Bisector fields and pencils of conics

TL;DR

This paper introduces bisector fields as collections of line pairs with consistent midpoints, characterizes their relation to pencils of affine conics, and explores their properties over various fields.

Contribution

It establishes a novel connection between bisector fields and pencils of affine conics, providing a full characterization of their correspondence.

Findings

Every bisector field corresponds to a pencil of affine conics.

Pairs of asymptotes of hyperbolas in a pencil form a bisector field.

The results hold over fields with characteristic not equal to 2.

Abstract

We introduce the notion of a bisector field, which is a maximal collection of pairs of lines such that for each line in each pair, the midpoint of the points where the line crosses every pair is the same, regardless of choice of pair. We use this to study asymptotic properties of pencils of affine conics over fields and show that pairs of lines in the plane that occur as the asymptotes of hyperbolas from a pencil of affine conics belong to a bisector field. By including also pairs of parallel lines arising from degenerate parabolas in the pencil, we obtain a full characterization: Every bisector field arises from a pencil of affine conics, and vice versa, every nontrivial pencil of affine conics is asymptotically a bisector field. Our main results are valid over any field of characteristic other than $2$ and hence hold in the classical Euclidean setting as well as in Galois geometries.