen.

Let $P = \tuple {x, y}$ be an arbitrary point of $M$.

We have:

\(\displaystyle P_1 P\) | \(=\) | \(\displaystyle \sqrt {\paren {x - a}^2 + y^2}\) | Distance Formula | ||||||||||

\(\displaystyle P_2 P\) | \(=\) | \(\displaystyle \sqrt {\paren {x + a}^2 + y^2}\) | Distance Formula | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \paren {P_1 P} \paren {P_2 P}\) | \(=\) | \(\displaystyle \sqrt {\paren {x + a}^2 + y^2} \sqrt {\paren {x - a}^2 + y^2}\) | Definition of Ovals of Cassini | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle b^2\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle b^4\) | \(=\) | \(\displaystyle \paren {\paren {x + a}^2 + y^2} \paren {\paren {x - a}^2 + y^2}\) | squaring both sides | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {x^2 + y^2 + a^2 + 2 a x} \paren {x^2 + y^2 + a^2 - 2 a x}\) | multiplying out and rearranging | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {x^2 + y^2 + a^2}^2 - 4 a^2 x^2\) | Difference of Two Squares |

Hence the result.

$\blacksquare$

## Also see

## Source of Name

This entry was named for Giovanni Domenico Cassini.

## Sources

- Weisstein, Eric W. "Cassini Ovals." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/CassiniOvals.html