Let $\mathbb K$ be a field and $\mathbb A^2_{\mathbb K}:=\mathbb K^2$. An affine plane curve is a set of the form $$C_f(\mathbb K)=\{(a, b)\in\mathbb A_{\mathbb K}^2: f(a, b)=0\}$$ where $f\in \mathbb K[x, y]$. Analogously a projective plane curve is a set of the form:
$$P_f(\mathbb K)=\{(a: b: c)\in \mathbb P^2_{\mathbb K}: f(a, b, c)=0 \}$$ where $\mathbb P_{\mathbb K}^2$ is the 2-dimensional projective plane and $f\in \mathbb K[x, y, z]$ is homogeneous.
It is easy to see that there is a bijection $$\mathbb P^2_{\mathbb K}\simeq \mathbb A_{\mathbb K}^2\sqcup \mathbb P_k^1.$$ Furtheremore, every $f\in \mathbb K[x, y]$ induces an homogeneous polynomial $F\in \mathbb K[x, y, z]$ such that $$F(x, y, 1)=f(x, y).$$ I believe the above data is enough to relate affine plane curves and projective plane curves. Does anyone know how do $C_f(\mathbb K)$ and $P_f(\mathbb K)$ relate?
Thanks.