Lemniscate of Bernoulli is Special Case of Ovals of Cassini

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Theorem

The lemniscate of Bernoulli is a special case of the ovals of Cassini.


Proof from Geometric Definition

The ovals of Cassini are defined as follows:


Let $P_1$ and $P_2$ be points in the plane such that $P_1 P_2 = 2 a$ for some constant $a$.

The ovals of Cassini are the loci of points $M$ in the plane such that:

$P_1 M \times P_2 M = b^2$

for a real constant $b$.


The lemniscate of Bernoulli is defined geometrically as:


Let $P_1$ and $P_2$ be points in the plane such that $P_1 P_2 = 2 a$ for some constant $a$.

The lemniscate of Bernoulli is the locus of points $M$ in the plane such that:

$P_1 M \times P_2 M = a^2$


It follows that the lemniscate of Bernoulli is an oval of Cassini where $b = a$.

$\blacksquare$


Proof from Cartesian Definition

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 $b = a$:

\(\displaystyle \paren {x^2 + y^2 + a^2}^2 - 4 a^2 x^2\) \(=\) \(\displaystyle a^4\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \paren {x^2 + y^2}^2 + 2 \paren {x^2 + y^2} a^2 + a^4 - 4 a^2 x^2\) \(=\) \(\displaystyle a^4\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \paren {x^2 + y^2}^2\) \(=\) \(\displaystyle 4 a^2 x^2 - 2 a^2 x^2 - 2 a^2 y^2\) simplifying and rearranging
\(\displaystyle \) \(=\) \(\displaystyle 2 a^2 \paren {x^2 - y^2}\) further simplification

It follows that the lemniscate of Bernoulli is an oval of Cassini where $b = a$.

$\blacksquare$


Proof from Polar Definition

The ovals of Cassini can be defined by a polar equation as follows:


The polar equation:

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

describes the ovals of Cassini.


The lemniscate of Bernoulli can be defined by a polar equation as follows:


The lemniscate of Bernoulli is the curve defined by the polar equation:

$r^2 = 2 a^2 \cos 2 \theta$


Setting $b = a$:

\(\displaystyle r^4 + a^4 - 2 a^2 r^2 \cos 2 \theta\) \(=\) \(\displaystyle a^4\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle r^4 - 2 a^2 r^2 \cos 2 \theta\) \(=\) \(\displaystyle 0\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle r^2\) \(=\) \(\displaystyle 2 a^2 \cos 2 \theta\) simplifying and rearranging

It follows that the lemniscate of Bernoulli is an oval of Cassini where $b = a$.

$\blacksquare$


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