= Parametrization of a conic by projection
For a nonsingular <conic> with a point $p$ and a <projective line> $L$ not containing $p$, send $q\in L$ to the second intersection of $pq$ with the <conic>, counted with multiplicity. The tangent direction gives $p$ itself. The inverse sends any other <conic> point to its joining line's intersection with $L$, and sends $p$ to the tangent's intersection. The <Bézout theorem> makes these maps inverse. With data over a <field>, the construction preserves rationality: a known rational root of the restricted quadratic equation leaves its other root rational too. A <conic> with a rational point is therefore rationally parametrized by a <projective line>.
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