Steiner conic

Summary

A Steiner conic is a conic built synthetically from two pencils of lines. Take two points $U$ and $V$ and a projective map from the lines through $U$ to the lines through $V$. The points where each line meets its image form the Steiner conic. In a projective plane over a field, Steiner conics are exactly the non-degenerate projective conics, so this gives a definition of conics with no equations. The construction goes back to Jakob Steiner.