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.
Let two triples of projective lines meet in nine distinct points. Any plane cubic containing eight contains all nine. To prove this, write the triples as and , with the missing point on . The candidate cubic agrees with a scalar multiple of on , so . Its three known zeros on force . Two remaining known zeros on force the linear factor to be a multiple of . Hence and vanishes at the missing point.
A two-cell-thick strip of triangles with degree-six vertices in a dual arrangement of a planar point set produces three indexed primal point families with collinearities whenever . A plane cubic can be fitted to nine initial points because its homogeneous coefficient space has dimension ten. Overlapping nine-point configurations then force each next point onto the same cubic by eight-point cubic completion for two triples of lines. Safe neighbourhoods guarantee the distinctness needed in these local completion steps. A possible seed is . Adjacent completion blocks force , followed by the alternating continuation along the two long transverse families.
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The Cayley–Bacharach theorem is a result in algebraic geometry that deals with the intersection of divisors on a projective space. It is particularly relevant in the study of linear systems of divisors and their properties. In its classical form, the theorem states the following: Let \( C \) be a non-singular irreducible curve of degree \( d \) in the projective plane \( \mathbb{P}^2 \).