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On Distinct Angles in the Plane

Sergei V. Konyagin, Jonathan Passant, Misha Rudnev

TL;DR

The paper addresses how many distinct angles a set of $N$ points in the plane can determine when the points are in convex position, proving a lower bound of $\\Omega(N^{5/4})$ unless many points lie on a common circle. It introduces an order-based partition of the point set into $P_1$ and $P_2$, reducing the angle-count problem to incidences between $P_1$ and a multiset of cubic curves $\\gamma_{pqst}$, while carefully handling reducible components and multiplicities. A key dichotomy is established: either the angle count scales as $\\Omega(N^{1/4})$ or there is a circle-based obstruction with $\\\Omega(N/K)$ cocircular points, which yields the same lower bound away from circle configurations. The work connects additive combinatorics (convexity versus sumsets) with incidence geometry, leveraging a wealth of tools from cubic-curve incidences, bisector-energy bounds, and the Elekes-Rónyai framework to advance understanding of planar angle configurations and their rigidity.

Abstract

We prove that if $N$ points lie in convex position in the plane then they determine $Ω(N^{5/4})$ distinct angles, provided that the points do not lie on a common circle. This is derived from a more general claim that if $N$ points in the convex position in the real plane determine $KN$ distinct angles, then $K=Ω(N^{1/4})$ or $Ω(N/K)$ points are co-circular. The proof makes use of the implicit order one can give to points in convex position and relies on a slightly more general order assumption. The assumption enables one to reduce the issue to counting incidences between points and a multiset of cubic curves, with special attention being paid to the case when the curves are reducible.

On Distinct Angles in the Plane

TL;DR

The paper addresses how many distinct angles a set of points in the plane can determine when the points are in convex position, proving a lower bound of unless many points lie on a common circle. It introduces an order-based partition of the point set into and , reducing the angle-count problem to incidences between and a multiset of cubic curves , while carefully handling reducible components and multiplicities. A key dichotomy is established: either the angle count scales as or there is a circle-based obstruction with cocircular points, which yields the same lower bound away from circle configurations. The work connects additive combinatorics (convexity versus sumsets) with incidence geometry, leveraging a wealth of tools from cubic-curve incidences, bisector-energy bounds, and the Elekes-Rónyai framework to advance understanding of planar angle configurations and their rigidity.

Abstract

We prove that if points lie in convex position in the plane then they determine distinct angles, provided that the points do not lie on a common circle. This is derived from a more general claim that if points in the convex position in the real plane determine distinct angles, then or points are co-circular. The proof makes use of the implicit order one can give to points in convex position and relies on a slightly more general order assumption. The assumption enables one to reduce the issue to counting incidences between points and a multiset of cubic curves, with special attention being paid to the case when the curves are reducible.
Paper Structure (16 sections, 22 theorems, 98 equations, 10 figures)

This paper contains 16 sections, 22 theorems, 98 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Discussion
  3. Generalisations
  4. Proof of Theorem \ref{['t:main']}
  5. Definitions
  6. Lower bound: Pigeonholing
  7. Upper bound: Representing $|G|$ via incidences of curves and points
  8. Incidences with Irreducible Curves
  9. Incidences With Circular Components.
  10. Circles whose centre lies on a bisecting line.
  11. Circles which contain all four points $p,q,s$ and $t$.
  12. Incidences With Linear Components
  13. Proof of Theorem \ref{['t:unc']}
  14. Invoking Theorem \ref{['th:conv']}
  15. Conclusion of the proof of Theorem \ref{['t:unc']}
  16. ...and 1 more sections

Key Result

Theorem 1.2

Suppose, a $N$-point set $P\subset \Bbb R^2$, satisfies the order assumption with $|P_1|,\,|P_2|=\Omega(N)$, and $P$ determines $KN$ distinct angles. Then $K=\Omega(N^{1/4})$ or there are $\,\Omega(N/K)$ co-circular points in $P$.

Figures (10)

  • Figure 1: Left: Angles in arithmetic progression on a circle. Right: Angles in arithmetic progression on a line.
  • Figure 2: Point set $P=P_1\cup P_2$ satisfies the order assumption. The sets $P_1$ and $P_2$ are mutually avoiding.
  • Figure 3: One plays the angles created at the shaded points off against each other. The convexity of the circle ensures that the angles created cannot share additive structure, so one must grow.
  • Figure 4: A direction $\overrightarrow{xp}$ represented as the oriented angle $\theta_{xp}$.
  • Figure 5: The angle $\angle pxq$ is the difference $\theta_{xq}-\theta_{xp}$, where $\theta_{xp}, \theta_{xq} \in D_x$.
  • ...and 5 more figures

Theorems & Definitions (33)

  • Definition 1.1: Order assumption
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Remark 1.6
  • Definition 2.1
  • Lemma 2.1
  • Proposition 2.2
  • Corollary 2.3
  • ...and 23 more