Equivalence of Definitions of Lemniscate of Bernoulli

Geometric Definition equivalent to Cartesian Definition
Let $M$ be a lemniscate of Bernoulli by the geometric definition.

Then by definition:

Let $P_1 = \tuple {a, 0}$ and $P_2 = \tuple {-a, 0}$.

Let $p = \tuple {x, y}$ be an arbitrary point of $M$.

We have:

Thus $M$ is a lemniscate of Bernoulli by the Cartesian definition.

Geometric Definition equivalent to Polar Definition
Let $M$ be a lemniscate of Bernoulli by the geometric definition.

Then by definition:

Let $M$ be embedded in a polar coordinate plane whose origin is at $O$ and such that $P_1 = \polar {a, 0}$ and $P_2 = \polar {a, \pi}$.



Consider an arbitrary point $p = \polar {r, \theta}$.

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

We have:

Parametric Definition equivalent to Cartesian Definition
Let $M$ be a lemniscate of Bernoulli by the parametric definition.

Then by definition:

We have:

Then: