Rulings of a smooth quadric surface
= Rulings of a smooth quadric surface
Under $Q\cong\mathbb P^1\times\mathbb P^1$, the fibres of the two <coordinate projection>[coordinate projections] are the two rulings of the <smooth quadric surface>. Two distinct fibres in one ruling are disjoint <projective line>[projective lines], while one fibre from each ruling meets in one point.