Cubic Plane Curve

Basics

A cubic plane curve, or simply a cubic is basically the set of the points in the Euclidean plane whose Cartesian coordinates are zeros of a polynomial of degree 3 in two variables: $${\displaystyle f(x,y)=a_{0}+a_{1}x+a_{2}y+a_{3}x^{2}+a_{4}xy+a_{5}y^{2}+a_{6}x^{3}+a_{7}x^{2}y+a_{8}xy^{2}+a_{9}y^{3}}$$