Let $C= V(F, H) \subset \mathbb{P}^3$ be a reduced space curve defined by homogeneous polynomials $F, H$ and $p: \mathbb{P}^3 \dashrightarrow \mathbb{P}^2$ a rational projection from a general point $Q \in \mathbb{P}^3$. As $Q$ we assumed to be general, $p_*C \subset \mathbb{P}^2$ is a (reduced) plane curve $\subset \mathbb{P}^2$.
If we know the explicit equation for $F(x,y,z,w)$ and $H(x,y,z,w) \in K[x,y,z,w]$, can we also write down an explicit homogeneous polynomial $G \in K[x,y,z]$ which defines $p_*C$ dependent on $F$ & $H$ ?