[Paper Review] Ordinary lines in space

Frank de Zeeuw|arXiv (Cornell University)|Mar 26, 2018

Computational Geometry and Mesh Generation6 references3 citations

TL;DR

This paper proves that finite point sets in ℝ³ with at most αn points on any plane (for α < 1) span at least cₐn² ordinary lines, establishing a quadratic lower bound. The proof uses a projection argument inspired by Kelly’s method and combines it with Beck’s theorem on line spans in the plane to show that truly three-dimensional configurations must generate significantly more ordinary lines than planar ones.

ABSTRACT

We prove that if a finite point set in real space does not have too many points on a plane, then it spans a quadratic number of ordinary lines. This answers the real case of a question of Basit, Dvir, Saraf, and Wolf. It shows that there is a significant difference in terms of ordinary lines between planar point sets, which may span a linear number of ordinary lines, and truly three-dimensional point sets. Our proof uses a projection argument of Kelly combined with a theorem of Beck on the number of spanned lines of a planar point set.

Motivation & Objective

• To resolve the real case of a question by Basit, Dvir, Saraf, and Wolf on the minimum number of ordinary lines in 3D point sets.
• To establish a significant quantitative difference between planar and truly three-dimensional point sets in terms of ordinary line counts.
• To show that if no plane contains more than αn points (α < 1), then the number of ordinary lines is Ω(n²), improving upon prior linear bounds.
• To explore the sharpness of the bound and its dependence on the coplanarity parameter α, and to discuss implications for configurations with few non-coplanar points.

Proposed method

• Apply a generic projection from a point outside all planes spanned by the point set to map the 3D configuration into a 2D plane while preserving collinearity.
• Use Beck’s theorem on line spans in the plane: if no line contains more than βn points, then at least γn² lines are spanned.
• Leverage the projection to transfer the line-spanning lower bound from the plane back to 3D, ensuring that the original set spans Ω(n²) lines.
• Combine the projected line count with a case analysis on the number of points on a single plane to derive a quadratic lower bound on ordinary lines.
• Use a construction of n/2 points on each of two skew lines to establish a lower bound of n²/4 ordinary lines, serving as a benchmark for the theoretical bound.
• Apply a recursive argument for configurations with at most n−k points on any plane, using Theorem 1.2 for planar subconfigurations and Theorem 1.3 for the remainder.

Experimental results

Research questions

• RQ1 What is the minimum number of ordinary lines that a finite point set in ℝ³ can span if no plane contains more than αn points?
• RQ2 How does the number of ordinary lines in 3D configurations differ from that in planar configurations, where linear bounds are known?
• RQ3 Can a quadratic lower bound on ordinary lines be established for 3D point sets with bounded coplanarity, and what is the best possible constant?
• RQ4 To what extent can the proof technique be extended to the complex case ℂ³, and what are the key obstructions?

Key findings

• For any α < 1, there exists cₐ > 0 such that any n-point set in ℝ³ with at most αn points on any single plane spans at least cₐn² ordinary lines.
• The bound is quadratic in n, showing a fundamental difference between planar and truly three-dimensional configurations in terms of ordinary line counts.
• The best known construction—n/2 points on each of two skew lines—produces exactly n²/4 ordinary lines, indicating that the theoretical bound is not tight in terms of constants.
• When at most n−k points are coplanar, the number of ordinary lines is at least (k + 1/2)(n−k) − (k choose 2) for sufficiently large n, which matches known constructions up to lower-order terms.
• The result implies that configurations with few non-coplanar points still generate a linear number of ordinary lines, but the constant grows with k.
• The proof fails over ℂ³ primarily due to the failure of the Sylvester–Gallai theorem, and two conjectures are proposed to potentially extend the result to the complex case.