Equation of Ovals of Cassini/Polar Form
Jump to navigation
Jump to search
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 $a \ne 0$.
The polar equation:
- $r^4 + a^4 - 2 a^2 r^2 \cos 2 \theta = b^4$
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 $M$.
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
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 11$: Special Plane Curves: Ovals of Cassini: $11.31$
- Weisstein, Eric W. "Cassini Ovals." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CassiniOvals.html