Calculate properties of a projective curve

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I'm confused as to how to calculate certain properties of a given projective curve (e.g. multiplicities & tangent lines at a singular point) and how to think about these. I've read that multiplicities & tangent lines are local properties, and that we can calculate them by dehomogenizing the curve, but I don't really understand why.

For example, say that we want to calculate multiplicities and tangent lines of a projective curve $C$ defined by a polynomial $F(X,Y,Z)$ at a singular point $x = [a : b : 1] \in \mathbb P^2$. As far as I understand, we can dehomogenize $F$ as $F_*(X,Y) = F(X,Y,1)$, calculate the multiplicity $m_{(a,b)}(F_*)$ and tangent lines $L_1, \dots, L_n$ (in $\mathbb A^2$), and then homogenize these tangent lines to acquire the tangent lines in projective space.

I'm having a hard time phrasing my question since I don't feel like I understand these concepts, but in summary, I'm confused as to why we can simply calculate the properties in affine space and then homogenize.

Thanks!