Equation of Ovals of Cassini/Cartesian Form

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Theorem

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

Let $b$ be a real constant.


The Cartesian equation:

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

describes the ovals of Cassini.


Ovals-of-Cassini.png


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:

\(\ds P_1 P\) \(=\) \(\ds \sqrt {\paren {x - a}^2 + y^2}\) Distance Formula
\(\ds P_2 P\) \(=\) \(\ds \sqrt {\paren {x + a}^2 + y^2}\) Distance Formula
\(\ds \leadsto \ \ \) \(\ds \paren {P_1 P} \paren {P_2 P}\) \(=\) \(\ds \sqrt {\paren {x + a}^2 + y^2} \sqrt {\paren {x - a}^2 + y^2}\) Definition of Ovals of Cassini
\(\ds \) \(=\) \(\ds b^2\)
\(\ds \leadsto \ \ \) \(\ds b^4\) \(=\) \(\ds \paren {\paren {x + a}^2 + y^2} \paren {\paren {x - a}^2 + y^2}\) squaring both sides
\(\ds \) \(=\) \(\ds \paren {x^2 + y^2 + a^2 + 2 a x} \paren {x^2 + y^2 + a^2 - 2 a x}\) multiplying out and rearranging
\(\ds \) \(=\) \(\ds \paren {x^2 + y^2 + a^2}^2 - 4 a^2 x^2\) Difference of Two Squares

Hence the result.

$\blacksquare$


Also presented as

Some sources present the equation defining the ovals of Cassini as:

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


Also see


Source of Name

This entry was named for Giovanni Domenico Cassini.


Sources

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