Finite cylinder approximation of ruled sets

We approximate compact ruled families of segments by finitely many cylinders with square intercept tiles and perpendicular-base area at most their integral projection cost plus any positive error. The velocity field is C${}^1$, and its differential has opposite real eigenvalues whose magnitudes are strictly below the inverse segment half-length; both eigenvalues may vanish. The result includes square-zero differentials with unrestricted shear and fields that pass between the two regimes. It holds for arbitrary compact label sets.