In a synthetic (Pappian) projective plane, one can define a cubic in various clever ways not referring to coordinates. For instance, if $f$ is a projectivity from the pencil of lines through a point $A$ to the pencil of lines through another point $B$, then the locus of intersections $m \cap f(m)$ is a conic (which is nondegenerate if $f$ is not a perspectivity), and all conics are obtaned in this way.

Is there any analogous way to define curves of higher order? Say cubics, for definiteness?

In a synthetic (Pappian) projective plane, one can define a conic in various clever ways not referring to coordinates. For instance, if $f$ is a projectivity from the pencil of lines through a point $A$ to the pencil of lines through another point $B$, then the locus of intersections $m \cap f(m)$ is a conic (which is nondegenerate if $f$ is not a perspectivity), and all conics are obtaned in this way.

Is there any analogous way to define curves of higher order? Say cubics, for definiteness?