= Rational parametrization of a cubic surface from two skew lines
{title2=$L\times M\dashrightarrow S$}
For two <skew lines> $L,M$ on a <smooth cubic surface>, their joining line meets the surface at a third point generically. In coordinates with $L=\{x_2=x_3=0\}$ and $M=\{x_0=x_1=0\}$, write $F(\alpha A+\beta B)=\alpha\beta(\alpha U+\beta V)$. The third point is $[VA-UB]$ whenever $UV\ne0$. Neither $U$ nor $V$ is identically zero, because that would make one of the two lines singular on the surface. The inverse records $[x_0:x_1]$ and $[x_2:x_3]$ of a general surface point. Thus the surface is <birational> to $\mathbb P^1\times\mathbb P^1$, hence to $\mathbb P^2$.
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