What is the Intuition Behind Unicursal Curves and Their Double Points?

[bolbteppa]

To quote Goursat:

It is shown in treatises on Analytic Geometry that every unicursal curve of degree n has $\frac{(n-1)(n-2)}{2}$ double points, and, conversely, that every curve of degree n which has this number of double points is unicursal.
P222

Would somebody mind developing some intuition for this statement, along with an example or four (if not an intuitive proof), that would help motivate me to pick up classical books on analytical geometry & encourage me to wade through hundreds of pages to get to results like this one? I know so little about topics like these that I'm still trying to figure out the intuition for deriving multiple points, since it seems like authors do it in different ways each time, thanks.

 

[fresh_42]

This is a difficult question as algebraic curves are a wide field and subject to an entire branch of mathematics: algebraic geometry which involves ring theory and algebraic varieties. They are normally not investigated by analytical means as they lose uniqueness at the singularities and analysis demands locally Euclidean spaces, which they are not at the singularities.

You will find a lot of images if you google images for "algebraic curves".