Equation of Ovals of Cassini/Polar Form

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Theorem

Let $P_1$ and $P_2$ be points in a polar coordinate plane located at $\polar {a, 0}$ and $\polar {a, \pi}$ for some constant $a \ne 0$.

Let $b$ be a real constant.


The polar equation:

$r^4 + a^4 - 2 a^2 r^2 \cos 2 \theta = b^4$

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 $M$.

Let $d_1 = \size {P_1 P}$ and $d_2 = \size {P_2 P}$.

We have:

\(\ds b^2\) \(=\) \(\ds d_1 d_2\) Definition of Ovals of Cassini
\(\ds \) \(=\) \(\ds \sqrt {r^2 + a^2 - 2 a r \cos \theta} \times \sqrt {r^2 + a^2 - 2 a r \, \map \cos {\pi - \theta} }\) Cosine Rule
\(\ds \leadsto \ \ \) \(\ds \paren {b^2}^2\) \(=\) \(\ds \paren {r^2 + a^2 - 2 a r \cos \theta} \paren {r^2 + a^2 + 2 a r \cos \theta}\) Cosine of Supplementary Angle, and squaring throughout
\(\ds \) \(=\) \(\ds \paren {r^2 + a^2}^2 - \paren {2 a r \cos \theta}^2\) Difference of Two Squares
\(\ds \) \(=\) \(\ds \paren {r^2}^2 + 2 a^2 r^2 + \paren {a^2}^2 - 4 a^2 r^2 \cos^2 \theta\) Square of Sum
\(\ds \) \(=\) \(\ds r^4 + a^4 - 2 a^2 r^2 \paren {2 \cos^2 \theta - 1}\) simplifying
\(\ds \) \(=\) \(\ds r^4 + a^4 - 2 a^2 r^2 \cos 2 \theta\) Double Angle Formula for Cosine: Corollary $1$

Hence the result.

$\blacksquare$


Also see


Source of Name

This entry was named for Giovanni Domenico Cassini.


Sources