Ordinary line

Relative to a finite point set, an ordinary line is a line containing exactly two points of that set. It is an exact-richness condition, not a count of unordered pairs alone: a line with three points contains three point pairs but is not ordinary. Point-line duality turns ordinary lines into intersections incident to exactly two dual lines.

If a finite planar point set has at most $Kn$ ordinary lines, its points can be covered by $O(K+1)$ possibly reducible cubics. The Euler defect identity for a projective line arrangement gives $O(Kn)$ bad edges in the dual. Bounded-radius propagation of edge defects and averaging select one dual line with $O(K+1)$ unsafe edges. Cubic propagation along a triangular strip covers each safe run; each exceptional intersection vertex represents a primal line, covered by a degenerate cubic containing a line. A dual line with very few intersection vertices instead yields a direct covering by few primal lines.