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On polynomials associated to Voronoi diagrams of point sets and crossing numbers

Mercè Claverol, Andrea de las Heras-Parrilla, David Flores-Peñaloza, Clemens Huemer, David Orden

Abstract

Three polynomials are defined for given sets $S$ of $n$ points in general position in the plane: The Voronoi polynomial with coefficients the numbers of vertices of the order-$k$ Voronoi diagrams of $S$, the circle polynomial with coefficients the numbers of circles through three points of $S$ enclosing $k$ points of $S$, and the $E_{\leq k}$ polynomial with coefficients the numbers of (at most $k$)-edges of $S$. We present several formulas for the rectilinear crossing number of $S$ in terms of these polynomials and their roots. We also prove that the roots of the Voronoi polynomial lie on the unit circle if, and only if, $S$ is in convex position. Further, we present bounds on the location of the roots of these polynomials.

On polynomials associated to Voronoi diagrams of point sets and crossing numbers

Abstract

Three polynomials are defined for given sets of points in general position in the plane: The Voronoi polynomial with coefficients the numbers of vertices of the order- Voronoi diagrams of , the circle polynomial with coefficients the numbers of circles through three points of enclosing points of , and the polynomial with coefficients the numbers of (at most )-edges of . We present several formulas for the rectilinear crossing number of in terms of these polynomials and their roots. We also prove that the roots of the Voronoi polynomial lie on the unit circle if, and only if, is in convex position. Further, we present bounds on the location of the roots of these polynomials.
Paper Structure (5 sections, 18 theorems, 52 equations, 3 figures)

This paper contains 5 sections, 18 theorems, 52 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Known relations
  3. Properties of the Voronoi, circle and $E_{\leq_k}$ polynomials
  4. On the roots of Voronoi, circle and $E_{\leq_k}$ polynomials
  5. Discussion

Key Result

Proposition 1

For every set $S$ of $n$ points in general position, the circle polynomial $p_C(z)=\sum_{k=0}^{n-3} c_k z^{k}$ and the Voronoi polynomial $p_V(z)=\sum_{k=1}^{n-1} v_k z^{k-1}$ satisfy:

Figures (3)

  • Figure 1: Left: 1591-th entry of the order type database for 8 points, from OrderTypesDBurl. With complex stream plots of its Voronoi polynomial (center): $p_V(z)=10 + 23z + 27z^2 + 24z^3 + 17z^4 + 9z^5 + 2z^6$, and its $E_{\leq k}$ polynomial (right): $p_E(z)=4 + 13z + 22z^2 + 34z^3 + 43z^4 + 52z^5$; roots are red points.
  • Figure 2: The sector $T= \{z \in \mathbb{C}\backslash\{0\} \ | \ \ |arg(z) | < \frac{\pi}{2(n-3)}\}$ is empty of roots of $p_C(z)$. The region $\mathbb{C} \backslash ( D^s \cup T)$ contains a root of $p_C(z)$, with modulus at least $d$. The unit circle is drawn dotted.
  • Figure 3: Left: Best known example for minimizing $\overline{cr}(S)$ for $|S|=100$, from Aurl. Roots of its Voronoi (center), and $E_{\leq k}$ (right) polynomials, with circles illustrating, respectively, the bounds of Theorems \ref{['theorem:VPRootModulus']} and \ref{['theorem:atmjedgeRootModulusBound']}.

Theorems & Definitions (30)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Theorem 5: Aziz and Mohammad, 1980
  • Proposition 6
  • ...and 20 more