The essence is the Taylor expansion, what we want to study the curve about is to find series of approximating algebraic functions on various degrees.

Note to add on gap knowledge of using discriminate to tell what type of curve a quadratic function represents:

Now to apply this Lagrange algebraic approach in dissecting Bernoulli Lemniscate curve/function

on the same process, we can go to higher dimension to study f(x, y, z) = 0 that’s in 3D.