Cubic Lines

The Mathematics

Every smooth cubic surface contains exactly twenty-seven straight lines — a classical fact first proved by Cayley and Salmon in 1849. This print path-traces one such surface photorealistically, resolving all twenty-seven lines at once. Beside it sits the projective plane blown up at six points, the construction from which every cubic surface arises: the six points, together with the lines and conics through them, account combinatorially for all 27 lines on the surface.

Technique

We wrote custom software for the entire pipeline. Beginning with six points in the projective plane, it computes the rational parametrization P² ⇢ P³ that realizes the cubic surface as the blow-up of those points. From this it derives all twenty-seven lines — first computing the points, lines, and quartics they arise from in P², and then feeding through the parameterization. Then it solves numerically for the coefficients of the cubic whose surface contains exactly these lines. With the implicit surface in hand, we render it with a path tracer I coded myself: it simulates the physics of light through glass — refraction, reflection, and dispersion — tracing many rays per pixel until the image converges.