Lemniscate of Bernoulli is Special Case of Ovals of Cassini/Cartesian Proof
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Theorem
The lemniscate of Bernoulli is a special case of the ovals of Cassini.
Proof
The ovals of Cassini can be defined by a Cartesian equation as follows:
The Cartesian equation:
- $\paren {x^2 + y^2 + a^2}^2 - 4 a^2 x^2 = b^4$
describes the ovals of Cassini.
The lemniscate of Bernoulli can be defined by a Cartesian equation as follows:
The lemniscate of Bernoulli is the curve defined by the Cartesian equation:
- $\paren {x^2 + y^2}^2 = 2 a^2 \paren {x^2 - y^2}$
Setting $b = a$:
\(\ds \paren {x^2 + y^2 + a^2}^2 - 4 a^2 x^2\) | \(=\) | \(\ds a^4\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {x^2 + y^2}^2 + 2 \paren {x^2 + y^2} a^2 + a^4 - 4 a^2 x^2\) | \(=\) | \(\ds a^4\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {x^2 + y^2}^2\) | \(=\) | \(\ds 4 a^2 x^2 - 2 a^2 x^2 - 2 a^2 y^2\) | simplifying and rearranging | ||||||||||
\(\ds \) | \(=\) | \(\ds 2 a^2 \paren {x^2 - y^2}\) | further simplification |
It follows that the lemniscate of Bernoulli is an oval of Cassini where $b = a$.
$\blacksquare$