Equation of Ovals of Cassini/Cartesian Form
Jump to navigation
Jump to search
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 $a \ne 0$.
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:
\(\ds P_1 P\) | \(=\) | \(\ds \sqrt {\paren {x - a}^2 + y^2}\) | 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
- 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