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In its classical nine-point cubic form, the Cayley-Bacharach theorem says that a cubic through eight of the nine distinct intersections of two plane cubics must contain the ninth. The eight-point cubic completion for two triples of lines is the elementary case needed for triangular-strip propagation in incidence geometry. The phenomenon explains why interpolation conditions at such a complete intersection are dependent.

A projective line $L=0$ is contained in the zero locus of the nonzero cubic $L^3$. The union of three lines is given by their product, and a conic together with a line is likewise a plane cubic. Reducibility and repeated factors are permitted in polynomial coverings used in incidence geometry; a covering by cubics need not consist of smooth irreducible curves.

For finite sets of points $P$ and geometric curves $\mathcal{C}$, a point-curve incidence is a pair $(p,\gamma )$ with $p\in \gamma$. The incidence count is $I(P,\mathcal{C})={\sum }_{\gamma \in \mathcal{C}}|P\cap \gamma |$. Bounds in incidence geometry exploit restrictions on how curves can meet points, rather than treating the incidence relation as an arbitrary bipartite graph.

A linear space is a structure in incidence geometry with a set of points and a collection of subsets called lines: any two distinct points lie on exactly one line, and every line contains at least two points. The lines here are incidence-theoretic objects, without a required coordinate realization. A finite linear space imposes finiteness; its nontrivial form excludes a line containing all points. This is distinct from a vector space, whose structure involves addition and scalar multiplication.

A point $[a:b:c]$ of the projective plane corresponds to the dual line $aX+bY+cZ=0$. A line given by those coefficients corresponds back to the point $[a:b:c]$. Incidence is reversed: $p\in \ell$ if and only if ${\ell }^{\ast }\in p^{\ast }$. Thus collinear point triples become concurrent line triples, preserving the essential combinatorics of incidence geometry.