(I'll ask the question over $\mathbb{R}$, but feel free to change fields if that makes the answer more straightforward or more interesting.)
Let $Q$ denote a bivariate quadratic:
$$Q(x,y) = Ax^2 + Bxy + Cy^2+Dx+Ey+F.$$
If the vanishing set $V(Q) \subseteq \mathbb{A}^2$ is a irreducible, then it will be either an ellipse, a hyperbola, or a parabola. One way of classifying what type $V(Q)$ is is by looking at points at infinity over the complex numbers; two distinct real points at infinity means $V(Q)$ is hyperbola, two distinct complex points at infinity means $V(Q)$ is an ellipse, and two equal points at infinity means $V(Q)$ is a parabola.
To find the points at infinity, we ignore terms lower than degree $2$. In more detail, looking at points at infinity is equivalent to homogenizing $Q$, yielding the equation $$Ax^2 + Bxy + Cy^2+Dxz+Eyz+Fz^2 = 0$$ and then setting $z = 0$, yielding the equation $$Ax^2 + Bxy + Cy^2 = 0$$ which can readily be solved by completing the square.
In summary: to find the points at infinity (of the projectivization), which tell us interesting things about the original (affine curve), we ignore terms lower than degree $2$.
It also seems interesting to ignore the constant term while keeping the linear terms. The points at infinity are basically asymptotes of the affine curve, except they go through the origin and hence have "forgotten" their affine position. In order to remember their affine position, it seems to be interesting to retain degree $1$ terms in our analysis, while disregarding the constant term. That is, we're interested in the "equation" $$Ax^2 + Bxy + Cy^2+Dx+Ey \sim 0,$$ where $\sim$ is defined such that $$P \sim Q \iff \mathrm{deg}(P-Q) \leq 1.$$
This could feasibly be extended to higher degree curves, higher dimensional spaces etc., and perhaps to the study of vanishing sets of functions defined on algebraic varieties other than $\mathbb{A}^n$.
Question. Are there some accepted definitions or terminology that could get an interested reader started in learning about such things?
